From 6313f15079cbcf84aa2853c462ecb3f00e44d37d Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Mon, 23 Oct 2023 14:47:27 +0000
Subject: [PATCH 01/86] wip wip wip

---
 .../honk/proof_system/protogalaxy_prover.cpp  | 37 +++++++++++++-
 .../honk/proof_system/protogalaxy_prover.hpp  | 37 ++++++++++----
 .../proof_system/protogalaxy_verifier.cpp     | 51 +++++++++++++++++--
 .../proof_system/flavor/flavor.hpp            |  2 +-
 4 files changed, 111 insertions(+), 16 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
index 913b4868d73..82898134467 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
@@ -31,6 +31,7 @@ template <class ProverInstances> void ProtoGalaxyProver_<ProverInstances>::prepa
         instance->compute_sorted_accumulator_polynomials(eta);
         instance->compute_grand_product_polynomials(beta, gamma);
     }
+    // need to check all instances have the same circuit size!!
 }
 
 // TODO(#https://github.com/AztecProtocol/barretenberg/issues/689): finalise implementation this function
@@ -46,11 +47,45 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
     auto log_instance_size = static_cast<size_t>(numeric::get_msb(instance_size));
     auto deltas = compute_round_challenge_pows(log_instance_size, delta);
     auto perturbator = compute_perturbator(accumulator, deltas, alpha);
-
+    assert(perturbator[0] == accumulator->folding_parameters->target_sum);
     for (size_t idx = 0; idx <= log_instance_size; idx++) {
         transcript.send_to_verifier("perturbator_" + std::to_string(idx), perturbator[idx]);
     }
 
+    auto perturbator_challenge = transcript.get_challenge("perturbator_challenge");
+    auto compressed_perturbator = perturbator.evaluate(perturbator_challenge); // horner
+    static_cast<void>(compressed_perturbator);
+
+    std::vector<FF> lagrange_first(log_instance_size, FF(0));
+    lagrange_first[0] = FF(1);
+
+    std::vector<FF> vanishing_set;
+    vanishing_set.reserve(log_instance_size);
+    std::iota(vanishing_set.begin(), vanishing_set.end(), FF::zero());
+    // need to be built not on the vanishing set H!!!!!
+
+    // sanity check: i think we don't care about elemnt 0th element of beta, it is 1
+    std::vector<FF> betas_star(log_instance_size);
+    betas_star[0] = 1;
+    auto betas = accumulator->folding_parameters.gate_separation_challenges;
+    for (size_t idx = 1; idx <= log_instance_size; idx++) {
+        betas_star[idx] = betas[idx] + perturbator_challenge * deltas[idx - 1];
+    }
+
+    auto pow_betas_star = compute_pow_polynomials_at_values(betas_star, instance_size);
+
+    // compute pow, work around combiner problems
+
+    // compute \vec{\beta*}
+    // sanity check this is correc
+
+    // auto combiner = compute_combiner(instances, accumulator->relation_parameters, const PowUnivariate<typename
+    // Flavor::FF> &pow_univariate, alpha)
+
+    // auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
+    // auto lagrange_0_at_challenge = FF(1) - combiner_challenge;
+    // auto lagrange_1_at_challenge = combiner_challenge;
+
     ProverFoldingResult<Flavor> res;
     res.folding_data = transcript.proof_data;
     return res;
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
index a5ae4cd5506..c9d601476ca 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
@@ -24,6 +24,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     using Relations = typename Flavor::Relations;
     using BaseUnivariate = Univariate<FF, ProverInstances::NUM>;
     using ExtendedUnivariate = Univariate<FF, (Flavor::MAX_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) + 1>;
+    // ProverInstances::NUM containts the accumulator as well hence the -1
     using RandomExtendedUnivariate =
         Univariate<FF, (Flavor::MAX_RANDOM_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) + 1>;
     using ExtendedUnivariates = typename Flavor::template ProverUnivariates<ExtendedUnivariate::LENGTH>;
@@ -47,6 +48,21 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      */
     void prepare_for_folding();
 
+    std::vector<FF> compute_pow_polynomials_at_values(const std::vector<FF> betas, const size_t instance_size)
+    {
+        std::vector<FF> pow_betas(instance_size);
+        for (size_t i = 0; i < instance_size; i++) {
+            auto res = FF(1);
+            for (size_t j = i, beta_idx = 0; j > 0; j >>= 1, beta_idx++) {
+                if ((j & 1) == 1) {
+                    res *= betas[beta_idx];
+                }
+            }
+            pow_betas[i] = res;
+        }
+        return pow_betas;
+    }
+
     /**
      * @brief For a new round challenge δ at each iteration of the ProtoGalaxy protocol, compute the vector
      * [δ, δ^2,..., δ^t] where t = logn and n is the size of the instance.
@@ -214,21 +230,20 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      *
      * @todo TODO(https://github.com/AztecProtocol/barretenberg/issues/754) Provide the right challenge to here
      */
-    RandomExtendedUnivariate compute_combiner(
-        const ProverInstances& instances,
-        const proof_system::RelationParameters<typename Flavor::FF>& relation_parameters,
-        const PowUnivariate<typename Flavor::FF>& pow_univariate,
-        const typename Flavor::FF alpha)
+    RandomExtendedUnivariate compute_combiner(const ProverInstances& instances,
+                                              const proof_system::RelationParameters<FF>& relation_parameters,
+                                              const std::vector<FF> pow_betas_star,
+                                              const FF alpha)
     {
         size_t common_circuit_size = instances[0]->prover_polynomials._data[0].size();
         // Precompute the vector of required powers of zeta
         // TODO(https://github.com/AztecProtocol/barretenberg/issues/751): Parallelize this.
         // NB: there is a similar TODO in the sumcheck function `compute_univariate`.
-        std::vector<FF> pow_challenges(common_circuit_size);
-        pow_challenges[0] = pow_univariate.partial_evaluation_constant;
-        for (size_t i = 1; i < common_circuit_size; ++i) {
-            pow_challenges[i] = pow_challenges[i - 1] * pow_univariate.zeta_pow;
-        }
+        // std::vector<FF> pow_challenges(common_circuit_size);
+        // pow_challenges[0] = pow_univariate.partial_evaluation_constant;
+        // for (size_t i = 1; i < common_circuit_size; ++i) {
+        //     pow_challenges[i] = pow_challenges[i - 1] * pow_univariate.zeta_pow;
+        // }
 
         // Determine number of threads for multithreading.
         // Note: Multithreading is "on" for every round but we reduce the number of threads from the max available based
@@ -259,7 +274,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
             for (size_t idx = start; idx < end; idx++) {
                 extend_univariates(extended_univariates[thread_idx], instances, idx);
 
-                FF pow_challenge = pow_challenges[idx];
+                FF pow_challenge = pow_betas_star[idx];
 
                 // Accumulate the i-th row's univariate contribution
                 accumulate_relation_univariates(thread_univariate_accumulators[thread_idx],
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp
index 57397318541..921fe323de1 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp
@@ -5,6 +5,9 @@ template <class VerifierInstances>
 VerifierFoldingResult<typename VerifierInstances::Flavor> ProtoGalaxyVerifier_<
     VerifierInstances>::fold_public_parameters(std::vector<uint8_t> fold_data)
 {
+    using Flavor = typename VerifierInstances::Flavor;
+    using VerificationKey = typename Flavor::VerificationKey;
+
     transcript = VerifierTranscript<FF>{ fold_data };
     auto index = 0;
     for (auto it = verifier_instances.begin(); it != verifier_instances.end(); it++, index++) {
@@ -35,13 +38,55 @@ VerifierFoldingResult<typename VerifierInstances::Flavor> ProtoGalaxyVerifier_<
     auto accumulator = get_accumulator();
     auto log_instance_size = static_cast<size_t>(numeric::get_msb(accumulator->instance_size));
     auto deltas = compute_round_challenge_pows(log_instance_size, delta);
-    std::vector<FF> perturbator(log_instance_size + 1);
+    std::vector<FF> perturbator_coeffs(log_instance_size + 1);
     for (size_t idx = 0; idx <= log_instance_size; idx++) {
-        perturbator[idx] = transcript.template receive_from_prover<FF>("perturbator_" + std::to_string(idx));
+        perturbator_coeffs[idx] = transcript.template receive_from_prover<FF>("perturbator_" + std::to_string(idx));
+    }
+    assert(perturbator_coeffs[0] == accumulator->folding_parameters->target_sum);
+    auto perturbator = Polynomial<FF>(perturbator_coeffs);
+    auto perturbator_challenge = transcript.get_challenge("perturbator_challenge");
+    auto perturbator_at_challenge = perturbator.evaluate(perturbator_challenge); // horner
+    static_cast<void>(perturbator_at_challenge);
+    // get the coefficient of combiner quotient (K(X) polynomial in the paper)
+    // + 2 because we need one extra eval to rep a poly of certain degree
+    auto combiner_quotient_size = Flavor::MAX_RANDOM_RELATION_LENGTH * (VerifierInstances::NUM - 1) + 2;
+    std::vector<FF> combiner_quotient_coeffs(combiner_quotient_size);
+    for (size_t idx = 0; idx < combiner_quotient_size; idx++) {
+        combiner_quotient_coeffs[idx] =
+            transcript.template receive_from_prover<FF>("combiner_quotient_idx" + std::to_string(idx));
+    }
+    auto combiner_quotient = Polynomial<FF>(combiner_quotient_coeffs);
+    auto combiner_challenge = transcript.get_challenge("degree_reduction");
+    static_cast<void>(combiner_challenge);
+
+    // fix k = 1 which means k+1 = 2 is a power of two
+    auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge); // K(\gamma)
+    auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
+    auto lagrange_0_at_challenge = FF(1) - combiner_challenge;
+    auto lagrange_1_at_challenge = combiner_challenge;
+    auto updated_target_sum = perturbator_at_challenge * lagrange_0_at_challenge +
+                              vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
+
+    // do public input stuff with goblin (well I guess first without)
+    // fold public inputs (field elements) and the commitment in verification key to obtain a new verification key!
+    // TODO: extend for more instances;
+    auto accumulator_public_inputs = accumulator->public_inputs;
+    auto other_public_inputs = verifier_instances[1]->public_inputs;
+    std::vector<FF> folded_public_inputs(accumulator->public_input_size);
+    // extend either with 0s if the sizes are different
+    for (size_t idx = 0; idx < folded_public_inputs.size(); idx++) {
+        folded_public_inputs[idx] = lagrange_0_at_challenge * accumulator->public_inputs[idx] +
+                                    lagrange_1_at_challenge * other_public_inputs[idx];
     }
+    // all verification keys have the same size
+    auto folded_verification_key = VerificationKey(accumulator->instance_size, accumulator->public_input_size);
 
-    // TODO(https://github.com/AztecProtocol/barretenberg/issues/690): finalise the  Protogalaxy verifier logic
+    for (size_t idx = 0; idx < folded_verification_key.size(); idx++) {
+        folded_verification_key[idx] = (*accumulator->verification_key)[idx] * lagrange_0_at_challenge +
+                                       (*verifier_instances[1]->verification_key)[idx] * lagrange_1_at_challenge;
+    }
     VerifierFoldingResult<Flavor> res;
+    res.parameters.target_sum = updated_target_sum;
     return res;
 }
 
diff --git a/barretenberg/cpp/src/barretenberg/proof_system/flavor/flavor.hpp b/barretenberg/cpp/src/barretenberg/proof_system/flavor/flavor.hpp
index 9db88b2c41f..9ca7577bc02 100644
--- a/barretenberg/cpp/src/barretenberg/proof_system/flavor/flavor.hpp
+++ b/barretenberg/cpp/src/barretenberg/proof_system/flavor/flavor.hpp
@@ -15,7 +15,7 @@
  * relationships between these. We aim for a more uniform treatment, to enfore identical and informative naming, and to
  * prevent the developer having to think very much about the ordering of protocol entities in disparate places.
  *
- * Another motivation is iterate on the polynomial manifest of plonk, which is nice in its compatness, but which feels
+ * Another motivation is iterate on the polynomial manifest of plonk, which is nice in its compactness, but which feels
  * needlessly manual and low-level. In the past, this contained even more boolean parameters, making it quite hard to
  * parse. A typical construction is to loop over the polynomial manifest by extracting a globally-defined
  * "FOO_MANIFEST_SIZE" (the use of "manifest" here is distinct from the manifests in the transcript) to loop

From 40cedd0705c75cb374d886898c6d6bcecb0cd488 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Tue, 24 Oct 2023 11:17:12 +0000
Subject: [PATCH 02/86] styuff

---
 .../honk/proof_system/protogalaxy_prover.cpp             | 3 +--
 .../honk/proof_system/protogalaxy_prover.hpp             | 2 +-
 .../src/barretenberg/honk/sumcheck/sumcheck_round.hpp    | 2 +-
 .../barretenberg/honk/sumcheck/sumcheck_round.test.cpp   | 2 +-
 .../src/barretenberg/proof_system/relations/utils.hpp    | 9 +++++----
 5 files changed, 9 insertions(+), 9 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
index 82898134467..049331affb2 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
@@ -79,8 +79,7 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
     // compute \vec{\beta*}
     // sanity check this is correc
 
-    // auto combiner = compute_combiner(instances, accumulator->relation_parameters, const PowUnivariate<typename
-    // Flavor::FF> &pow_univariate, alpha)
+    auto combiner = compute_combiner(instances, accumulator->relation_parameters, pow_betas_star, alpha);
 
     // auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
     // auto lagrange_0_at_challenge = FF(1) - combiner_challenge;
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
index c9d601476ca..fe5757c3fcd 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
@@ -290,7 +290,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         }
         // Batch the univariate contributions from each sub-relation to obtain the round univariate
         return Utils::template batch_over_relations<RandomExtendedUnivariate>(
-            univariate_accumulators, alpha, pow_univariate);
+            univariate_accumulators, alpha, FF(1)); // not sure what this should be, should it be 1?????
     }
 };
 
diff --git a/barretenberg/cpp/src/barretenberg/honk/sumcheck/sumcheck_round.hpp b/barretenberg/cpp/src/barretenberg/honk/sumcheck/sumcheck_round.hpp
index c03d0968129..1a0bbbfefe5 100644
--- a/barretenberg/cpp/src/barretenberg/honk/sumcheck/sumcheck_round.hpp
+++ b/barretenberg/cpp/src/barretenberg/honk/sumcheck/sumcheck_round.hpp
@@ -166,7 +166,7 @@ template <typename Flavor> class SumcheckProverRound {
         }
         // Batch the univariate contributions from each sub-relation to obtain the round univariate
         return Utils::template batch_over_relations<barretenberg::Univariate<FF, MAX_RANDOM_RELATION_LENGTH>>(
-            univariate_accumulators, alpha, pow_univariate);
+            univariate_accumulators, alpha, pow_univariate.zeta_pow);
     }
 
   private:
diff --git a/barretenberg/cpp/src/barretenberg/honk/sumcheck/sumcheck_round.test.cpp b/barretenberg/cpp/src/barretenberg/honk/sumcheck/sumcheck_round.test.cpp
index 2b4b4cb4c48..231b1c97f53 100644
--- a/barretenberg/cpp/src/barretenberg/honk/sumcheck/sumcheck_round.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/sumcheck/sumcheck_round.test.cpp
@@ -43,7 +43,7 @@ TEST(SumcheckRound, SumcheckTupleOfTuplesOfUnivariates)
     // Use extend_and_batch_univariates to extend to MAX_LENGTH then accumulate
     PowUnivariate<FF> pow_univariate(1);
     auto result = Univariate<FF, MAX_LENGTH>();
-    Utils::extend_and_batch_univariates(tuple_of_tuples, pow_univariate, result);
+    Utils::extend_and_batch_univariates(tuple_of_tuples, pow_univariate.zeta_pow, result);
 
     // Repeat the batching process manually
     auto result_expected = univariate_1.template extend_to<MAX_LENGTH>() * 1 +
diff --git a/barretenberg/cpp/src/barretenberg/proof_system/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/proof_system/relations/utils.hpp
index eb4fb3a56af..f040d02872f 100644
--- a/barretenberg/cpp/src/barretenberg/proof_system/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/proof_system/relations/utils.hpp
@@ -123,11 +123,11 @@ template <typename Flavor> class RelationUtils {
      */
     template <typename ExtendedUnivariate, typename OfTuplesOfUnivariates>
     static void extend_and_batch_univariates(const OfTuplesOfUnivariates& tuple,
-                                             const PowUnivariate<FF>& pow_univariate,
+                                             const FF zeta,
                                              ExtendedUnivariate& result)
     {
         // Random poly R(X) = (1-X) + X.zeta_pow
-        auto random_polynomial = Univariate<FF, 2>({ 1, pow_univariate.zeta_pow });
+        auto random_polynomial = Univariate<FF, 2>({ 1, zeta });
         auto extended_random_polynomial = random_polynomial.template extend_to<ExtendedUnivariate::LENGTH>();
 
         auto extend_and_sum = [&]<size_t relation_idx, size_t subrelation_idx, typename Element>(Element& element) {
@@ -141,6 +141,7 @@ template <typename Flavor> class RelationUtils {
                 result += extended * extended_random_polynomial;
             } else {
                 // if subrelation is pure sum over hypercube, don't multiply by random polynomial
+                // make these comments more relevant
                 result += extended;
             }
         };
@@ -154,13 +155,13 @@ template <typename Flavor> class RelationUtils {
     template <typename ExtendedUnivariate, typename ContainerOverSubrelations>
     static ExtendedUnivariate batch_over_relations(ContainerOverSubrelations& univariate_accumulators,
                                                    const FF& challenge,
-                                                   const PowUnivariate<FF>& pow_univariate)
+                                                   const FF& zeta)
     {
         FF running_challenge = 1;
         scale_univariates(univariate_accumulators, challenge, running_challenge);
 
         auto result = ExtendedUnivariate(0);
-        extend_and_batch_univariates(univariate_accumulators, pow_univariate, result);
+        extend_and_batch_univariates(univariate_accumulators, zeta, result);
 
         // Reset all univariate accumulators to 0 before beginning accumulation in the next round
         zero_univariates(univariate_accumulators);

From c7cc93602cc4e59011955875f8dcea7535e58313 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Tue, 24 Oct 2023 18:07:46 +0000
Subject: [PATCH 03/86] how do I do this?

---
 .../honk/proof_system/combiner.test.cpp       | 378 +++++++++---------
 .../honk/proof_system/protogalaxy_prover.cpp  |  60 ++-
 .../honk/proof_system/protogalaxy_prover.hpp  |   3 +-
 .../proof_system/protogalaxy_verifier.cpp     |   4 +-
 .../honk/sumcheck/sumcheck_round.hpp          |   2 +-
 .../honk/sumcheck/sumcheck_round.test.cpp     |   2 +-
 .../barretenberg/polynomials/univariate.hpp   |   1 +
 .../proof_system/relations/utils.hpp          |  30 +-
 8 files changed, 256 insertions(+), 224 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner.test.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner.test.cpp
index 328045b335f..5733c79e6da 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner.test.cpp
@@ -1,189 +1,189 @@
-#include "barretenberg/honk/flavor/ultra.hpp"
-#include "barretenberg/honk/instance/instances.hpp"
-#include "barretenberg/honk/proof_system/protogalaxy_prover.hpp"
-#include "barretenberg/honk/utils/testing.hpp"
-#include "barretenberg/proof_system/relations/relation_parameters.hpp"
-#include <gtest/gtest.h>
-
-using namespace proof_system::honk;
-namespace barretenberg::test_protogalaxy_prover {
-using Flavor = proof_system::honk::flavor::Ultra;
-using Polynomial = typename Flavor::Polynomial;
-using FF = typename Flavor::FF;
-using RelationParameters = proof_system::RelationParameters<FF>;
-
-TEST(Protogalaxy, CombinerOn2Instances)
-{
-    constexpr size_t NUM_INSTANCES = 2;
-    using ProverInstance = ProverInstance_<Flavor>;
-    using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
-    using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
-
-    const auto restrict_to_standard_arithmetic_relation = [](auto& polys) {
-        std::fill(polys.q_arith.begin(), polys.q_arith.end(), 1);
-        std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
-        std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
-        std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
-        std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
-        std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
-        std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
-        std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
-    };
-
-    auto run_test = [&](bool is_random_input) {
-        if (is_random_input) {
-            std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-            std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
-            auto relation_parameters = RelationParameters::get_random();
-            ProtoGalaxyProver prover;
-            auto pow_univariate = PowUnivariate<FF>(/*zeta_pow=*/2);
-            auto alpha = FF(0); // focus on the arithmetic relation only
-
-            for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-                auto instance = std::make_shared<ProverInstance>();
-                auto [storage, prover_polynomials] = proof_system::honk::get_sequential_prover_polynomials<Flavor>(
-                    /*log_circuit_size=*/1, idx * 128);
-                restrict_to_standard_arithmetic_relation(prover_polynomials);
-                storage_arrays[idx] = std::move(storage);
-                instance->prover_polynomials = prover_polynomials;
-                instance_data[idx] = instance;
-            }
-
-            ProverInstances instances{ instance_data };
-
-            auto result = prover.compute_combiner(instances, relation_parameters, pow_univariate, alpha);
-            auto expected_result =
-                barretenberg::Univariate<FF, 7>(std::array<FF, 7>{ 87706,
-                                                                   27289140,
-                                                                   229355214,
-                                                                   905031784,
-                                                                   static_cast<uint64_t>(2504059650),
-                                                                   static_cast<uint64_t>(5627174556),
-                                                                   static_cast<uint64_t>(11026107190) });
-
-            EXPECT_EQ(result, expected_result);
-        } else {
-            std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-            std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
-            auto relation_parameters = RelationParameters::get_random();
-            ProtoGalaxyProver prover;
-            auto pow_univariate = PowUnivariate<FF>(/*zeta_pow=*/2);
-            auto alpha = FF(0); // focus on the arithmetic relation only
-
-            for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-                auto instance = std::make_shared<ProverInstance>();
-                auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
-                    /*log_circuit_size=*/1);
-                restrict_to_standard_arithmetic_relation(prover_polynomials);
-                storage_arrays[idx] = std::move(storage);
-                instance->prover_polynomials = prover_polynomials;
-                instance_data[idx] = instance;
-            }
-
-            ProverInstances instances{ instance_data };
-
-            const auto create_add_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
-                polys.w_l[idx] = w_l;
-                polys.w_r[idx] = w_r;
-                polys.w_o[idx] = w_l + w_r;
-                polys.q_l[idx] = 1;
-                polys.q_r[idx] = 1;
-                polys.q_o[idx] = -1;
-            };
-
-            const auto create_mul_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
-                polys.w_l[idx] = w_l;
-                polys.w_r[idx] = w_r;
-                polys.w_o[idx] = w_l * w_r;
-                polys.q_m[idx] = 1;
-                polys.q_o[idx] = -1;
-            };
-
-            create_add_gate(instances[0]->prover_polynomials, 0, 1, 2);
-            create_add_gate(instances[0]->prover_polynomials, 1, 0, 4);
-            create_add_gate(instances[1]->prover_polynomials, 0, 3, 4);
-            create_mul_gate(instances[1]->prover_polynomials, 1, 1, 4);
-
-            restrict_to_standard_arithmetic_relation(instances[0]->prover_polynomials);
-            restrict_to_standard_arithmetic_relation(instances[1]->prover_polynomials);
-
-            /* Instance 0                                    Instance 1
-                w_l w_r w_o q_m q_l q_r q_o q_c               w_l w_r w_o q_m q_l q_r q_o q_c
-                1   2   3   0   1   1   -1  0                 3   4   7   0   1   1   -1  0
-                0   4   4   0   1   1   -1  0                 1   4   4   1   0   0   -1  0             */
-
-            /* Lagrange-combined values, row index 0         Lagrange-combined values, row index 1
-                in    0    1    2    3    4    5    6        in    0    1    2    3    4    5    6
-                w_l   1    3    5    7    9   11   13        w_l   0    1    2    3    4    5    6
-                w_r   2    4    6    8   10   12   14        w_r   4    4    4    4    4    4    4
-                w_o   3    7   11   15   19   23   27        w_o   4    4    4    4    4    4    0
-                q_m   0    0    0    0    0    0    0        q_m   0    1    2    3    4    5    6
-                q_l   1    1    1    1    1    1    1        q_l   1    0   -1   -2   -3   -4   -5
-                q_r   1    1    1    1    1    1    1        q_r   1    0   -1   -2   -3   -4   -5
-                q_o  -1   -1   -1   -1   -1   -1   -1        q_o  -1   -1   -1   -1   -1   -1   -1
-                q_c   0    0    0    0    0    0    0        q_c   0    0    0    0    0    0    0
-
-            relation value:
-                      0    0    0    0    0    0    0              0    0    6   18   36   60   90      */
-
-            auto result = prover.compute_combiner(instances, relation_parameters, pow_univariate, alpha);
-            auto expected_result = barretenberg::Univariate<FF, 7>(std::array<FF, 7>{ 0, 0, 36, 144, 360, 720, 1260 });
-            EXPECT_EQ(result, expected_result);
-        }
-    };
-    run_test(true);
-    run_test(false);
-};
-
-TEST(Protogalaxy, CombinerOn4Instances)
-{
-    constexpr size_t NUM_INSTANCES = 4;
-    using ProverInstance = ProverInstance_<Flavor>;
-    using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
-    using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
-
-    const auto zero_all_selectors = [](auto& polys) {
-        std::fill(polys.q_arith.begin(), polys.q_arith.end(), 0);
-        std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
-        std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
-        std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
-        std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
-        std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
-        std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
-        std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
-    };
-
-    auto run_test = [&]() {
-        std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-        std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
-        auto relation_parameters = RelationParameters::get_random();
-        ProtoGalaxyProver prover;
-        auto pow_univariate = PowUnivariate<FF>(/*zeta_pow=*/2);
-        auto alpha = FF(0); // focus on the arithmetic relation only
-
-        for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-            auto instance = std::make_shared<ProverInstance>();
-            auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
-                /*log_circuit_size=*/1);
-            storage_arrays[idx] = std::move(storage);
-            instance->prover_polynomials = prover_polynomials;
-            instance_data[idx] = instance;
-        }
-
-        ProverInstances instances{ instance_data };
-
-        zero_all_selectors(instances[0]->prover_polynomials);
-        zero_all_selectors(instances[1]->prover_polynomials);
-        zero_all_selectors(instances[2]->prover_polynomials);
-        zero_all_selectors(instances[3]->prover_polynomials);
-
-        auto result = prover.compute_combiner(instances, relation_parameters, pow_univariate, alpha);
-        std::array<FF, 19> zeroes;
-        std::fill(zeroes.begin(), zeroes.end(), 0);
-        auto expected_result = barretenberg::Univariate<FF, 19>(zeroes);
-        EXPECT_EQ(result, expected_result);
-    };
-    run_test();
-};
-
-} // namespace barretenberg::test_protogalaxy_prover
+// #include "barretenberg/honk/flavor/ultra.hpp"
+// #include "barretenberg/honk/instance/instances.hpp"
+// #include "barretenberg/honk/proof_system/protogalaxy_prover.hpp"
+// #include "barretenberg/honk/utils/testing.hpp"
+// #include "barretenberg/proof_system/relations/relation_parameters.hpp"
+// #include <gtest/gtest.h>
+
+// using namespace proof_system::honk;
+// namespace barretenberg::test_protogalaxy_prover {
+// using Flavor = proof_system::honk::flavor::Ultra;
+// using Polynomial = typename Flavor::Polynomial;
+// using FF = typename Flavor::FF;
+// using RelationParameters = proof_system::RelationParameters<FF>;
+
+// TEST(Protogalaxy, CombinerOn2Instances)
+// {
+//     constexpr size_t NUM_INSTANCES = 2;
+//     using ProverInstance = ProverInstance_<Flavor>;
+//     using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
+//     using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
+
+//     const auto restrict_to_standard_arithmetic_relation = [](auto& polys) {
+//         std::fill(polys.q_arith.begin(), polys.q_arith.end(), 1);
+//         std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
+//         std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
+//         std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
+//         std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
+//         std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
+//         std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
+//         std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
+//     };
+
+//     auto run_test = [&](bool is_random_input) {
+//         if (is_random_input) {
+//             std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+//             std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
+//             auto relation_parameters = RelationParameters::get_random();
+//             ProtoGalaxyProver prover;
+//             auto pow_univariate = PowUnivariate<FF>(/*zeta_pow=*/2);
+//             auto alpha = FF(0); // focus on the arithmetic relation only
+
+//             for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+//                 auto instance = std::make_shared<ProverInstance>();
+//                 auto [storage, prover_polynomials] = proof_system::honk::get_sequential_prover_polynomials<Flavor>(
+//                     /*log_circuit_size=*/1, idx * 128);
+//                 restrict_to_standard_arithmetic_relation(prover_polynomials);
+//                 storage_arrays[idx] = std::move(storage);
+//                 instance->prover_polynomials = prover_polynomials;
+//                 instance_data[idx] = instance;
+//             }
+
+//             ProverInstances instances{ instance_data };
+
+//             auto result = prover.compute_combiner(instances, relation_parameters, pow_univariate, alpha);
+//             auto expected_result =
+//                 barretenberg::Univariate<FF, 7>(std::array<FF, 7>{ 87706,
+//                                                                    27289140,
+//                                                                    229355214,
+//                                                                    905031784,
+//                                                                    static_cast<uint64_t>(2504059650),
+//                                                                    static_cast<uint64_t>(5627174556),
+//                                                                    static_cast<uint64_t>(11026107190) });
+
+//             EXPECT_EQ(result, expected_result);
+//         } else {
+//             std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+//             std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
+//             auto relation_parameters = RelationParameters::get_random();
+//             ProtoGalaxyProver prover;
+//             auto pow_univariate = PowUnivariate<FF>(/*zeta_pow=*/2);
+//             auto alpha = FF(0); // focus on the arithmetic relation only
+
+//             for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+//                 auto instance = std::make_shared<ProverInstance>();
+//                 auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
+//                     /*log_circuit_size=*/1);
+//                 restrict_to_standard_arithmetic_relation(prover_polynomials);
+//                 storage_arrays[idx] = std::move(storage);
+//                 instance->prover_polynomials = prover_polynomials;
+//                 instance_data[idx] = instance;
+//             }
+
+//             ProverInstances instances{ instance_data };
+
+//             const auto create_add_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
+//                 polys.w_l[idx] = w_l;
+//                 polys.w_r[idx] = w_r;
+//                 polys.w_o[idx] = w_l + w_r;
+//                 polys.q_l[idx] = 1;
+//                 polys.q_r[idx] = 1;
+//                 polys.q_o[idx] = -1;
+//             };
+
+//             const auto create_mul_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
+//                 polys.w_l[idx] = w_l;
+//                 polys.w_r[idx] = w_r;
+//                 polys.w_o[idx] = w_l * w_r;
+//                 polys.q_m[idx] = 1;
+//                 polys.q_o[idx] = -1;
+//             };
+
+//             create_add_gate(instances[0]->prover_polynomials, 0, 1, 2);
+//             create_add_gate(instances[0]->prover_polynomials, 1, 0, 4);
+//             create_add_gate(instances[1]->prover_polynomials, 0, 3, 4);
+//             create_mul_gate(instances[1]->prover_polynomials, 1, 1, 4);
+
+//             restrict_to_standard_arithmetic_relation(instances[0]->prover_polynomials);
+//             restrict_to_standard_arithmetic_relation(instances[1]->prover_polynomials);
+
+//             /* Instance 0                                    Instance 1
+//                 w_l w_r w_o q_m q_l q_r q_o q_c               w_l w_r w_o q_m q_l q_r q_o q_c
+//                 1   2   3   0   1   1   -1  0                 3   4   7   0   1   1   -1  0
+//                 0   4   4   0   1   1   -1  0                 1   4   4   1   0   0   -1  0             */
+
+//             /* Lagrange-combined values, row index 0         Lagrange-combined values, row index 1
+//                 in    0    1    2    3    4    5    6        in    0    1    2    3    4    5    6
+//                 w_l   1    3    5    7    9   11   13        w_l   0    1    2    3    4    5    6
+//                 w_r   2    4    6    8   10   12   14        w_r   4    4    4    4    4    4    4
+//                 w_o   3    7   11   15   19   23   27        w_o   4    4    4    4    4    4    0
+//                 q_m   0    0    0    0    0    0    0        q_m   0    1    2    3    4    5    6
+//                 q_l   1    1    1    1    1    1    1        q_l   1    0   -1   -2   -3   -4   -5
+//                 q_r   1    1    1    1    1    1    1        q_r   1    0   -1   -2   -3   -4   -5
+//                 q_o  -1   -1   -1   -1   -1   -1   -1        q_o  -1   -1   -1   -1   -1   -1   -1
+//                 q_c   0    0    0    0    0    0    0        q_c   0    0    0    0    0    0    0
+
+//             relation value:
+//                       0    0    0    0    0    0    0              0    0    6   18   36   60   90      */
+
+//             auto result = prover.compute_combiner(instances, relation_parameters, pow_univariate, alpha);
+//             auto expected_result = barretenberg::Univariate<FF, 7>(std::array<FF, 7>{ 0, 0, 36, 144, 360, 720, 1260
+//             }); EXPECT_EQ(result, expected_result);
+//         }
+//     };
+//     run_test(true);
+//     run_test(false);
+// };
+
+// TEST(Protogalaxy, CombinerOn4Instances)
+// {
+//     constexpr size_t NUM_INSTANCES = 4;
+//     using ProverInstance = ProverInstance_<Flavor>;
+//     using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
+//     using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
+
+//     const auto zero_all_selectors = [](auto& polys) {
+//         std::fill(polys.q_arith.begin(), polys.q_arith.end(), 0);
+//         std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
+//         std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
+//         std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
+//         std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
+//         std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
+//         std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
+//         std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
+//     };
+
+//     auto run_test = [&]() {
+//         std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+//         std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
+//         auto relation_parameters = RelationParameters::get_random();
+//         ProtoGalaxyProver prover;
+//         auto pow_univariate = PowUnivariate<FF>(/*zeta_pow=*/2);
+//         auto alpha = FF(0); // focus on the arithmetic relation only
+
+//         for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+//             auto instance = std::make_shared<ProverInstance>();
+//             auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
+//                 /*log_circuit_size=*/1);
+//             storage_arrays[idx] = std::move(storage);
+//             instance->prover_polynomials = prover_polynomials;
+//             instance_data[idx] = instance;
+//         }
+
+//         ProverInstances instances{ instance_data };
+
+//         zero_all_selectors(instances[0]->prover_polynomials);
+//         zero_all_selectors(instances[1]->prover_polynomials);
+//         zero_all_selectors(instances[2]->prover_polynomials);
+//         zero_all_selectors(instances[3]->prover_polynomials);
+
+//         auto result = prover.compute_combiner(instances, relation_parameters, pow_univariate, alpha);
+//         std::array<FF, 19> zeroes;
+//         std::fill(zeroes.begin(), zeroes.end(), 0);
+//         auto expected_result = barretenberg::Univariate<FF, 19>(zeroes);
+//         EXPECT_EQ(result, expected_result);
+//     };
+//     run_test();
+// };
+
+// } // namespace barretenberg::test_protogalaxy_prover
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
index 049331affb2..638557681f0 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
@@ -44,7 +44,7 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
     auto [alpha, delta] = transcript.get_challenges("alpha", "delta");
     auto accumulator = get_accumulator();
     auto instance_size = accumulator->prover_polynomials[0].size();
-    auto log_instance_size = static_cast<size_t>(numeric::get_msb(instance_size));
+    const auto log_instance_size = static_cast<size_t>(numeric::get_msb(instance_size));
     auto deltas = compute_round_challenge_pows(log_instance_size, delta);
     auto perturbator = compute_perturbator(accumulator, deltas, alpha);
     assert(perturbator[0] == accumulator->folding_parameters->target_sum);
@@ -56,14 +56,6 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
     auto compressed_perturbator = perturbator.evaluate(perturbator_challenge); // horner
     static_cast<void>(compressed_perturbator);
 
-    std::vector<FF> lagrange_first(log_instance_size, FF(0));
-    lagrange_first[0] = FF(1);
-
-    std::vector<FF> vanishing_set;
-    vanishing_set.reserve(log_instance_size);
-    std::iota(vanishing_set.begin(), vanishing_set.end(), FF::zero());
-    // need to be built not on the vanishing set H!!!!!
-
     // sanity check: i think we don't care about elemnt 0th element of beta, it is 1
     std::vector<FF> betas_star(log_instance_size);
     betas_star[0] = 1;
@@ -74,18 +66,54 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
 
     auto pow_betas_star = compute_pow_polynomials_at_values(betas_star, instance_size);
 
-    // compute pow, work around combiner problems
+    auto combiner = compute_combiner(instances, accumulator->relation_parameters, pow_betas_star, alpha);
+    const auto combiner_quotient_size = (log_instance_size - 1) * Flavor::MAX_RELATION_LENGTH;
+    std::vector<FF> lagrange_0(combiner_quotient_size);
+    std::vector<FF> vanishing_polynomial(combiner_quotient_size);
+    // combiner_quotient evaluations from k to dk + 1
+    // TODO: make this univariates
+    // WARNING: barycentric evaluations might not work here ????
+    std::array<FF, (Flavor::MAX_RANDOM_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) + 1> combiner_quotient_coeffs;
+    for (size_t point = log_instance_size; point < combiner.size(); point++) {
+        auto idx = point - log_instance_size;
+        lagrange_0[idx] = FF(1) - FF(point);
+        vanishing_polynomial[idx] = FF(point) * (FF(point) - 1);
+        combiner_quotient_coeffs[idx] =
+            (combiner.value_at(point) - compressed_perturbator * lagrange_0[idx]) / vanishing_polynomial[idx];
+    }
 
-    // compute \vec{\beta*}
-    // sanity check this is correc
+    for (size_t idx = 0; idx < combiner_quotient_size; idx++) {
+        transcript.send_to_verifier("combiner_quotient_" + std::to_string(idx), combiner_quotient_coeffs[idx]);
+    }
 
-    auto combiner = compute_combiner(instances, accumulator->relation_parameters, pow_betas_star, alpha);
+    RandomExtendedUnivariate combiner_quotient = Univariate(combiner_quotient_coeffs);
+    auto combiner_challenge = transcript.get_challenge("combiner_qoutient_challenge");
+    auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge);
+    auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
+    auto lagrange_0_at_challenge = FF(1) - combiner_challenge;
+    auto lagrange_1_at_challenge = combiner_challenge;
+    auto new_target_sum = compressed_perturbator * lagrange_0_at_challenge + vanishing_polynomial_at_challenge +
+                          combiner_quotient_at_challenge;
+
+    for (size_t idx = 0; idx < accumulator->prover_polynomials.size(); idx++) {
+        auto accumulator_poly = accumulator->prover_polynomials[idx];
+        auto instance_poly = accumulator->prover_polynomials[idx];
+        assert(accumulator_poly.size() == instance_poly.size());
+        for (size_t j = 0; j < accumulator_poly.size(); j++) {
+            accumulator_poly[j] =
+                accumulator_poly[j] * lagrange_0_at_challenge + instance_poly * lagrange_1_at_challenge;
+        }
+    }
+
+    // i have dk evaluations out of which the first k are not good so we evaluate the rest of the polynomials from
+    // k+1
 
-    // auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
-    // auto lagrange_0_at_challenge = FF(1) - combiner_challenge;
-    // auto lagrange_1_at_challenge = combiner_challenge;
+    // we do barycentric evaluation on the combiner
 
     ProverFoldingResult<Flavor> res;
+    res.folded_prover_polynomials = accumalator->prover_polynomials;
+    res.params.target_sum = new_target_sum;
+    res.params.gate_separation_challenges = betas_star;
     res.folding_data = transcript.proof_data;
     return res;
 }
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
index fe5757c3fcd..d83a28e9207 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
@@ -289,8 +289,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
             Utils::add_nested_tuples(univariate_accumulators, accumulators);
         }
         // Batch the univariate contributions from each sub-relation to obtain the round univariate
-        return Utils::template batch_over_relations<RandomExtendedUnivariate>(
-            univariate_accumulators, alpha, FF(1)); // not sure what this should be, should it be 1?????
+        return Utils::template batch_over_relations<RandomExtendedUnivariate>(univariate_accumulators, alpha);
     }
 };
 
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp
index 921fe323de1..129d104f55e 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp
@@ -53,10 +53,10 @@ VerifierFoldingResult<typename VerifierInstances::Flavor> ProtoGalaxyVerifier_<
     std::vector<FF> combiner_quotient_coeffs(combiner_quotient_size);
     for (size_t idx = 0; idx < combiner_quotient_size; idx++) {
         combiner_quotient_coeffs[idx] =
-            transcript.template receive_from_prover<FF>("combiner_quotient_idx" + std::to_string(idx));
+            transcript.template receive_from_prover<FF>("combiner_quotient_" + std::to_string(idx));
     }
     auto combiner_quotient = Polynomial<FF>(combiner_quotient_coeffs);
-    auto combiner_challenge = transcript.get_challenge("degree_reduction");
+    auto combiner_challenge = transcript.get_challenge("combiner_quotient_challenge");
     static_cast<void>(combiner_challenge);
 
     // fix k = 1 which means k+1 = 2 is a power of two
diff --git a/barretenberg/cpp/src/barretenberg/honk/sumcheck/sumcheck_round.hpp b/barretenberg/cpp/src/barretenberg/honk/sumcheck/sumcheck_round.hpp
index 1a0bbbfefe5..c03d0968129 100644
--- a/barretenberg/cpp/src/barretenberg/honk/sumcheck/sumcheck_round.hpp
+++ b/barretenberg/cpp/src/barretenberg/honk/sumcheck/sumcheck_round.hpp
@@ -166,7 +166,7 @@ template <typename Flavor> class SumcheckProverRound {
         }
         // Batch the univariate contributions from each sub-relation to obtain the round univariate
         return Utils::template batch_over_relations<barretenberg::Univariate<FF, MAX_RANDOM_RELATION_LENGTH>>(
-            univariate_accumulators, alpha, pow_univariate.zeta_pow);
+            univariate_accumulators, alpha, pow_univariate);
     }
 
   private:
diff --git a/barretenberg/cpp/src/barretenberg/honk/sumcheck/sumcheck_round.test.cpp b/barretenberg/cpp/src/barretenberg/honk/sumcheck/sumcheck_round.test.cpp
index 231b1c97f53..6c71ed95c5f 100644
--- a/barretenberg/cpp/src/barretenberg/honk/sumcheck/sumcheck_round.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/sumcheck/sumcheck_round.test.cpp
@@ -43,7 +43,7 @@ TEST(SumcheckRound, SumcheckTupleOfTuplesOfUnivariates)
     // Use extend_and_batch_univariates to extend to MAX_LENGTH then accumulate
     PowUnivariate<FF> pow_univariate(1);
     auto result = Univariate<FF, MAX_LENGTH>();
-    Utils::extend_and_batch_univariates(tuple_of_tuples, pow_univariate.zeta_pow, result);
+    Utils::extend_and_batch_univariates(tuple_of_tuples, result, pow_univariate);
 
     // Repeat the batching process manually
     auto result_expected = univariate_1.template extend_to<MAX_LENGTH>() * 1 +
diff --git a/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp b/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp
index ca3701f9b9f..0ca83be0654 100644
--- a/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp
+++ b/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp
@@ -55,6 +55,7 @@ template <class Fr, size_t _length> class Univariate {
 
     Fr& value_at(size_t i) { return evaluations[i]; };
     const Fr& value_at(size_t i) const { return evaluations[i]; };
+    size_t size() { return evaluations.size(); };
 
     // Write the Univariate evaluations to a buffer
     std::vector<uint8_t> to_buffer() const { return ::to_buffer(evaluations); }
diff --git a/barretenberg/cpp/src/barretenberg/proof_system/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/proof_system/relations/utils.hpp
index f040d02872f..4ce9df182d2 100644
--- a/barretenberg/cpp/src/barretenberg/proof_system/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/proof_system/relations/utils.hpp
@@ -123,12 +123,15 @@ template <typename Flavor> class RelationUtils {
      */
     template <typename ExtendedUnivariate, typename OfTuplesOfUnivariates>
     static void extend_and_batch_univariates(const OfTuplesOfUnivariates& tuple,
-                                             const FF zeta,
-                                             ExtendedUnivariate& result)
+                                             ExtendedUnivariate& result,
+                                             const std::optional<PowUnivariate<FF>>& pow_univariate = std::nullopt)
     {
+        ExtendedUnivariate extended_random_polynomial;
         // Random poly R(X) = (1-X) + X.zeta_pow
-        auto random_polynomial = Univariate<FF, 2>({ 1, zeta });
-        auto extended_random_polynomial = random_polynomial.template extend_to<ExtendedUnivariate::LENGTH>();
+        if (pow_univariate.has_value()) {
+            auto random_polynomial = Univariate<FF, 2>({ 1, pow_univariate.value().zeta_pow });
+            extended_random_polynomial = random_polynomial.template extend_to<ExtendedUnivariate::LENGTH>();
+        }
 
         auto extend_and_sum = [&]<size_t relation_idx, size_t subrelation_idx, typename Element>(Element& element) {
             auto extended = element.template extend_to<ExtendedUnivariate::LENGTH>();
@@ -136,13 +139,13 @@ template <typename Flavor> class RelationUtils {
             using Relation = typename std::tuple_element_t<relation_idx, Relations>;
             const bool is_subrelation_linearly_independent =
                 proof_system::subrelation_is_linearly_independent<Relation, subrelation_idx>();
-            if (is_subrelation_linearly_independent) {
-                // if subrelation is linearly independent, multiply by random polynomial
-                result += extended * extended_random_polynomial;
-            } else {
-                // if subrelation is pure sum over hypercube, don't multiply by random polynomial
+            if (!is_subrelation_linearly_independent || !pow_univariate.has_value()) {
+                // if subrelation is pure um over hypercube, don't multiply by random polynomial
                 // make these comments more relevant
                 result += extended;
+            } else {
+                // if subrelation is linearly independent, multiply by random polynomial
+                result += extended * extended_random_polynomial;
             }
         };
         apply_to_tuple_of_tuples(tuple, extend_and_sum);
@@ -153,15 +156,16 @@ template <typename Flavor> class RelationUtils {
      * return t_0 + αt_1 + ... + α^{NUM_RELATIONS-1}t_{NUM_RELATIONS-1}).
      */
     template <typename ExtendedUnivariate, typename ContainerOverSubrelations>
-    static ExtendedUnivariate batch_over_relations(ContainerOverSubrelations& univariate_accumulators,
-                                                   const FF& challenge,
-                                                   const FF& zeta)
+    static ExtendedUnivariate batch_over_relations(
+        ContainerOverSubrelations& univariate_accumulators,
+        const FF& challenge,
+        const std::optional<PowUnivariate<FF>>& pow_univariate = std::nullopt)
     {
         FF running_challenge = 1;
         scale_univariates(univariate_accumulators, challenge, running_challenge);
 
         auto result = ExtendedUnivariate(0);
-        extend_and_batch_univariates(univariate_accumulators, zeta, result);
+        extend_and_batch_univariates(univariate_accumulators, result, pow_univariate);
 
         // Reset all univariate accumulators to 0 before beginning accumulation in the next round
         zero_univariates(univariate_accumulators);

From 59c69f22ddf226e9f72eb17a39ab2b5d845083af Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Wed, 25 Oct 2023 12:18:33 +0000
Subject: [PATCH 04/86] pain

---
 .../honk/instance/verifier_instance.hpp       |   2 +-
 .../honk/proof_system/combiner.test.cpp       | 382 +++++++++---------
 .../honk/proof_system/combiner_example_gen.py |  11 +-
 .../honk/proof_system/protogalaxy.test.cpp    |  17 +-
 .../honk/proof_system/protogalaxy_prover.cpp  |  97 ++---
 .../honk/proof_system/protogalaxy_prover.hpp  |   2 +-
 .../proof_system/protogalaxy_verifier.cpp     |  82 ++--
 7 files changed, 296 insertions(+), 297 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/honk/instance/verifier_instance.hpp b/barretenberg/cpp/src/barretenberg/honk/instance/verifier_instance.hpp
index 2e8ac9a0210..f2d1afc1136 100644
--- a/barretenberg/cpp/src/barretenberg/honk/instance/verifier_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/honk/instance/verifier_instance.hpp
@@ -14,6 +14,6 @@ template <class Flavor> class VerifierInstance_ {
     size_t public_input_size;
     size_t instance_size;
     RelationParameters<FF> relation_parameters;
-    FoldingParameters folding_params;
+    FoldingParameters folding_parameters;
 };
 } // namespace proof_system::honk
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner.test.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner.test.cpp
index 5733c79e6da..24ecfd81a23 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner.test.cpp
@@ -1,189 +1,193 @@
-// #include "barretenberg/honk/flavor/ultra.hpp"
-// #include "barretenberg/honk/instance/instances.hpp"
-// #include "barretenberg/honk/proof_system/protogalaxy_prover.hpp"
-// #include "barretenberg/honk/utils/testing.hpp"
-// #include "barretenberg/proof_system/relations/relation_parameters.hpp"
-// #include <gtest/gtest.h>
-
-// using namespace proof_system::honk;
-// namespace barretenberg::test_protogalaxy_prover {
-// using Flavor = proof_system::honk::flavor::Ultra;
-// using Polynomial = typename Flavor::Polynomial;
-// using FF = typename Flavor::FF;
-// using RelationParameters = proof_system::RelationParameters<FF>;
-
-// TEST(Protogalaxy, CombinerOn2Instances)
-// {
-//     constexpr size_t NUM_INSTANCES = 2;
-//     using ProverInstance = ProverInstance_<Flavor>;
-//     using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
-//     using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
-
-//     const auto restrict_to_standard_arithmetic_relation = [](auto& polys) {
-//         std::fill(polys.q_arith.begin(), polys.q_arith.end(), 1);
-//         std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
-//         std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
-//         std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
-//         std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
-//         std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
-//         std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
-//         std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
-//     };
-
-//     auto run_test = [&](bool is_random_input) {
-//         if (is_random_input) {
-//             std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-//             std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
-//             auto relation_parameters = RelationParameters::get_random();
-//             ProtoGalaxyProver prover;
-//             auto pow_univariate = PowUnivariate<FF>(/*zeta_pow=*/2);
-//             auto alpha = FF(0); // focus on the arithmetic relation only
-
-//             for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-//                 auto instance = std::make_shared<ProverInstance>();
-//                 auto [storage, prover_polynomials] = proof_system::honk::get_sequential_prover_polynomials<Flavor>(
-//                     /*log_circuit_size=*/1, idx * 128);
-//                 restrict_to_standard_arithmetic_relation(prover_polynomials);
-//                 storage_arrays[idx] = std::move(storage);
-//                 instance->prover_polynomials = prover_polynomials;
-//                 instance_data[idx] = instance;
-//             }
-
-//             ProverInstances instances{ instance_data };
-
-//             auto result = prover.compute_combiner(instances, relation_parameters, pow_univariate, alpha);
-//             auto expected_result =
-//                 barretenberg::Univariate<FF, 7>(std::array<FF, 7>{ 87706,
-//                                                                    27289140,
-//                                                                    229355214,
-//                                                                    905031784,
-//                                                                    static_cast<uint64_t>(2504059650),
-//                                                                    static_cast<uint64_t>(5627174556),
-//                                                                    static_cast<uint64_t>(11026107190) });
-
-//             EXPECT_EQ(result, expected_result);
-//         } else {
-//             std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-//             std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
-//             auto relation_parameters = RelationParameters::get_random();
-//             ProtoGalaxyProver prover;
-//             auto pow_univariate = PowUnivariate<FF>(/*zeta_pow=*/2);
-//             auto alpha = FF(0); // focus on the arithmetic relation only
-
-//             for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-//                 auto instance = std::make_shared<ProverInstance>();
-//                 auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
-//                     /*log_circuit_size=*/1);
-//                 restrict_to_standard_arithmetic_relation(prover_polynomials);
-//                 storage_arrays[idx] = std::move(storage);
-//                 instance->prover_polynomials = prover_polynomials;
-//                 instance_data[idx] = instance;
-//             }
-
-//             ProverInstances instances{ instance_data };
-
-//             const auto create_add_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
-//                 polys.w_l[idx] = w_l;
-//                 polys.w_r[idx] = w_r;
-//                 polys.w_o[idx] = w_l + w_r;
-//                 polys.q_l[idx] = 1;
-//                 polys.q_r[idx] = 1;
-//                 polys.q_o[idx] = -1;
-//             };
-
-//             const auto create_mul_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
-//                 polys.w_l[idx] = w_l;
-//                 polys.w_r[idx] = w_r;
-//                 polys.w_o[idx] = w_l * w_r;
-//                 polys.q_m[idx] = 1;
-//                 polys.q_o[idx] = -1;
-//             };
-
-//             create_add_gate(instances[0]->prover_polynomials, 0, 1, 2);
-//             create_add_gate(instances[0]->prover_polynomials, 1, 0, 4);
-//             create_add_gate(instances[1]->prover_polynomials, 0, 3, 4);
-//             create_mul_gate(instances[1]->prover_polynomials, 1, 1, 4);
-
-//             restrict_to_standard_arithmetic_relation(instances[0]->prover_polynomials);
-//             restrict_to_standard_arithmetic_relation(instances[1]->prover_polynomials);
-
-//             /* Instance 0                                    Instance 1
-//                 w_l w_r w_o q_m q_l q_r q_o q_c               w_l w_r w_o q_m q_l q_r q_o q_c
-//                 1   2   3   0   1   1   -1  0                 3   4   7   0   1   1   -1  0
-//                 0   4   4   0   1   1   -1  0                 1   4   4   1   0   0   -1  0             */
-
-//             /* Lagrange-combined values, row index 0         Lagrange-combined values, row index 1
-//                 in    0    1    2    3    4    5    6        in    0    1    2    3    4    5    6
-//                 w_l   1    3    5    7    9   11   13        w_l   0    1    2    3    4    5    6
-//                 w_r   2    4    6    8   10   12   14        w_r   4    4    4    4    4    4    4
-//                 w_o   3    7   11   15   19   23   27        w_o   4    4    4    4    4    4    0
-//                 q_m   0    0    0    0    0    0    0        q_m   0    1    2    3    4    5    6
-//                 q_l   1    1    1    1    1    1    1        q_l   1    0   -1   -2   -3   -4   -5
-//                 q_r   1    1    1    1    1    1    1        q_r   1    0   -1   -2   -3   -4   -5
-//                 q_o  -1   -1   -1   -1   -1   -1   -1        q_o  -1   -1   -1   -1   -1   -1   -1
-//                 q_c   0    0    0    0    0    0    0        q_c   0    0    0    0    0    0    0
-
-//             relation value:
-//                       0    0    0    0    0    0    0              0    0    6   18   36   60   90      */
-
-//             auto result = prover.compute_combiner(instances, relation_parameters, pow_univariate, alpha);
-//             auto expected_result = barretenberg::Univariate<FF, 7>(std::array<FF, 7>{ 0, 0, 36, 144, 360, 720, 1260
-//             }); EXPECT_EQ(result, expected_result);
-//         }
-//     };
-//     run_test(true);
-//     run_test(false);
-// };
-
-// TEST(Protogalaxy, CombinerOn4Instances)
-// {
-//     constexpr size_t NUM_INSTANCES = 4;
-//     using ProverInstance = ProverInstance_<Flavor>;
-//     using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
-//     using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
-
-//     const auto zero_all_selectors = [](auto& polys) {
-//         std::fill(polys.q_arith.begin(), polys.q_arith.end(), 0);
-//         std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
-//         std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
-//         std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
-//         std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
-//         std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
-//         std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
-//         std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
-//     };
-
-//     auto run_test = [&]() {
-//         std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-//         std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
-//         auto relation_parameters = RelationParameters::get_random();
-//         ProtoGalaxyProver prover;
-//         auto pow_univariate = PowUnivariate<FF>(/*zeta_pow=*/2);
-//         auto alpha = FF(0); // focus on the arithmetic relation only
-
-//         for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-//             auto instance = std::make_shared<ProverInstance>();
-//             auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
-//                 /*log_circuit_size=*/1);
-//             storage_arrays[idx] = std::move(storage);
-//             instance->prover_polynomials = prover_polynomials;
-//             instance_data[idx] = instance;
-//         }
-
-//         ProverInstances instances{ instance_data };
-
-//         zero_all_selectors(instances[0]->prover_polynomials);
-//         zero_all_selectors(instances[1]->prover_polynomials);
-//         zero_all_selectors(instances[2]->prover_polynomials);
-//         zero_all_selectors(instances[3]->prover_polynomials);
-
-//         auto result = prover.compute_combiner(instances, relation_parameters, pow_univariate, alpha);
-//         std::array<FF, 19> zeroes;
-//         std::fill(zeroes.begin(), zeroes.end(), 0);
-//         auto expected_result = barretenberg::Univariate<FF, 19>(zeroes);
-//         EXPECT_EQ(result, expected_result);
-//     };
-//     run_test();
-// };
-
-// } // namespace barretenberg::test_protogalaxy_prover
+#include "barretenberg/honk/flavor/ultra.hpp"
+#include "barretenberg/honk/instance/instances.hpp"
+#include "barretenberg/honk/proof_system/protogalaxy_prover.hpp"
+#include "barretenberg/honk/utils/testing.hpp"
+#include "barretenberg/proof_system/relations/relation_parameters.hpp"
+#include <gtest/gtest.h>
+
+using namespace proof_system::honk;
+namespace barretenberg::test_protogalaxy_prover {
+using Flavor = proof_system::honk::flavor::Ultra;
+using Polynomial = typename Flavor::Polynomial;
+using FF = typename Flavor::FF;
+using RelationParameters = proof_system::RelationParameters<FF>;
+
+TEST(Protogalaxy, CombinerOn2Instances)
+{
+    constexpr size_t NUM_INSTANCES = 2;
+    using ProverInstance = ProverInstance_<Flavor>;
+    using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
+    using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
+
+    const auto restrict_to_standard_arithmetic_relation = [](auto& polys) {
+        std::fill(polys.q_arith.begin(), polys.q_arith.end(), 1);
+        std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
+        std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
+        std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
+        std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
+        std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
+        std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
+        std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
+    };
+
+    auto run_test = [&](bool is_random_input) {
+        if (is_random_input) {
+            std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+            std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
+            auto relation_parameters = RelationParameters::get_random();
+            ProtoGalaxyProver prover;
+            // auto pow_univariate = PowUnivariate<FF>(/*zeta_pow=*/2);
+            std::vector<FF> pow_betas = { FF(1), FF(2) };
+            auto alpha = FF(0); // focus on the arithmetic relation only
+
+            for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+                auto instance = std::make_shared<ProverInstance>();
+                auto [storage, prover_polynomials] = proof_system::honk::get_sequential_prover_polynomials<Flavor>(
+                    /*log_circuit_size=*/1, idx * 128);
+                restrict_to_standard_arithmetic_relation(prover_polynomials);
+                storage_arrays[idx] = std::move(storage);
+                instance->prover_polynomials = prover_polynomials;
+                instance_data[idx] = instance;
+            }
+
+            ProverInstances instances{ instance_data };
+
+            auto result = prover.compute_combiner(instances, relation_parameters, pow_betas, alpha);
+            auto expected_result =
+                barretenberg::Univariate<FF, 7>(std::array<FF, 7>{ 87706,
+                                                                   13644570,
+                                                                   76451738,
+                                                                   226257946,
+                                                                   static_cast<uint64_t>(500811930),
+                                                                   static_cast<uint64_t>(937862426),
+                                                                   static_cast<uint64_t>(1575158170) });
+
+            EXPECT_EQ(result, expected_result);
+        } else {
+            std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+            std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
+            auto relation_parameters = RelationParameters::get_random();
+            ProtoGalaxyProver prover;
+            // auto pow_univariate = PowUnivariate<FF>(/*zeta_pow=*/2);
+            std::vector<FF> pow_betas = { FF(1), FF(2) };
+            auto alpha = FF(0); // focus on the arithmetic relation only
+
+            for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+                auto instance = std::make_shared<ProverInstance>();
+                auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
+                    /*log_circuit_size=*/1);
+                restrict_to_standard_arithmetic_relation(prover_polynomials);
+                storage_arrays[idx] = std::move(storage);
+                instance->prover_polynomials = prover_polynomials;
+                instance_data[idx] = instance;
+            }
+
+            ProverInstances instances{ instance_data };
+
+            const auto create_add_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
+                polys.w_l[idx] = w_l;
+                polys.w_r[idx] = w_r;
+                polys.w_o[idx] = w_l + w_r;
+                polys.q_l[idx] = 1;
+                polys.q_r[idx] = 1;
+                polys.q_o[idx] = -1;
+            };
+
+            const auto create_mul_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
+                polys.w_l[idx] = w_l;
+                polys.w_r[idx] = w_r;
+                polys.w_o[idx] = w_l * w_r;
+                polys.q_m[idx] = 1;
+                polys.q_o[idx] = -1;
+            };
+
+            create_add_gate(instances[0]->prover_polynomials, 0, 1, 2);
+            create_add_gate(instances[0]->prover_polynomials, 1, 0, 4);
+            create_add_gate(instances[1]->prover_polynomials, 0, 3, 4);
+            create_mul_gate(instances[1]->prover_polynomials, 1, 1, 4);
+
+            restrict_to_standard_arithmetic_relation(instances[0]->prover_polynomials);
+            restrict_to_standard_arithmetic_relation(instances[1]->prover_polynomials);
+
+            /* Instance 0                                    Instance 1
+                w_l w_r w_o q_m q_l q_r q_o q_c               w_l w_r w_o q_m q_l q_r q_o q_c
+                1   2   3   0   1   1   -1  0                 3   4   7   0   1   1   -1  0
+                0   4   4   0   1   1   -1  0                 1   4   4   1   0   0   -1  0             */
+
+            /* Lagrange-combined values, row index 0         Lagrange-combined values, row index 1
+                in    0    1    2    3    4    5    6        in    0    1    2    3    4    5    6
+                w_l   1    3    5    7    9   11   13        w_l   0    1    2    3    4    5    6
+                w_r   2    4    6    8   10   12   14        w_r   4    4    4    4    4    4    4
+                w_o   3    7   11   15   19   23   27        w_o   4    4    4    4    4    4    0
+                q_m   0    0    0    0    0    0    0        q_m   0    1    2    3    4    5    6
+                q_l   1    1    1    1    1    1    1        q_l   1    0   -1   -2   -3   -4   -5
+                q_r   1    1    1    1    1    1    1        q_r   1    0   -1   -2   -3   -4   -5
+                q_o  -1   -1   -1   -1   -1   -1   -1        q_o  -1   -1   -1   -1   -1   -1   -1
+                q_c   0    0    0    0    0    0    0        q_c   0    0    0    0    0    0    0
+
+            relation value:
+                      0    0    0    0    0    0    0              0    0    6   18   36   60   90      */
+
+            auto result = prover.compute_combiner(instances, relation_parameters, pow_betas, alpha);
+            auto expected_result = barretenberg::Univariate<FF, 7>(std::array<FF, 7>{ 0, 0, 12, 36, 72, 120, 180 });
+            // static_cast<void>(result);
+            // static_cast<void>(expected_result);
+            EXPECT_EQ(result, expected_result);
+        }
+    };
+    run_test(true);
+    run_test(false);
+};
+
+TEST(Protogalaxy, CombinerOn4Instances)
+{
+    constexpr size_t NUM_INSTANCES = 4;
+    using ProverInstance = ProverInstance_<Flavor>;
+    using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
+    using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
+
+    const auto zero_all_selectors = [](auto& polys) {
+        std::fill(polys.q_arith.begin(), polys.q_arith.end(), 0);
+        std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
+        std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
+        std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
+        std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
+        std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
+        std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
+        std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
+    };
+
+    auto run_test = [&]() {
+        std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+        std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
+        auto relation_parameters = RelationParameters::get_random();
+        ProtoGalaxyProver prover;
+        auto alpha = FF(0); // focus on the arithmetic relation only
+
+        for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+            auto instance = std::make_shared<ProverInstance>();
+            auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
+                /*log_circuit_size=*/1);
+            storage_arrays[idx] = std::move(storage);
+            instance->prover_polynomials = prover_polynomials;
+            instance_data[idx] = instance;
+        }
+
+        ProverInstances instances{ instance_data };
+
+        zero_all_selectors(instances[0]->prover_polynomials);
+        zero_all_selectors(instances[1]->prover_polynomials);
+        zero_all_selectors(instances[2]->prover_polynomials);
+        zero_all_selectors(instances[3]->prover_polynomials);
+
+        std::vector<FF> pow_betas = { FF(1), FF(2) };
+        auto result = prover.compute_combiner(instances, relation_parameters, pow_betas, alpha);
+        std::array<FF, 19> zeroes;
+        std::fill(zeroes.begin(), zeroes.end(), 0);
+        auto expected_result = barretenberg::Univariate<FF, 19>(zeroes);
+        EXPECT_EQ(result, expected_result);
+    };
+    run_test();
+};
+
+} // namespace barretenberg::test_protogalaxy_prover
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner_example_gen.py b/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner_example_gen.py
index 2c37304ae97..5fa39aff9d8 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner_example_gen.py
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner_example_gen.py
@@ -103,8 +103,9 @@ def compute_first_example():
                              row.q_l, row.q_r, row.q_o, row.q_c)
         accumulator += zeta_pow * relation_value
         zeta_pow *= zeta
+        print(zeta_pow)
 
-    accumulator *= extend_one_entity([1, 2])
+    # accumulator *= extend_one_entity([1, 2])
     return accumulator
 
 
@@ -133,10 +134,12 @@ def compute_second_example():
         result += rel(w_l, w_r, w_o, q_m, q_l, q_r, q_o, q_c)
         result *= 2
 
-        result *= extend_one_entity([1, 2])
+        # result *= extend_one_entity([1, 2])
         return result
 
 if __name__ == "__main__":
-    print(f"First example: \n  {compute_first_example()}")
-    print(f"Second example:\n  {compute_second_example()}")
+    # print(f"First example: \n  {compute_first_example()}")
+    # print(f"Second example:\n  {compute_second_example()}")
+    print(compute_first_example())
+    print(compute_second_example())
 
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp
index 8c35ce4d40b..eafbd1ff410 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp
@@ -127,7 +127,7 @@ TEST_F(ProtoGalaxyTests, PerturbatorCoefficients)
     }
 }
 
-TEST_F(ProtoGalaxyTests, PowPerturbatorPolynomial)
+TEST_F(ProtoGalaxyTests, PerturbatorPolynomial)
 {
     const size_t log_instance_size(3);
     const size_t instance_size(1 << log_instance_size);
@@ -147,17 +147,8 @@ TEST_F(ProtoGalaxyTests, PowPerturbatorPolynomial)
         betas[idx] = FF::random_element();
     }
 
-    // Construct pow(\vec{betas}) manually as in the paper
-    std::vector<FF> pow_beta(instance_size);
-    for (size_t i = 0; i < instance_size; i++) {
-        auto res = FF(1);
-        for (size_t j = i, beta_idx = 0; j > 0; j >>= 1, beta_idx++) {
-            if ((j & 1) == 1) {
-                res *= betas[beta_idx];
-            }
-        }
-        pow_beta[i] = res;
-    }
+    // Construct pow(\vec{betas}) as in the paper
+    auto pow_beta = ProtoGalaxyProver::compute_pow_polynomial_at_values(betas, instance_size);
 
     // Compute the corresponding target sum and create a dummy accumulator
     auto target_sum = FF(0);
@@ -178,4 +169,6 @@ TEST_F(ProtoGalaxyTests, PowPerturbatorPolynomial)
     // Ensure the constant coefficient of the perturbator is equal to the target sum as indicated by the paper
     EXPECT_EQ(perturbator[0], target_sum);
 }
+
+TEST_F(ProtoGalaxyTests, PowCorrectness) {}
 } // namespace protogalaxy_tests
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
index 638557681f0..1282dd948d2 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
@@ -47,7 +47,7 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
     const auto log_instance_size = static_cast<size_t>(numeric::get_msb(instance_size));
     auto deltas = compute_round_challenge_pows(log_instance_size, delta);
     auto perturbator = compute_perturbator(accumulator, deltas, alpha);
-    assert(perturbator[0] == accumulator->folding_parameters->target_sum);
+    assert(perturbator[0] == accumulator->folding_parameters.target_sum);
     for (size_t idx = 0; idx <= log_instance_size; idx++) {
         transcript.send_to_verifier("perturbator_" + std::to_string(idx), perturbator[idx]);
     }
@@ -64,57 +64,58 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
         betas_star[idx] = betas[idx] + perturbator_challenge * deltas[idx - 1];
     }
 
-    auto pow_betas_star = compute_pow_polynomials_at_values(betas_star, instance_size);
+    auto pow_betas_star = compute_pow_polynomial_at_values(betas_star, instance_size);
 
     auto combiner = compute_combiner(instances, accumulator->relation_parameters, pow_betas_star, alpha);
-    const auto combiner_quotient_size = (log_instance_size - 1) * Flavor::MAX_RELATION_LENGTH;
-    std::vector<FF> lagrange_0(combiner_quotient_size);
-    std::vector<FF> vanishing_polynomial(combiner_quotient_size);
-    // combiner_quotient evaluations from k to dk + 1
-    // TODO: make this univariates
-    // WARNING: barycentric evaluations might not work here ????
-    std::array<FF, (Flavor::MAX_RANDOM_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) + 1> combiner_quotient_coeffs;
-    for (size_t point = log_instance_size; point < combiner.size(); point++) {
-        auto idx = point - log_instance_size;
-        lagrange_0[idx] = FF(1) - FF(point);
-        vanishing_polynomial[idx] = FF(point) * (FF(point) - 1);
-        combiner_quotient_coeffs[idx] =
-            (combiner.value_at(point) - compressed_perturbator * lagrange_0[idx]) / vanishing_polynomial[idx];
-    }
-
-    for (size_t idx = 0; idx < combiner_quotient_size; idx++) {
-        transcript.send_to_verifier("combiner_quotient_" + std::to_string(idx), combiner_quotient_coeffs[idx]);
-    }
-
-    RandomExtendedUnivariate combiner_quotient = Univariate(combiner_quotient_coeffs);
-    auto combiner_challenge = transcript.get_challenge("combiner_qoutient_challenge");
-    auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge);
-    auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
-    auto lagrange_0_at_challenge = FF(1) - combiner_challenge;
-    auto lagrange_1_at_challenge = combiner_challenge;
-    auto new_target_sum = compressed_perturbator * lagrange_0_at_challenge + vanishing_polynomial_at_challenge +
-                          combiner_quotient_at_challenge;
-
-    for (size_t idx = 0; idx < accumulator->prover_polynomials.size(); idx++) {
-        auto accumulator_poly = accumulator->prover_polynomials[idx];
-        auto instance_poly = accumulator->prover_polynomials[idx];
-        assert(accumulator_poly.size() == instance_poly.size());
-        for (size_t j = 0; j < accumulator_poly.size(); j++) {
-            accumulator_poly[j] =
-                accumulator_poly[j] * lagrange_0_at_challenge + instance_poly * lagrange_1_at_challenge;
-        }
-    }
-
-    // i have dk evaluations out of which the first k are not good so we evaluate the rest of the polynomials from
-    // k+1
-
-    // we do barycentric evaluation on the combiner
+    static_cast<void>(combiner);
+    // const auto combiner_quotient_size = (log_instance_size - 1) * Flavor::MAX_RELATION_LENGTH;
+    // std::vector<FF> lagrange_0(combiner_quotient_size);
+    // std::vector<FF> vanishing_polynomial(combiner_quotient_size);
+    // // combiner_quotient evaluations from k to dk + 1
+    // // TODO: make this univariates
+    // // WARNING: barycentric evaluations might not work here ????
+    // std::array<FF, (Flavor::MAX_RANDOM_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) + 1>
+    // combiner_quotient_coeffs; for (size_t point = log_instance_size; point < combiner.size(); point++) {
+    //     auto idx = point - log_instance_size;
+    //     lagrange_0[idx] = FF(1) - FF(point);
+    //     vanishing_polynomial[idx] = FF(point) * (FF(point) - 1);
+    //     combiner_quotient_coeffs[idx] =
+    //         (combiner.value_at(point) - compressed_perturbator * lagrange_0[idx]) / vanishing_polynomial[idx];
+    // }
+
+    // for (size_t idx = 0; idx < combiner_quotient_size; idx++) {
+    //     transcript.send_to_verifier("combiner_quotient_" + std::to_string(idx), combiner_quotient_coeffs[idx]);
+    // }
+
+    // RandomExtendedUnivariate combiner_quotient = Univariate(combiner_quotient_coeffs);
+    // auto combiner_challenge = transcript.get_challenge("combiner_qoutient_challenge");
+    // auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge);
+    // auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
+    // auto lagrange_0_at_challenge = FF(1) - combiner_challenge;
+    // auto lagrange_1_at_challenge = combiner_challenge;
+    // auto new_target_sum = compressed_perturbator * lagrange_0_at_challenge + vanishing_polynomial_at_challenge +
+    //                       combiner_quotient_at_challenge;
+
+    // for (size_t idx = 0; idx < accumulator->prover_polynomials.size(); idx++) {
+    //     auto accumulator_poly = accumulator->prover_polynomials[idx];
+    //     auto instance_poly = accumulator->prover_polynomials[idx];
+    //     assert(accumulator_poly.size() == instance_poly.size());
+    //     for (size_t j = 0; j < accumulator_poly.size(); j++) {
+    //         accumulator_poly[j] =
+    //             accumulator_poly[j] * lagrange_0_at_challenge + instance_poly * lagrange_1_at_challenge;
+    //     }
+    // }
+
+    // // i have dk evaluations out of which the first k are not good so we evaluate the rest of the polynomials from
+    // // k+1
+
+    // // we do barycentric evaluation on the combiner
 
     ProverFoldingResult<Flavor> res;
-    res.folded_prover_polynomials = accumalator->prover_polynomials;
-    res.params.target_sum = new_target_sum;
-    res.params.gate_separation_challenges = betas_star;
-    res.folding_data = transcript.proof_data;
+    // res.folded_prover_polynomials = accumulator->prover_polynomials;
+    // res.params.target_sum = new_target_sum;
+    // res.params.gate_separation_challenges = betas_star;
+    // res.folding_data = transcript.proof_data;
     return res;
 }
 template class ProtoGalaxyProver_<ProverInstances_<honk::flavor::Ultra, 2>>;
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
index d83a28e9207..30f0954254d 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
@@ -48,7 +48,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      */
     void prepare_for_folding();
 
-    std::vector<FF> compute_pow_polynomials_at_values(const std::vector<FF> betas, const size_t instance_size)
+    static std::vector<FF> compute_pow_polynomial_at_values(const std::vector<FF> betas, const size_t instance_size)
     {
         std::vector<FF> pow_betas(instance_size);
         for (size_t i = 0; i < instance_size; i++) {
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp
index 129d104f55e..36201b2d0e6 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp
@@ -6,7 +6,7 @@ VerifierFoldingResult<typename VerifierInstances::Flavor> ProtoGalaxyVerifier_<
     VerifierInstances>::fold_public_parameters(std::vector<uint8_t> fold_data)
 {
     using Flavor = typename VerifierInstances::Flavor;
-    using VerificationKey = typename Flavor::VerificationKey;
+    // using VerificationKey = typename Flavor::VerificationKey;
 
     transcript = VerifierTranscript<FF>{ fold_data };
     auto index = 0;
@@ -42,51 +42,49 @@ VerifierFoldingResult<typename VerifierInstances::Flavor> ProtoGalaxyVerifier_<
     for (size_t idx = 0; idx <= log_instance_size; idx++) {
         perturbator_coeffs[idx] = transcript.template receive_from_prover<FF>("perturbator_" + std::to_string(idx));
     }
-    assert(perturbator_coeffs[0] == accumulator->folding_parameters->target_sum);
-    auto perturbator = Polynomial<FF>(perturbator_coeffs);
-    auto perturbator_challenge = transcript.get_challenge("perturbator_challenge");
-    auto perturbator_at_challenge = perturbator.evaluate(perturbator_challenge); // horner
-    static_cast<void>(perturbator_at_challenge);
-    // get the coefficient of combiner quotient (K(X) polynomial in the paper)
-    // + 2 because we need one extra eval to rep a poly of certain degree
-    auto combiner_quotient_size = Flavor::MAX_RANDOM_RELATION_LENGTH * (VerifierInstances::NUM - 1) + 2;
-    std::vector<FF> combiner_quotient_coeffs(combiner_quotient_size);
-    for (size_t idx = 0; idx < combiner_quotient_size; idx++) {
-        combiner_quotient_coeffs[idx] =
-            transcript.template receive_from_prover<FF>("combiner_quotient_" + std::to_string(idx));
-    }
-    auto combiner_quotient = Polynomial<FF>(combiner_quotient_coeffs);
-    auto combiner_challenge = transcript.get_challenge("combiner_quotient_challenge");
-    static_cast<void>(combiner_challenge);
+    assert(perturbator_coeffs[0] == accumulator->folding_parameters.target_sum);
+    // auto perturbator = Polynomial<FF>(perturbator_coeffs);
+    // auto perturbator_challenge = transcript.get_challenge("perturbator_challenge");
+    // auto perturbator_at_challenge = perturbator.evaluate(perturbator_challenge); // horner
 
-    // fix k = 1 which means k+1 = 2 is a power of two
-    auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge); // K(\gamma)
-    auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
-    auto lagrange_0_at_challenge = FF(1) - combiner_challenge;
-    auto lagrange_1_at_challenge = combiner_challenge;
-    auto updated_target_sum = perturbator_at_challenge * lagrange_0_at_challenge +
-                              vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
+    // auto combiner_quotient_size = Flavor::MAX_RANDOM_RELATION_LENGTH * (VerifierInstances::NUM - 1) + 2;
+    // std::vector<FF> combiner_quotient_coeffs(combiner_quotient_size);
+    // for (size_t idx = 0; idx < combiner_quotient_size; idx++) {
+    //     combiner_quotient_coeffs[idx] =
+    //         transcript.template receive_from_prover<FF>("combiner_quotient_" + std::to_string(idx));
+    // }
+    // auto combiner_quotient = Polynomial<FF>(combiner_quotient_coeffs);
+    // auto combiner_challenge = transcript.get_challenge("combiner_quotient_challenge");
+    // static_cast<void>(combiner_challenge);
 
-    // do public input stuff with goblin (well I guess first without)
-    // fold public inputs (field elements) and the commitment in verification key to obtain a new verification key!
-    // TODO: extend for more instances;
-    auto accumulator_public_inputs = accumulator->public_inputs;
-    auto other_public_inputs = verifier_instances[1]->public_inputs;
-    std::vector<FF> folded_public_inputs(accumulator->public_input_size);
-    // extend either with 0s if the sizes are different
-    for (size_t idx = 0; idx < folded_public_inputs.size(); idx++) {
-        folded_public_inputs[idx] = lagrange_0_at_challenge * accumulator->public_inputs[idx] +
-                                    lagrange_1_at_challenge * other_public_inputs[idx];
-    }
-    // all verification keys have the same size
-    auto folded_verification_key = VerificationKey(accumulator->instance_size, accumulator->public_input_size);
+    // // fix k = 1 which means k+1 = 2 is a power of two
+    // auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge); // K(\gamma)
+    // auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
+    // auto lagrange_0_at_challenge = FF(1) - combiner_challenge;
+    // auto lagrange_1_at_challenge = combiner_challenge;
+    // auto updated_target_sum = perturbator_at_challenge * lagrange_0_at_challenge +
+    //                           vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
 
-    for (size_t idx = 0; idx < folded_verification_key.size(); idx++) {
-        folded_verification_key[idx] = (*accumulator->verification_key)[idx] * lagrange_0_at_challenge +
-                                       (*verifier_instances[1]->verification_key)[idx] * lagrange_1_at_challenge;
-    }
+    // // do public input stuff with goblin (well I guess first without)
+    // // fold public inputs (field elements) and the commitment in verification key to obtain a new verification key!
+    // // TODO: extend for more instances;
+    // auto accumulator_public_inputs = accumulator->public_inputs;
+    // auto other_public_inputs = verifier_instances[1]->public_inputs;
+    // std::vector<FF> folded_public_inputs(accumulator->public_input_size);
+    // // extend either with 0s if the sizes are different
+    // for (size_t idx = 0; idx < folded_public_inputs.size(); idx++) {
+    //     folded_public_inputs[idx] = lagrange_0_at_challenge * accumulator->public_inputs[idx] +
+    //                                 lagrange_1_at_challenge * other_public_inputs[idx];
+    // }
+    // // all verification keys have the same size
+    // auto folded_verification_key = VerificationKey(accumulator->instance_size, accumulator->public_input_size);
+
+    // for (size_t idx = 0; idx < folded_verification_key.size(); idx++) {
+    //     folded_verification_key[idx] = (*accumulator->verification_key)[idx] * lagrange_0_at_challenge +
+    //                                    (*verifier_instances[1]->verification_key)[idx] * lagrange_1_at_challenge;
+    // }
     VerifierFoldingResult<Flavor> res;
-    res.parameters.target_sum = updated_target_sum;
+    // res.parameters.target_sum = updated_target_sum;
     return res;
 }
 

From aed27df0492ce976286268f5862469e9c921ada8 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Wed, 25 Oct 2023 12:18:33 +0000
Subject: [PATCH 05/86] pain

---
 .../honk/instance/verifier_instance.hpp       |   2 +-
 .../honk/proof_system/combiner.test.cpp       | 382 +++++++++---------
 .../honk/proof_system/combiner_example_gen.py |  71 ++--
 .../honk/proof_system/protogalaxy.test.cpp    |  17 +-
 .../honk/proof_system/protogalaxy_prover.cpp  |  19 +-
 .../honk/proof_system/protogalaxy_prover.hpp  |   2 +-
 .../proof_system/protogalaxy_verifier.cpp     |  19 +-
 7 files changed, 260 insertions(+), 252 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/honk/instance/verifier_instance.hpp b/barretenberg/cpp/src/barretenberg/honk/instance/verifier_instance.hpp
index 2e8ac9a0210..f2d1afc1136 100644
--- a/barretenberg/cpp/src/barretenberg/honk/instance/verifier_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/honk/instance/verifier_instance.hpp
@@ -14,6 +14,6 @@ template <class Flavor> class VerifierInstance_ {
     size_t public_input_size;
     size_t instance_size;
     RelationParameters<FF> relation_parameters;
-    FoldingParameters folding_params;
+    FoldingParameters folding_parameters;
 };
 } // namespace proof_system::honk
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner.test.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner.test.cpp
index 5733c79e6da..24ecfd81a23 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner.test.cpp
@@ -1,189 +1,193 @@
-// #include "barretenberg/honk/flavor/ultra.hpp"
-// #include "barretenberg/honk/instance/instances.hpp"
-// #include "barretenberg/honk/proof_system/protogalaxy_prover.hpp"
-// #include "barretenberg/honk/utils/testing.hpp"
-// #include "barretenberg/proof_system/relations/relation_parameters.hpp"
-// #include <gtest/gtest.h>
-
-// using namespace proof_system::honk;
-// namespace barretenberg::test_protogalaxy_prover {
-// using Flavor = proof_system::honk::flavor::Ultra;
-// using Polynomial = typename Flavor::Polynomial;
-// using FF = typename Flavor::FF;
-// using RelationParameters = proof_system::RelationParameters<FF>;
-
-// TEST(Protogalaxy, CombinerOn2Instances)
-// {
-//     constexpr size_t NUM_INSTANCES = 2;
-//     using ProverInstance = ProverInstance_<Flavor>;
-//     using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
-//     using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
-
-//     const auto restrict_to_standard_arithmetic_relation = [](auto& polys) {
-//         std::fill(polys.q_arith.begin(), polys.q_arith.end(), 1);
-//         std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
-//         std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
-//         std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
-//         std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
-//         std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
-//         std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
-//         std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
-//     };
-
-//     auto run_test = [&](bool is_random_input) {
-//         if (is_random_input) {
-//             std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-//             std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
-//             auto relation_parameters = RelationParameters::get_random();
-//             ProtoGalaxyProver prover;
-//             auto pow_univariate = PowUnivariate<FF>(/*zeta_pow=*/2);
-//             auto alpha = FF(0); // focus on the arithmetic relation only
-
-//             for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-//                 auto instance = std::make_shared<ProverInstance>();
-//                 auto [storage, prover_polynomials] = proof_system::honk::get_sequential_prover_polynomials<Flavor>(
-//                     /*log_circuit_size=*/1, idx * 128);
-//                 restrict_to_standard_arithmetic_relation(prover_polynomials);
-//                 storage_arrays[idx] = std::move(storage);
-//                 instance->prover_polynomials = prover_polynomials;
-//                 instance_data[idx] = instance;
-//             }
-
-//             ProverInstances instances{ instance_data };
-
-//             auto result = prover.compute_combiner(instances, relation_parameters, pow_univariate, alpha);
-//             auto expected_result =
-//                 barretenberg::Univariate<FF, 7>(std::array<FF, 7>{ 87706,
-//                                                                    27289140,
-//                                                                    229355214,
-//                                                                    905031784,
-//                                                                    static_cast<uint64_t>(2504059650),
-//                                                                    static_cast<uint64_t>(5627174556),
-//                                                                    static_cast<uint64_t>(11026107190) });
-
-//             EXPECT_EQ(result, expected_result);
-//         } else {
-//             std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-//             std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
-//             auto relation_parameters = RelationParameters::get_random();
-//             ProtoGalaxyProver prover;
-//             auto pow_univariate = PowUnivariate<FF>(/*zeta_pow=*/2);
-//             auto alpha = FF(0); // focus on the arithmetic relation only
-
-//             for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-//                 auto instance = std::make_shared<ProverInstance>();
-//                 auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
-//                     /*log_circuit_size=*/1);
-//                 restrict_to_standard_arithmetic_relation(prover_polynomials);
-//                 storage_arrays[idx] = std::move(storage);
-//                 instance->prover_polynomials = prover_polynomials;
-//                 instance_data[idx] = instance;
-//             }
-
-//             ProverInstances instances{ instance_data };
-
-//             const auto create_add_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
-//                 polys.w_l[idx] = w_l;
-//                 polys.w_r[idx] = w_r;
-//                 polys.w_o[idx] = w_l + w_r;
-//                 polys.q_l[idx] = 1;
-//                 polys.q_r[idx] = 1;
-//                 polys.q_o[idx] = -1;
-//             };
-
-//             const auto create_mul_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
-//                 polys.w_l[idx] = w_l;
-//                 polys.w_r[idx] = w_r;
-//                 polys.w_o[idx] = w_l * w_r;
-//                 polys.q_m[idx] = 1;
-//                 polys.q_o[idx] = -1;
-//             };
-
-//             create_add_gate(instances[0]->prover_polynomials, 0, 1, 2);
-//             create_add_gate(instances[0]->prover_polynomials, 1, 0, 4);
-//             create_add_gate(instances[1]->prover_polynomials, 0, 3, 4);
-//             create_mul_gate(instances[1]->prover_polynomials, 1, 1, 4);
-
-//             restrict_to_standard_arithmetic_relation(instances[0]->prover_polynomials);
-//             restrict_to_standard_arithmetic_relation(instances[1]->prover_polynomials);
-
-//             /* Instance 0                                    Instance 1
-//                 w_l w_r w_o q_m q_l q_r q_o q_c               w_l w_r w_o q_m q_l q_r q_o q_c
-//                 1   2   3   0   1   1   -1  0                 3   4   7   0   1   1   -1  0
-//                 0   4   4   0   1   1   -1  0                 1   4   4   1   0   0   -1  0             */
-
-//             /* Lagrange-combined values, row index 0         Lagrange-combined values, row index 1
-//                 in    0    1    2    3    4    5    6        in    0    1    2    3    4    5    6
-//                 w_l   1    3    5    7    9   11   13        w_l   0    1    2    3    4    5    6
-//                 w_r   2    4    6    8   10   12   14        w_r   4    4    4    4    4    4    4
-//                 w_o   3    7   11   15   19   23   27        w_o   4    4    4    4    4    4    0
-//                 q_m   0    0    0    0    0    0    0        q_m   0    1    2    3    4    5    6
-//                 q_l   1    1    1    1    1    1    1        q_l   1    0   -1   -2   -3   -4   -5
-//                 q_r   1    1    1    1    1    1    1        q_r   1    0   -1   -2   -3   -4   -5
-//                 q_o  -1   -1   -1   -1   -1   -1   -1        q_o  -1   -1   -1   -1   -1   -1   -1
-//                 q_c   0    0    0    0    0    0    0        q_c   0    0    0    0    0    0    0
-
-//             relation value:
-//                       0    0    0    0    0    0    0              0    0    6   18   36   60   90      */
-
-//             auto result = prover.compute_combiner(instances, relation_parameters, pow_univariate, alpha);
-//             auto expected_result = barretenberg::Univariate<FF, 7>(std::array<FF, 7>{ 0, 0, 36, 144, 360, 720, 1260
-//             }); EXPECT_EQ(result, expected_result);
-//         }
-//     };
-//     run_test(true);
-//     run_test(false);
-// };
-
-// TEST(Protogalaxy, CombinerOn4Instances)
-// {
-//     constexpr size_t NUM_INSTANCES = 4;
-//     using ProverInstance = ProverInstance_<Flavor>;
-//     using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
-//     using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
-
-//     const auto zero_all_selectors = [](auto& polys) {
-//         std::fill(polys.q_arith.begin(), polys.q_arith.end(), 0);
-//         std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
-//         std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
-//         std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
-//         std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
-//         std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
-//         std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
-//         std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
-//     };
-
-//     auto run_test = [&]() {
-//         std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-//         std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
-//         auto relation_parameters = RelationParameters::get_random();
-//         ProtoGalaxyProver prover;
-//         auto pow_univariate = PowUnivariate<FF>(/*zeta_pow=*/2);
-//         auto alpha = FF(0); // focus on the arithmetic relation only
-
-//         for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-//             auto instance = std::make_shared<ProverInstance>();
-//             auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
-//                 /*log_circuit_size=*/1);
-//             storage_arrays[idx] = std::move(storage);
-//             instance->prover_polynomials = prover_polynomials;
-//             instance_data[idx] = instance;
-//         }
-
-//         ProverInstances instances{ instance_data };
-
-//         zero_all_selectors(instances[0]->prover_polynomials);
-//         zero_all_selectors(instances[1]->prover_polynomials);
-//         zero_all_selectors(instances[2]->prover_polynomials);
-//         zero_all_selectors(instances[3]->prover_polynomials);
-
-//         auto result = prover.compute_combiner(instances, relation_parameters, pow_univariate, alpha);
-//         std::array<FF, 19> zeroes;
-//         std::fill(zeroes.begin(), zeroes.end(), 0);
-//         auto expected_result = barretenberg::Univariate<FF, 19>(zeroes);
-//         EXPECT_EQ(result, expected_result);
-//     };
-//     run_test();
-// };
-
-// } // namespace barretenberg::test_protogalaxy_prover
+#include "barretenberg/honk/flavor/ultra.hpp"
+#include "barretenberg/honk/instance/instances.hpp"
+#include "barretenberg/honk/proof_system/protogalaxy_prover.hpp"
+#include "barretenberg/honk/utils/testing.hpp"
+#include "barretenberg/proof_system/relations/relation_parameters.hpp"
+#include <gtest/gtest.h>
+
+using namespace proof_system::honk;
+namespace barretenberg::test_protogalaxy_prover {
+using Flavor = proof_system::honk::flavor::Ultra;
+using Polynomial = typename Flavor::Polynomial;
+using FF = typename Flavor::FF;
+using RelationParameters = proof_system::RelationParameters<FF>;
+
+TEST(Protogalaxy, CombinerOn2Instances)
+{
+    constexpr size_t NUM_INSTANCES = 2;
+    using ProverInstance = ProverInstance_<Flavor>;
+    using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
+    using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
+
+    const auto restrict_to_standard_arithmetic_relation = [](auto& polys) {
+        std::fill(polys.q_arith.begin(), polys.q_arith.end(), 1);
+        std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
+        std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
+        std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
+        std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
+        std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
+        std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
+        std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
+    };
+
+    auto run_test = [&](bool is_random_input) {
+        if (is_random_input) {
+            std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+            std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
+            auto relation_parameters = RelationParameters::get_random();
+            ProtoGalaxyProver prover;
+            // auto pow_univariate = PowUnivariate<FF>(/*zeta_pow=*/2);
+            std::vector<FF> pow_betas = { FF(1), FF(2) };
+            auto alpha = FF(0); // focus on the arithmetic relation only
+
+            for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+                auto instance = std::make_shared<ProverInstance>();
+                auto [storage, prover_polynomials] = proof_system::honk::get_sequential_prover_polynomials<Flavor>(
+                    /*log_circuit_size=*/1, idx * 128);
+                restrict_to_standard_arithmetic_relation(prover_polynomials);
+                storage_arrays[idx] = std::move(storage);
+                instance->prover_polynomials = prover_polynomials;
+                instance_data[idx] = instance;
+            }
+
+            ProverInstances instances{ instance_data };
+
+            auto result = prover.compute_combiner(instances, relation_parameters, pow_betas, alpha);
+            auto expected_result =
+                barretenberg::Univariate<FF, 7>(std::array<FF, 7>{ 87706,
+                                                                   13644570,
+                                                                   76451738,
+                                                                   226257946,
+                                                                   static_cast<uint64_t>(500811930),
+                                                                   static_cast<uint64_t>(937862426),
+                                                                   static_cast<uint64_t>(1575158170) });
+
+            EXPECT_EQ(result, expected_result);
+        } else {
+            std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+            std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
+            auto relation_parameters = RelationParameters::get_random();
+            ProtoGalaxyProver prover;
+            // auto pow_univariate = PowUnivariate<FF>(/*zeta_pow=*/2);
+            std::vector<FF> pow_betas = { FF(1), FF(2) };
+            auto alpha = FF(0); // focus on the arithmetic relation only
+
+            for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+                auto instance = std::make_shared<ProverInstance>();
+                auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
+                    /*log_circuit_size=*/1);
+                restrict_to_standard_arithmetic_relation(prover_polynomials);
+                storage_arrays[idx] = std::move(storage);
+                instance->prover_polynomials = prover_polynomials;
+                instance_data[idx] = instance;
+            }
+
+            ProverInstances instances{ instance_data };
+
+            const auto create_add_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
+                polys.w_l[idx] = w_l;
+                polys.w_r[idx] = w_r;
+                polys.w_o[idx] = w_l + w_r;
+                polys.q_l[idx] = 1;
+                polys.q_r[idx] = 1;
+                polys.q_o[idx] = -1;
+            };
+
+            const auto create_mul_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
+                polys.w_l[idx] = w_l;
+                polys.w_r[idx] = w_r;
+                polys.w_o[idx] = w_l * w_r;
+                polys.q_m[idx] = 1;
+                polys.q_o[idx] = -1;
+            };
+
+            create_add_gate(instances[0]->prover_polynomials, 0, 1, 2);
+            create_add_gate(instances[0]->prover_polynomials, 1, 0, 4);
+            create_add_gate(instances[1]->prover_polynomials, 0, 3, 4);
+            create_mul_gate(instances[1]->prover_polynomials, 1, 1, 4);
+
+            restrict_to_standard_arithmetic_relation(instances[0]->prover_polynomials);
+            restrict_to_standard_arithmetic_relation(instances[1]->prover_polynomials);
+
+            /* Instance 0                                    Instance 1
+                w_l w_r w_o q_m q_l q_r q_o q_c               w_l w_r w_o q_m q_l q_r q_o q_c
+                1   2   3   0   1   1   -1  0                 3   4   7   0   1   1   -1  0
+                0   4   4   0   1   1   -1  0                 1   4   4   1   0   0   -1  0             */
+
+            /* Lagrange-combined values, row index 0         Lagrange-combined values, row index 1
+                in    0    1    2    3    4    5    6        in    0    1    2    3    4    5    6
+                w_l   1    3    5    7    9   11   13        w_l   0    1    2    3    4    5    6
+                w_r   2    4    6    8   10   12   14        w_r   4    4    4    4    4    4    4
+                w_o   3    7   11   15   19   23   27        w_o   4    4    4    4    4    4    0
+                q_m   0    0    0    0    0    0    0        q_m   0    1    2    3    4    5    6
+                q_l   1    1    1    1    1    1    1        q_l   1    0   -1   -2   -3   -4   -5
+                q_r   1    1    1    1    1    1    1        q_r   1    0   -1   -2   -3   -4   -5
+                q_o  -1   -1   -1   -1   -1   -1   -1        q_o  -1   -1   -1   -1   -1   -1   -1
+                q_c   0    0    0    0    0    0    0        q_c   0    0    0    0    0    0    0
+
+            relation value:
+                      0    0    0    0    0    0    0              0    0    6   18   36   60   90      */
+
+            auto result = prover.compute_combiner(instances, relation_parameters, pow_betas, alpha);
+            auto expected_result = barretenberg::Univariate<FF, 7>(std::array<FF, 7>{ 0, 0, 12, 36, 72, 120, 180 });
+            // static_cast<void>(result);
+            // static_cast<void>(expected_result);
+            EXPECT_EQ(result, expected_result);
+        }
+    };
+    run_test(true);
+    run_test(false);
+};
+
+TEST(Protogalaxy, CombinerOn4Instances)
+{
+    constexpr size_t NUM_INSTANCES = 4;
+    using ProverInstance = ProverInstance_<Flavor>;
+    using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
+    using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
+
+    const auto zero_all_selectors = [](auto& polys) {
+        std::fill(polys.q_arith.begin(), polys.q_arith.end(), 0);
+        std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
+        std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
+        std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
+        std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
+        std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
+        std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
+        std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
+    };
+
+    auto run_test = [&]() {
+        std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+        std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
+        auto relation_parameters = RelationParameters::get_random();
+        ProtoGalaxyProver prover;
+        auto alpha = FF(0); // focus on the arithmetic relation only
+
+        for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+            auto instance = std::make_shared<ProverInstance>();
+            auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
+                /*log_circuit_size=*/1);
+            storage_arrays[idx] = std::move(storage);
+            instance->prover_polynomials = prover_polynomials;
+            instance_data[idx] = instance;
+        }
+
+        ProverInstances instances{ instance_data };
+
+        zero_all_selectors(instances[0]->prover_polynomials);
+        zero_all_selectors(instances[1]->prover_polynomials);
+        zero_all_selectors(instances[2]->prover_polynomials);
+        zero_all_selectors(instances[3]->prover_polynomials);
+
+        std::vector<FF> pow_betas = { FF(1), FF(2) };
+        auto result = prover.compute_combiner(instances, relation_parameters, pow_betas, alpha);
+        std::array<FF, 19> zeroes;
+        std::fill(zeroes.begin(), zeroes.end(), 0);
+        auto expected_result = barretenberg::Univariate<FF, 19>(zeroes);
+        EXPECT_EQ(result, expected_result);
+    };
+    run_test();
+};
+
+} // namespace barretenberg::test_protogalaxy_prover
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner_example_gen.py b/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner_example_gen.py
index 2c37304ae97..ec8d90a25dd 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner_example_gen.py
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner_example_gen.py
@@ -87,6 +87,13 @@ def get_extended_univariates(instances, row_idx):
         result = np.array(extend_one_entity(result))
         return result
 
+
+def compute_lagrange(points){
+    lagrange_0 = np.array([])
+    lagrange_1 = np.array([])
+    
+}
+
 def compute_first_example():
     i0 = Instance([Row(0), Row(1)])
     i1 = Instance([Row(128), Row(129)])
@@ -104,39 +111,43 @@ def compute_first_example():
         accumulator += zeta_pow * relation_value
         zeta_pow *= zeta
 
-    accumulator *= extend_one_entity([1, 2])
+
+
+    # accumulator *= extend_one_entity([1, 2])
     return accumulator
 
 
-def compute_second_example():
-        result = 0
-        w_l = np.array([ 1,  3,  5,  7,  9, 11, 13])
-        w_r = np.array([ 2,  4,  6,  8, 10, 12, 14])
-        w_o = np.array([ 3,  7, 11, 15, 19, 23, 27])
-        q_m = np.array([ 0,  0,  0,  0,  0,  0,  0])
-        q_l = np.array([ 1,  1,  1,  1,  1,  1,  1])
-        q_r = np.array([ 1,  1,  1,  1,  1,  1,  1])
-        q_o = np.array([-1, -1, -1, -1, -1, -1, -1])
-        q_c = np.array([ 0,  0,  0,  0,  0,  0,  0])
-        # contribution is zero, but why not?
-        result += rel(w_l, w_r, w_o, q_m, q_l, q_r, q_o, q_c)      
-
-        w_l = np.array([ 0,  1,  2,  3,  4,  5,  6])
-        w_r = np.array([ 4,  4,  4,  4,  4,  4,  4])
-        w_o = np.array([ 4,  4,  4,  4,  4,  4,  4])
-        q_m = np.array([ 0,  1,  2,  3,  4,  5,  6])
-        q_l = np.array([ 1,  0, -1, -2, -3, -4, -5])
-        q_r = np.array([ 1,  0, -1, -2, -3, -4, -5])
-        q_o = np.array([-1, -1, -1, -1, -1, -1, -1])
-        q_c = np.array([ 0,  0,  0,  0,  0,  0,  0])
-
-        result += rel(w_l, w_r, w_o, q_m, q_l, q_r, q_o, q_c)
-        result *= 2
-
-        result *= extend_one_entity([1, 2])
-        return result
+# def compute_second_example():
+#         result = 0
+#         w_l = np.array([ 1,  3,  5,  7,  9, 11, 13])
+#         w_r = np.array([ 2,  4,  6,  8, 10, 12, 14])
+#         w_o = np.array([ 3,  7, 11, 15, 19, 23, 27])
+#         q_m = np.array([ 0,  0,  0,  0,  0,  0,  0])
+#         q_l = np.array([ 1,  1,  1,  1,  1,  1,  1])
+#         q_r = np.array([ 1,  1,  1,  1,  1,  1,  1])
+#         q_o = np.array([-1, -1, -1, -1, -1, -1, -1])
+#         q_c = np.array([ 0,  0,  0,  0,  0,  0,  0])
+#         # contribution is zero, but why not?
+#         result += rel(w_l, w_r, w_o, q_m, q_l, q_r, q_o, q_c)      
+
+#         w_l = np.array([ 0,  1,  2,  3,  4,  5,  6])
+#         w_r = np.array([ 4,  4,  4,  4,  4,  4,  4])
+#         w_o = np.array([ 4,  4,  4,  4,  4,  4,  4])
+#         q_m = np.array([ 0,  1,  2,  3,  4,  5,  6])
+#         q_l = np.array([ 1,  0, -1, -2, -3, -4, -5])
+#         q_r = np.array([ 1,  0, -1, -2, -3, -4, -5])
+#         q_o = np.array([-1, -1, -1, -1, -1, -1, -1])
+#         q_c = np.array([ 0,  0,  0,  0,  0,  0,  0])
+
+#         result += rel(w_l, w_r, w_o, q_m, q_l, q_r, q_o, q_c)
+#         result *= 2
+
+#         # result *= extend_one_entity([1, 2])
+#         return result
 
 if __name__ == "__main__":
-    print(f"First example: \n  {compute_first_example()}")
-    print(f"Second example:\n  {compute_second_example()}")
+    # print(f"First example: \n  {compute_first_example()}")
+    # print(f"Second example:\n  {compute_second_example()}")
+    print(compute_first_example())
+    # print(compute_second_example())
 
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp
index 8c35ce4d40b..eafbd1ff410 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp
@@ -127,7 +127,7 @@ TEST_F(ProtoGalaxyTests, PerturbatorCoefficients)
     }
 }
 
-TEST_F(ProtoGalaxyTests, PowPerturbatorPolynomial)
+TEST_F(ProtoGalaxyTests, PerturbatorPolynomial)
 {
     const size_t log_instance_size(3);
     const size_t instance_size(1 << log_instance_size);
@@ -147,17 +147,8 @@ TEST_F(ProtoGalaxyTests, PowPerturbatorPolynomial)
         betas[idx] = FF::random_element();
     }
 
-    // Construct pow(\vec{betas}) manually as in the paper
-    std::vector<FF> pow_beta(instance_size);
-    for (size_t i = 0; i < instance_size; i++) {
-        auto res = FF(1);
-        for (size_t j = i, beta_idx = 0; j > 0; j >>= 1, beta_idx++) {
-            if ((j & 1) == 1) {
-                res *= betas[beta_idx];
-            }
-        }
-        pow_beta[i] = res;
-    }
+    // Construct pow(\vec{betas}) as in the paper
+    auto pow_beta = ProtoGalaxyProver::compute_pow_polynomial_at_values(betas, instance_size);
 
     // Compute the corresponding target sum and create a dummy accumulator
     auto target_sum = FF(0);
@@ -178,4 +169,6 @@ TEST_F(ProtoGalaxyTests, PowPerturbatorPolynomial)
     // Ensure the constant coefficient of the perturbator is equal to the target sum as indicated by the paper
     EXPECT_EQ(perturbator[0], target_sum);
 }
+
+TEST_F(ProtoGalaxyTests, PowCorrectness) {}
 } // namespace protogalaxy_tests
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
index 638557681f0..495c3b9ba18 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
@@ -47,14 +47,13 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
     const auto log_instance_size = static_cast<size_t>(numeric::get_msb(instance_size));
     auto deltas = compute_round_challenge_pows(log_instance_size, delta);
     auto perturbator = compute_perturbator(accumulator, deltas, alpha);
-    assert(perturbator[0] == accumulator->folding_parameters->target_sum);
+    assert(perturbator[0] == accumulator->folding_parameters.target_sum);
     for (size_t idx = 0; idx <= log_instance_size; idx++) {
         transcript.send_to_verifier("perturbator_" + std::to_string(idx), perturbator[idx]);
     }
 
     auto perturbator_challenge = transcript.get_challenge("perturbator_challenge");
     auto compressed_perturbator = perturbator.evaluate(perturbator_challenge); // horner
-    static_cast<void>(compressed_perturbator);
 
     // sanity check: i think we don't care about elemnt 0th element of beta, it is 1
     std::vector<FF> betas_star(log_instance_size);
@@ -64,16 +63,17 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
         betas_star[idx] = betas[idx] + perturbator_challenge * deltas[idx - 1];
     }
 
-    auto pow_betas_star = compute_pow_polynomials_at_values(betas_star, instance_size);
+    auto pow_betas_star = compute_pow_polynomial_at_values(betas_star, instance_size);
 
     auto combiner = compute_combiner(instances, accumulator->relation_parameters, pow_betas_star, alpha);
     const auto combiner_quotient_size = (log_instance_size - 1) * Flavor::MAX_RELATION_LENGTH;
     std::vector<FF> lagrange_0(combiner_quotient_size);
     std::vector<FF> vanishing_polynomial(combiner_quotient_size);
-    // combiner_quotient evaluations from k to dk + 1
-    // TODO: make this univariates
-    // WARNING: barycentric evaluations might not work here ????
+
+    // degree dk - k  = d(k - 1) so length is d(k-1) + 1
     std::array<FF, (Flavor::MAX_RANDOM_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) + 1> combiner_quotient_coeffs;
+
+    // we need to evaluate at values not part in the vanishing set
     for (size_t point = log_instance_size; point < combiner.size(); point++) {
         auto idx = point - log_instance_size;
         lagrange_0[idx] = FF(1) - FF(point);
@@ -86,7 +86,8 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
         transcript.send_to_verifier("combiner_quotient_" + std::to_string(idx), combiner_quotient_coeffs[idx]);
     }
 
-    RandomExtendedUnivariate combiner_quotient = Univariate(combiner_quotient_coeffs);
+    Univariate<FF, (Flavor::MAX_RANDOM_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) + 1> combiner_quotient(
+        combiner_quotient_coeffs);
     auto combiner_challenge = transcript.get_challenge("combiner_qoutient_challenge");
     auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge);
     auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
@@ -101,7 +102,7 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
         assert(accumulator_poly.size() == instance_poly.size());
         for (size_t j = 0; j < accumulator_poly.size(); j++) {
             accumulator_poly[j] =
-                accumulator_poly[j] * lagrange_0_at_challenge + instance_poly * lagrange_1_at_challenge;
+                accumulator_poly[j] * lagrange_0_at_challenge + instance_poly[j] * lagrange_1_at_challenge;
         }
     }
 
@@ -111,7 +112,7 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
     // we do barycentric evaluation on the combiner
 
     ProverFoldingResult<Flavor> res;
-    res.folded_prover_polynomials = accumalator->prover_polynomials;
+    res.folded_prover_polynomials = accumulator->prover_polynomials;
     res.params.target_sum = new_target_sum;
     res.params.gate_separation_challenges = betas_star;
     res.folding_data = transcript.proof_data;
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
index d83a28e9207..30f0954254d 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
@@ -48,7 +48,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      */
     void prepare_for_folding();
 
-    std::vector<FF> compute_pow_polynomials_at_values(const std::vector<FF> betas, const size_t instance_size)
+    static std::vector<FF> compute_pow_polynomial_at_values(const std::vector<FF> betas, const size_t instance_size)
     {
         std::vector<FF> pow_betas(instance_size);
         for (size_t i = 0; i < instance_size; i++) {
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp
index 129d104f55e..b3e937847e0 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp
@@ -6,7 +6,7 @@ VerifierFoldingResult<typename VerifierInstances::Flavor> ProtoGalaxyVerifier_<
     VerifierInstances>::fold_public_parameters(std::vector<uint8_t> fold_data)
 {
     using Flavor = typename VerifierInstances::Flavor;
-    using VerificationKey = typename Flavor::VerificationKey;
+    // using VerificationKey = typename Flavor::VerificationKey;
 
     transcript = VerifierTranscript<FF>{ fold_data };
     auto index = 0;
@@ -42,25 +42,24 @@ VerifierFoldingResult<typename VerifierInstances::Flavor> ProtoGalaxyVerifier_<
     for (size_t idx = 0; idx <= log_instance_size; idx++) {
         perturbator_coeffs[idx] = transcript.template receive_from_prover<FF>("perturbator_" + std::to_string(idx));
     }
-    assert(perturbator_coeffs[0] == accumulator->folding_parameters->target_sum);
+    assert(perturbator_coeffs[0] == accumulator->folding_parameters.target_sum);
     auto perturbator = Polynomial<FF>(perturbator_coeffs);
     auto perturbator_challenge = transcript.get_challenge("perturbator_challenge");
     auto perturbator_at_challenge = perturbator.evaluate(perturbator_challenge); // horner
-    static_cast<void>(perturbator_at_challenge);
-    // get the coefficient of combiner quotient (K(X) polynomial in the paper)
-    // + 2 because we need one extra eval to rep a poly of certain degree
-    auto combiner_quotient_size = Flavor::MAX_RANDOM_RELATION_LENGTH * (VerifierInstances::NUM - 1) + 2;
-    std::vector<FF> combiner_quotient_coeffs(combiner_quotient_size);
+
+    std::array<FF, (Flavor::MAX_RANDOM_RELATION_LENGTH - 1) * (VerifierInstances::NUM - 1) + 1>
+        combiner_quotient_coeffs;
+    auto combiner_quotient_size = (Flavor::MAX_RANDOM_RELATION_LENGTH - 1) * (VerifierInstances::NUM - 1) + 1;
     for (size_t idx = 0; idx < combiner_quotient_size; idx++) {
         combiner_quotient_coeffs[idx] =
             transcript.template receive_from_prover<FF>("combiner_quotient_" + std::to_string(idx));
     }
-    auto combiner_quotient = Polynomial<FF>(combiner_quotient_coeffs);
+    Univariate<FF, (Flavor::MAX_RANDOM_RELATION_LENGTH - 1) * (VerifierInstances::NUM - 1) + 1> combiner_quotient(
+        combiner_quotient_coeffs);
     auto combiner_challenge = transcript.get_challenge("combiner_quotient_challenge");
-    static_cast<void>(combiner_challenge);
 
-    // fix k = 1 which means k+1 = 2 is a power of two
     auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge); // K(\gamma)
+    // WORKTODO make functions for bigger Z and more lagrange basis
     auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
     auto lagrange_0_at_challenge = FF(1) - combiner_challenge;
     auto lagrange_1_at_challenge = combiner_challenge;

From 265f2485c15de3cb1d59e9307a31f71e232cd3f0 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Mon, 6 Nov 2023 12:02:31 +0000
Subject: [PATCH 06/86] wip that works

---
 .../honk/proof_system/folding_result.hpp      |   2 +-
 .../honk/proof_system/protogalaxy.test.cpp    |  93 +++++++++++-
 .../honk/proof_system/protogalaxy_prover.cpp  |  68 +++------
 .../honk/proof_system/protogalaxy_prover.hpp  |  25 +++
 .../proof_system/protogalaxy_verifier.cpp     |  80 +++++-----
 .../proof_system/protogalaxy_verifier.hpp     |   2 +
 .../honk/proof_system/ultra_verifier.hpp      |   2 +-
 .../honk/transcript/transcript.test.cpp       |  77 +++++-----
 .../barretenberg/polynomials/barycentric.hpp  |  26 ++--
 .../barretenberg/polynomials/univariate.hpp   | 143 +++++++++---------
 .../polynomials/univariate.test.cpp           |   8 +
 .../relations/nested_containers.hpp           |  26 ++--
 .../transcript/transcript.test.cpp            |   2 +-
 13 files changed, 338 insertions(+), 216 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/folding_result.hpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/folding_result.hpp
index 5888d866b04..490cc62c8a3 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/folding_result.hpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/folding_result.hpp
@@ -17,7 +17,7 @@ template <class Flavor> struct VerifierFoldingResult {
     using VerificationKey = typename Flavor::VerificationKey;
     using FoldingParameters = typename Flavor::FoldingParameters;
     std::vector<FF> folded_public_inputs;
-    VerificationKey folded_verification_key;
+    std::shared_ptr<VerificationKey> folded_verification_key;
     FoldingParameters parameters;
 };
 
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp
index eafbd1ff410..ff95f2630fe 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp
@@ -1,5 +1,6 @@
 // Note these tests are a bit hacky
 #include "barretenberg/honk/composer/ultra_composer.hpp"
+#include "barretenberg/honk/utils/testing.hpp"
 #include "protogalaxy_prover.hpp"
 #include <gtest/gtest.h>
 using namespace proof_system::honk;
@@ -10,7 +11,9 @@ using Instances = ProverInstances_<Flavor, 2>;
 using ProtoGalaxyProver = ProtoGalaxyProver_<Instances>;
 using FF = Flavor::FF;
 using Builder = Flavor::CircuitBuilder;
+using Polynomial = typename Flavor::Polynomial;
 using ProverPolynomials = Flavor::ProverPolynomials;
+using RelationParameters = proof_system::RelationParameters<FF>;
 const size_t NUM_POLYNOMIALS = Flavor::NUM_ALL_ENTITIES;
 
 namespace protogalaxy_tests {
@@ -170,5 +173,93 @@ TEST_F(ProtoGalaxyTests, PerturbatorPolynomial)
     EXPECT_EQ(perturbator[0], target_sum);
 }
 
-TEST_F(ProtoGalaxyTests, PowCorrectness) {}
+TEST_F(ProtoGalaxyTests, CombinerQuotient)
+{
+    auto compressed_perturbator = FF(2);
+    auto combiner = barretenberg::Univariate<FF, 7>(std::array<FF, 7>{
+        20,
+        21,
+        22,
+        23,
+        24,
+        25,
+        26,
+    });
+    auto combiner_quotient = ProtoGalaxyProver::compute_combiner_quotient(compressed_perturbator, combiner);
+
+    auto expected_evals = barretenberg::Univariate<FF, 7, 2>(std::array<FF, 5>{
+        (FF(22) - (FF(1) - FF(2)) * compressed_perturbator) / (FF(2) * FF(2 - 1)),
+        (FF(23) - (FF(1) - FF(3)) * compressed_perturbator) / (FF(3) * FF(3 - 1)),
+        (FF(24) - (FF(1) - FF(4)) * compressed_perturbator) / (FF(4) * FF(4 - 1)),
+        (FF(25) - (FF(1) - FF(5)) * compressed_perturbator) / (FF(5) * FF(5 - 1)),
+        (FF(26) - (FF(1) - FF(6)) * compressed_perturbator) / (FF(6) * FF(6 - 1)),
+    });
+
+    for (size_t idx = 2; idx < 7; idx++) {
+        EXPECT_EQ(combiner_quotient.value_at(idx), expected_evals.value_at(idx));
+    }
+}
+
+// TEST_F(ProtoGalaxyTests, Galaxify)
+// {
+//     auto composer = UltraComposer();
+
+//     // Create instance from actual circuit
+//     std::vector<std::shared_ptr<ProverInstance_<Flavor>>> insts(2);
+//     auto builder = proof_system::UltraCircuitBuilder();
+//     auto a = FF::random_element();
+//     auto b = FF::random_element();
+//     builder.add_variable(a);
+//     builder.add_public_variable(a);
+//     builder.add_public_variable(b);
+//     insts[1] = composer.create_instance(builder);
+
+//     auto instance_size = insts[1]->proving_key->circuit_size;
+//     auto log_instance_size = static_cast<size_t>(numeric::get_msb(instance_size));
+
+//     // Create fake accumulator
+//     std::array<barretenberg::Polynomial<FF>, NUM_POLYNOMIALS> random_polynomials;
+//     for (auto& poly : random_polynomials) {
+//         poly = get_random_polynomial(instance_size);
+//     }
+//     auto full_polynomials = construct_ultra_full_polynomials(random_polynomials);
+//     auto relation_parameters = proof_system::RelationParameters<FF>::get_random();
+//     auto alpha = FF::random_element();
+
+//     auto full_honk_evals =
+//         ProtoGalaxyProver::compute_full_honk_evaluations(full_polynomials, alpha, relation_parameters);
+//     std::vector<FF> betas(log_instance_size);
+//     for (size_t idx = 0; idx < log_instance_size; idx++) {
+//         betas[idx] = FF::random_element();
+//     }
+
+//     // Construct pow(\vec{betas}) as in the paper
+//     auto pow_beta = ProtoGalaxyProver::compute_pow_polynomial_at_values(betas, instance_size);
+
+//     // Compute the corresponding target sum and create a dummy accumulator
+//     auto target_sum = FF(0);
+//     for (size_t i = 0; i < instance_size; i++) {
+//         target_sum += full_honk_evals[i] * pow_beta[i];
+//     }
+
+//     insts[0] = std::make_shared<Instance>(
+//         FoldingResult<Flavor>{ .folded_prover_polynomials = full_polynomials,
+//                                .folded_public_inputs = std::vector<FF>{},
+//                                .verification_key = std::make_shared<Flavor::VerificationKey>(),
+//                                .folding_parameters = { betas, target_sum } });
+//     insts[0]->relation_parameters = relation_parameters;
+
+//     // artificially make first instance relaxed
+
+//     auto prover = composer.create_folding_prover(insts);
+//     auto verifier = composer.create_folding_verifier(insts);
+//     prover.prepare_for_folding();
+//     auto new_target_sum_prover = prover.galaxify(alpha)[0];
+
+//     verifier.prepare_for_folding(prover.transcript.proof_data);
+//     auto new_target_sum_verifier = verifier.galaxify(alpha)[0];
+//     info(new_target_sum_prover);
+//     info(new_target_sum_verifier);
+// }
+
 } // namespace protogalaxy_tests
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
index 495c3b9ba18..8c4f3d65b5a 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
@@ -4,7 +4,6 @@ namespace proof_system::honk {
 
 template <class ProverInstances> void ProtoGalaxyProver_<ProverInstances>::prepare_for_folding()
 {
-    // this doesnt work in the current format
     auto idx = 0;
     for (auto it = instances.begin(); it != instances.end(); it++, idx++) {
         auto instance = *it;
@@ -24,14 +23,13 @@ template <class ProverInstances> void ProtoGalaxyProver_<ProverInstances>::prepa
             transcript.send_to_verifier(domain_separator + "_public_input_" + std::to_string(i), public_input_i);
         }
 
-        // TODO(https://github.com/AztecProtocol/barretenberg/issues/752): establish whether we can use the same grand
-        // product parameters for all instances securely
+        // TODO(https://github.com/AztecProtocol/barretenberg/issues/752): establish whether we can use the same
+        // grand product parameters for all instances securely
         auto [eta, beta, gamma] = transcript.get_challenges(
             domain_separator + "_eta", domain_separator + "_beta", domain_separator + "_gamma");
         instance->compute_sorted_accumulator_polynomials(eta);
         instance->compute_grand_product_polynomials(beta, gamma);
     }
-    // need to check all instances have the same circuit size!!
 }
 
 // TODO(#https://github.com/AztecProtocol/barretenberg/issues/689): finalise implementation this function
@@ -46,65 +44,45 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
     auto instance_size = accumulator->prover_polynomials[0].size();
     const auto log_instance_size = static_cast<size_t>(numeric::get_msb(instance_size));
     auto deltas = compute_round_challenge_pows(log_instance_size, delta);
+    // This is not OK here
     auto perturbator = compute_perturbator(accumulator, deltas, alpha);
-    assert(perturbator[0] == accumulator->folding_parameters.target_sum);
     for (size_t idx = 0; idx <= log_instance_size; idx++) {
         transcript.send_to_verifier("perturbator_" + std::to_string(idx), perturbator[idx]);
     }
 
     auto perturbator_challenge = transcript.get_challenge("perturbator_challenge");
-    auto compressed_perturbator = perturbator.evaluate(perturbator_challenge); // horner
-
-    // sanity check: i think we don't care about elemnt 0th element of beta, it is 1
+    auto compressed_perturbator = perturbator.evaluate(perturbator_challenge);
     std::vector<FF> betas_star(log_instance_size);
     betas_star[0] = 1;
     auto betas = accumulator->folding_parameters.gate_separation_challenges;
-    for (size_t idx = 1; idx <= log_instance_size; idx++) {
+    for (size_t idx = 1; idx < log_instance_size; idx++) {
         betas_star[idx] = betas[idx] + perturbator_challenge * deltas[idx - 1];
     }
 
     auto pow_betas_star = compute_pow_polynomial_at_values(betas_star, instance_size);
 
     auto combiner = compute_combiner(instances, accumulator->relation_parameters, pow_betas_star, alpha);
-    const auto combiner_quotient_size = (log_instance_size - 1) * Flavor::MAX_RELATION_LENGTH;
-    std::vector<FF> lagrange_0(combiner_quotient_size);
-    std::vector<FF> vanishing_polynomial(combiner_quotient_size);
-
-    // degree dk - k  = d(k - 1) so length is d(k-1) + 1
-    std::array<FF, (Flavor::MAX_RANDOM_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) + 1> combiner_quotient_coeffs;
-
-    // we need to evaluate at values not part in the vanishing set
-    for (size_t point = log_instance_size; point < combiner.size(); point++) {
-        auto idx = point - log_instance_size;
-        lagrange_0[idx] = FF(1) - FF(point);
-        vanishing_polynomial[idx] = FF(point) * (FF(point) - 1);
-        combiner_quotient_coeffs[idx] =
-            (combiner.value_at(point) - compressed_perturbator * lagrange_0[idx]) / vanishing_polynomial[idx];
-    }
-
-    for (size_t idx = 0; idx < combiner_quotient_size; idx++) {
-        transcript.send_to_verifier("combiner_quotient_" + std::to_string(idx), combiner_quotient_coeffs[idx]);
+    auto combiner_quotient = compute_combiner_quotient(compressed_perturbator, combiner);
+    for (size_t idx = ProverInstances::NUM; idx < combiner.size(); idx++) {
+        transcript.send_to_verifier("combiner_quotient_" + std::to_string(idx), combiner_quotient.value_at(idx));
     }
-
-    Univariate<FF, (Flavor::MAX_RANDOM_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) + 1> combiner_quotient(
-        combiner_quotient_coeffs);
     auto combiner_challenge = transcript.get_challenge("combiner_qoutient_challenge");
     auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge);
     auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
     auto lagrange_0_at_challenge = FF(1) - combiner_challenge;
-    auto lagrange_1_at_challenge = combiner_challenge;
-    auto new_target_sum = compressed_perturbator * lagrange_0_at_challenge + vanishing_polynomial_at_challenge +
-                          combiner_quotient_at_challenge;
-
-    for (size_t idx = 0; idx < accumulator->prover_polynomials.size(); idx++) {
-        auto accumulator_poly = accumulator->prover_polynomials[idx];
-        auto instance_poly = accumulator->prover_polynomials[idx];
-        assert(accumulator_poly.size() == instance_poly.size());
-        for (size_t j = 0; j < accumulator_poly.size(); j++) {
-            accumulator_poly[j] =
-                accumulator_poly[j] * lagrange_0_at_challenge + instance_poly[j] * lagrange_1_at_challenge;
-        }
-    }
+    // auto lagrange_1_at_challenge = combiner_challenge;
+    auto new_target_sum = compressed_perturbator * lagrange_0_at_challenge +
+                          vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
+
+    // for (size_t idx = 0; idx < accumulator->prover_polynomials.size(); idx++) {
+    //     auto accumulator_poly = accumulator->prover_polynomials[idx];
+    //     auto instance_poly = accumulator->prover_polynomials[idx];
+    //     assert(accumulator_poly.size() == instance_poly.size());
+    //     for (size_t j = 0; j < accumulator_poly.size(); j++) {
+    //         accumulator_poly[j] =
+    //             accumulator_poly[j] * lagrange_0_at_challenge + instance_poly[j] * lagrange_1_at_challenge;
+    //     }
+    // }
 
     // i have dk evaluations out of which the first k are not good so we evaluate the rest of the polynomials from
     // k+1
@@ -112,9 +90,9 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
     // we do barycentric evaluation on the combiner
 
     ProverFoldingResult<Flavor> res;
-    res.folded_prover_polynomials = accumulator->prover_polynomials;
+    // res.folded_prover_polynomials = accumulator->prover_polynomials;
     res.params.target_sum = new_target_sum;
-    res.params.gate_separation_challenges = betas_star;
+    // res.params.gate_separation_challenges = betas_star;
     res.folding_data = transcript.proof_data;
     return res;
 }
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
index 30f0954254d..5037161968a 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
@@ -48,6 +48,8 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      */
     void prepare_for_folding();
 
+    std::array<FF, 3> galaxify(FF);
+
     static std::vector<FF> compute_pow_polynomial_at_values(const std::vector<FF> betas, const size_t instance_size)
     {
         std::vector<FF> pow_betas(instance_size);
@@ -291,6 +293,29 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         // Batch the univariate contributions from each sub-relation to obtain the round univariate
         return Utils::template batch_over_relations<RandomExtendedUnivariate>(univariate_accumulators, alpha);
     }
+
+    static Univariate<FF,
+                      (Flavor::MAX_RANDOM_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) + 1,
+                      ProverInstances::NUM>
+    compute_combiner_quotient(FF compressed_perturbator, RandomExtendedUnivariate combiner)
+    {
+        // degree dk - k - 1  = d(k - 1) - 1
+        std::array<FF, (Flavor::MAX_RANDOM_RELATION_LENGTH - 2) * (ProverInstances::NUM - 1)>
+            combiner_quotient_evals = {};
+
+        // we need to evaluate at values not part in the vanishing set
+        for (size_t point = ProverInstances::NUM; point < combiner.size(); point++) {
+            auto idx = point - ProverInstances::NUM;
+            auto lagrange_0 = FF(1) - FF(point);
+            auto vanishing_polynomial = FF(point) * (FF(point) - 1);
+            combiner_quotient_evals[idx] =
+                (combiner.value_at(point) - compressed_perturbator * lagrange_0) / vanishing_polynomial;
+        }
+
+        Univariate<FF, (Flavor::MAX_RANDOM_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) + 1, ProverInstances::NUM>
+            combiner_quotient(combiner_quotient_evals);
+        return combiner_quotient;
+    }
 };
 
 extern template class ProtoGalaxyProver_<ProverInstances_<honk::flavor::Ultra, 2>>;
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp
index b3e937847e0..b3db9945e50 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp
@@ -1,13 +1,10 @@
 #include "protogalaxy_verifier.hpp"
 #include "barretenberg/honk/utils/grand_product_delta.hpp"
 namespace proof_system::honk {
+
 template <class VerifierInstances>
-VerifierFoldingResult<typename VerifierInstances::Flavor> ProtoGalaxyVerifier_<
-    VerifierInstances>::fold_public_parameters(std::vector<uint8_t> fold_data)
+void ProtoGalaxyVerifier_<VerifierInstances>::prepare_for_folding(std::vector<uint8_t> fold_data)
 {
-    using Flavor = typename VerifierInstances::Flavor;
-    // using VerificationKey = typename Flavor::VerificationKey;
-
     transcript = VerifierTranscript<FF>{ fold_data };
     auto index = 0;
     for (auto it = verifier_instances.begin(); it != verifier_instances.end(); it++, index++) {
@@ -32,9 +29,16 @@ VerifierFoldingResult<typename VerifierInstances::Flavor> ProtoGalaxyVerifier_<
         inst->relation_parameters =
             RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
     }
+}
+
+template <class VerifierInstances>
+VerifierFoldingResult<typename VerifierInstances::Flavor> ProtoGalaxyVerifier_<
+    VerifierInstances>::fold_public_parameters(std::vector<uint8_t> fold_data)
+{
+    using Flavor = typename VerifierInstances::Flavor;
 
-    auto [alpha, delta] =
-        transcript.get_challenges("alpha", "delta"); // what does verifier do with this alpha which is from plonk paper?
+    prepare_for_folding(fold_data);
+    auto [alpha, delta] = transcript.get_challenges("alpha", "delta");
     auto accumulator = get_accumulator();
     auto log_instance_size = static_cast<size_t>(numeric::get_msb(accumulator->instance_size));
     auto deltas = compute_round_challenge_pows(log_instance_size, delta);
@@ -42,50 +46,46 @@ VerifierFoldingResult<typename VerifierInstances::Flavor> ProtoGalaxyVerifier_<
     for (size_t idx = 0; idx <= log_instance_size; idx++) {
         perturbator_coeffs[idx] = transcript.template receive_from_prover<FF>("perturbator_" + std::to_string(idx));
     }
-    assert(perturbator_coeffs[0] == accumulator->folding_parameters.target_sum);
     auto perturbator = Polynomial<FF>(perturbator_coeffs);
     auto perturbator_challenge = transcript.get_challenge("perturbator_challenge");
-    auto perturbator_at_challenge = perturbator.evaluate(perturbator_challenge); // horner
-
-    std::array<FF, (Flavor::MAX_RANDOM_RELATION_LENGTH - 1) * (VerifierInstances::NUM - 1) + 1>
-        combiner_quotient_coeffs;
-    auto combiner_quotient_size = (Flavor::MAX_RANDOM_RELATION_LENGTH - 1) * (VerifierInstances::NUM - 1) + 1;
-    for (size_t idx = 0; idx < combiner_quotient_size; idx++) {
-        combiner_quotient_coeffs[idx] =
-            transcript.template receive_from_prover<FF>("combiner_quotient_" + std::to_string(idx));
+    auto perturbator_at_challenge = perturbator.evaluate(perturbator_challenge);
+    std::array<FF, (Flavor::MAX_RANDOM_RELATION_LENGTH - 2) * (VerifierInstances::NUM - 1)>
+        combiner_quotient_evals = {};
+    for (size_t idx = 0; idx < (Flavor::MAX_RANDOM_RELATION_LENGTH - 2) * (VerifierInstances::NUM - 1); idx++) {
+        combiner_quotient_evals[idx] = transcript.template receive_from_prover<FF>(
+            "combiner_quotient_" + std::to_string(idx + VerifierInstances::NUM));
     }
-    Univariate<FF, (Flavor::MAX_RANDOM_RELATION_LENGTH - 1) * (VerifierInstances::NUM - 1) + 1> combiner_quotient(
-        combiner_quotient_coeffs);
+    Univariate<FF, (Flavor::MAX_RANDOM_RELATION_LENGTH - 1) * (VerifierInstances::NUM - 1) + 1, VerifierInstances::NUM>
+        combiner_quotient(combiner_quotient_evals);
     auto combiner_challenge = transcript.get_challenge("combiner_quotient_challenge");
-
     auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge); // K(\gamma)
     // WORKTODO make functions for bigger Z and more lagrange basis
     auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
     auto lagrange_0_at_challenge = FF(1) - combiner_challenge;
-    auto lagrange_1_at_challenge = combiner_challenge;
-    auto updated_target_sum = perturbator_at_challenge * lagrange_0_at_challenge +
-                              vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
+    // auto lagrange_1_at_challenge = combiner_challenge;
+    auto new_target_sum = perturbator_at_challenge * lagrange_0_at_challenge +
+                          vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
 
-    // do public input stuff with goblin (well I guess first without)
-    // fold public inputs (field elements) and the commitment in verification key to obtain a new verification key!
-    // TODO: extend for more instances;
-    auto accumulator_public_inputs = accumulator->public_inputs;
-    auto other_public_inputs = verifier_instances[1]->public_inputs;
-    std::vector<FF> folded_public_inputs(accumulator->public_input_size);
-    // extend either with 0s if the sizes are different
-    for (size_t idx = 0; idx < folded_public_inputs.size(); idx++) {
-        folded_public_inputs[idx] = lagrange_0_at_challenge * accumulator->public_inputs[idx] +
-                                    lagrange_1_at_challenge * other_public_inputs[idx];
-    }
-    // all verification keys have the same size
-    auto folded_verification_key = VerificationKey(accumulator->instance_size, accumulator->public_input_size);
+    // auto accumulator_public_inputs = accumulator->public_inputs;
+    // auto other_public_inputs = verifier_instances[1]->public_inputs;
+    // std::vector<FF> folded_public_inputs(accumulator->public_input_size);
+    // // extend either with 0s if the sizes are different
+    // for (size_t idx = 0; idx < folded_public_inputs.size(); idx++) {
+    //     folded_public_inputs[idx] = lagrange_0_at_challenge * accumulator->public_inputs[idx] +
+    //                                 lagrange_1_at_challenge * other_public_inputs[idx];
+    // }
+    // // all verification keys have the same size
+    // auto folded_verification_key =
+    //     std::make_shared<VerificationKey>(accumulator->instance_size, accumulator->public_input_size);
 
-    for (size_t idx = 0; idx < folded_verification_key.size(); idx++) {
-        folded_verification_key[idx] = (*accumulator->verification_key)[idx] * lagrange_0_at_challenge +
-                                       (*verifier_instances[1]->verification_key)[idx] * lagrange_1_at_challenge;
-    }
+    // for (size_t idx = 0; idx < (*folded_verification_key).size(); idx++) {
+    //     (*folded_verification_key)[idx] = (*accumulator->verification_key)[idx] * lagrange_0_at_challenge +
+    //                                       (*verifier_instances[1]->verification_key)[idx] * lagrange_1_at_challenge;
+    // }
     VerifierFoldingResult<Flavor> res;
-    res.parameters.target_sum = updated_target_sum;
+    res.parameters.target_sum = new_target_sum;
+    // res.folded_verification_key = folded_verification_key;
+    // res.folded_public_inputs = folded_public_inputs;
     return res;
 }
 
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.hpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.hpp
index 309fcadcdd5..590d336869c 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.hpp
@@ -35,6 +35,8 @@ template <class VerifierInstances> class ProtoGalaxyVerifier_ {
     }
     std::shared_ptr<Instance> get_accumulator() { return verifier_instances[0]; }
 
+    void prepare_for_folding(std::vector<uint8_t> fold_data);
+
     VerifierFoldingResult<Flavor> fold_public_parameters(std::vector<uint8_t> fold_data);
 };
 
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/ultra_verifier.hpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/ultra_verifier.hpp
index 34c524672d5..32e4c0f1fb7 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/ultra_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/ultra_verifier.hpp
@@ -14,7 +14,7 @@ template <typename Flavor> class UltraVerifier_ {
   public:
     explicit UltraVerifier_(std::shared_ptr<VerificationKey> verifier_key = nullptr);
     UltraVerifier_(UltraVerifier_&& other);
-    UltraVerifier_(const UltraVerifier_& other) = delete;
+
     UltraVerifier_& operator=(const UltraVerifier_& other) = delete;
     UltraVerifier_& operator=(UltraVerifier_&& other);
 
diff --git a/barretenberg/cpp/src/barretenberg/honk/transcript/transcript.test.cpp b/barretenberg/cpp/src/barretenberg/honk/transcript/transcript.test.cpp
index 3a0271bdd8d..2060a93df14 100644
--- a/barretenberg/cpp/src/barretenberg/honk/transcript/transcript.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/transcript/transcript.test.cpp
@@ -197,41 +197,42 @@ TEST_F(UltraTranscriptTests, ChallengeGenerationTest)
     ASSERT_NE(b, 0) << "Challenge a is 0";
 }
 
-TEST_F(UltraTranscriptTests, FoldingManifestTest)
-{
-    using Flavor = flavor::Ultra;
-    auto composer = UltraComposer();
-
-    std::vector<std::shared_ptr<ProverInstance_<Flavor>>> insts(2);
-    std::generate(insts.begin(), insts.end(), [&]() {
-        auto builder = proof_system::UltraCircuitBuilder();
-        auto a = FF::random_element();
-        auto b = FF::random_element();
-        builder.add_variable(a);
-        builder.add_public_variable(a);
-        builder.add_public_variable(b);
-        return composer.create_instance(builder);
-    });
-
-    // artificially make first instance relaxed
-    auto log_instance_size = static_cast<size_t>(numeric::get_msb(insts[0]->proving_key->circuit_size));
-    std::vector<FF> betas(log_instance_size);
-    for (size_t idx = 0; idx < log_instance_size; idx++) {
-        betas[idx] = FF::random_element();
-    }
-    insts[0]->folding_parameters = { betas, FF(1) };
-
-    auto prover = composer.create_folding_prover(insts);
-    auto verifier = composer.create_folding_verifier(insts);
-
-    auto prover_res = prover.fold_instances();
-    verifier.fold_public_parameters(prover_res.folding_data);
-
-    // Check consistency between the manifests generated by the prover and verifier
-    auto prover_manifest = prover.transcript.get_manifest();
-    auto verifier_manifest = verifier.transcript.get_manifest();
-    for (size_t round = 0; round < prover_manifest.size(); ++round) {
-        ASSERT_EQ(prover_manifest[round], verifier_manifest[round])
-            << "Prover/Verifier manifest discrepency in round " << round;
-    }
-}
+// TEST_F(UltraTranscriptTests, FoldingManifestTest)
+// {
+//     using Flavor = flavor::Ultra;
+//     auto composer = UltraComposer();
+
+//     std::vector<std::shared_ptr<ProverInstance_<Flavor>>> insts(2);
+//     std::generate(insts.begin(), insts.end(), [&]() {
+//         auto builder = proof_system::UltraCircuitBuilder();
+//         auto a = FF::random_element();
+//         auto b = FF::random_element();
+//         builder.add_variable(a);
+//         builder.add_public_variable(a);
+//         builder.add_public_variable(b);
+//         return composer.create_instance(builder);
+//     });
+
+//     // artificially make first instance relaxed
+//     auto log_instance_size = static_cast<size_t>(numeric::get_msb(insts[0]->proving_key->circuit_size));
+//     std::vector<FF> betas(log_instance_size);
+//     for (size_t idx = 0; idx < log_instance_size; idx++) {
+//         betas[idx] = FF::random_element();
+//     }
+//     insts[0]->folding_parameters = { betas, FF(1) };
+
+//     auto prover = composer.create_folding_prover(insts);
+//     auto verifier = composer.create_folding_verifier(insts);
+
+//     auto prover_res = prover.fold_instances();
+//     verifier.fold_public_parameters(prover_res.folding_data);
+
+//     // Check consistency between the manifests generated by the prover and verifier
+//     auto prover_manifest = prover.transcript.get_manifest();
+//     auto verifier_manifest = verifier.transcript.get_manifest();
+
+//     for (size_t round = 0; round < prover_manifest.size(); ++round) {
+//         ASSERT_EQ(prover_manifest[round], verifier_manifest[round])
+//             << "Prover/Verifier manifest discrepency in round " << round;
+//     }
+// }
diff --git a/barretenberg/cpp/src/barretenberg/polynomials/barycentric.hpp b/barretenberg/cpp/src/barretenberg/polynomials/barycentric.hpp
index a8ddb6d30d3..27cd1c8631b 100644
--- a/barretenberg/cpp/src/barretenberg/polynomials/barycentric.hpp
+++ b/barretenberg/cpp/src/barretenberg/polynomials/barycentric.hpp
@@ -20,22 +20,25 @@ namespace barretenberg {
 
 /**
  * @todo: TODO(https://github.com/AztecProtocol/barretenberg/issues/713) Optimize with lookup tables?
+ * @tparam domain_end, domain_start specify the given evaluation domain {domain_start,..., domain_end - 1}
+ * @tparam num_evals the number of evaluations that are computable with specific barycentric extension formula
  */
 
-template <class Fr, size_t domain_size, size_t num_evals> class BarycentricDataCompileTime {
+template <class Fr, size_t domain_end, size_t num_evals, size_t domain_start = 0> class BarycentricDataCompileTime {
   public:
+    static constexpr size_t domain_size = domain_end - domain_start;
     static constexpr size_t big_domain_size = std::max(domain_size, num_evals);
 
     /**
      * Static constexpr methods for computing arrays of precomputable data used for barycentric extension and evaluation
      */
 
-    // build big_domain, currently the set of x_i in {0, 1, ..., t-1}
+    // build big_domain, currently the set of x_i in {domain_start, ..., big_domain_end - 1 }
     static constexpr std::array<Fr, big_domain_size> construct_big_domain()
     {
         std::array<Fr, big_domain_size> result;
         for (size_t i = 0; i < big_domain_size; ++i) {
-            result[i] = static_cast<Fr>(i);
+            result[i] = static_cast<Fr>(i + domain_start);
         }
         return result;
     }
@@ -109,7 +112,7 @@ template <class Fr, size_t domain_size, size_t num_evals> class BarycentricDataC
         std::array<Fr, num_evals> result;
         for (size_t i = 0; i != num_evals; ++i) {
             result[i] = 1;
-            Fr v_i = i;
+            Fr v_i = i + domain_start;
             for (size_t j = 0; j != domain_size; ++j) {
                 result[i] *= v_i - big_domain[j];
             }
@@ -124,20 +127,21 @@ template <class Fr, size_t domain_size, size_t num_evals> class BarycentricDataC
     static constexpr auto full_numerator_values = construct_full_numerator_values(big_domain);
 };
 
-template <class Fr, size_t domain_size, size_t num_evals> class BarycentricDataRunTime {
+template <class Fr, size_t domain_end, size_t num_evals, size_t domain_start = 0> class BarycentricDataRunTime {
   public:
+    static constexpr size_t domain_size = domain_end - domain_start;
     static constexpr size_t big_domain_size = std::max(domain_size, num_evals);
 
     /**
      * Static constexpr methods for computing arrays of precomputable data used for barycentric extension and evaluation
      */
 
-    // build big_domain, currently the set of x_i in {0, 1, ..., t-1}
+    // build big_domain, currently the set of x_i in {domain_start, ..., big_domain_end - 1 }
     static std::array<Fr, big_domain_size> construct_big_domain()
     {
         std::array<Fr, big_domain_size> result;
         for (size_t i = 0; i < big_domain_size; ++i) {
-            result[i] = static_cast<Fr>(i);
+            result[i] = static_cast<Fr>(i + domain_start);
         }
         return result;
     }
@@ -210,7 +214,7 @@ template <class Fr, size_t domain_size, size_t num_evals> class BarycentricDataR
         std::array<Fr, num_evals> result;
         for (size_t i = 0; i != num_evals; ++i) {
             result[i] = 1;
-            Fr v_i = i;
+            Fr v_i = i + domain_start;
             for (size_t j = 0; j != domain_size; ++j) {
                 result[i] *= v_i - big_domain[j];
             }
@@ -247,9 +251,9 @@ template <typename T> inline constexpr bool is_field_type_v = is_field_type<T>::
  * @tparam domain_size
  * @tparam num_evals
  */
-template <class Fr, size_t domain_size, size_t num_evals>
+template <class Fr, size_t domain_end, size_t num_evals, size_t domain_start = 0>
 using BarycentricData = std::conditional_t<is_field_type_v<Fr>,
-                                           BarycentricDataCompileTime<Fr, domain_size, num_evals>,
-                                           BarycentricDataRunTime<Fr, domain_size, num_evals>>;
+                                           BarycentricDataCompileTime<Fr, domain_end, num_evals, domain_start>,
+                                           BarycentricDataRunTime<Fr, domain_end, num_evals, domain_start>>;
 
 } // namespace barretenberg
diff --git a/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp b/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp
index 0ca83be0654..93773dd5685 100644
--- a/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp
+++ b/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp
@@ -13,22 +13,24 @@ namespace barretenberg {
  * of the data in those univariates. We do that by taking a view of those elements and then, as needed, using this to
  * populate new containers.
  */
-template <class Fr, size_t view_length> class UnivariateView;
+template <class Fr, size_t view_domain_end, size_t view_domain_start> class UnivariateView;
 
 /**
- * @brief A univariate polynomial represented by its values on {0,1,..., _length-1}
+ * @brief A univariate polynomial represented by its values on {domain_start, domain_start + 1,..., domain_end - 1}. For
+ * memory efficiency purposes, we store the evaluations in an array starting from 0 and make the mapping to the right
+ * domain under the hood.
  */
-template <class Fr, size_t _length> class Univariate {
+template <class Fr, size_t domain_end, size_t domain_start = 0> class Univariate {
   public:
-    static constexpr size_t LENGTH = _length;
-    using View = UnivariateView<Fr, LENGTH>;
+    static constexpr size_t LENGTH = domain_end - domain_start;
+    using View = UnivariateView<Fr, domain_end, domain_start>;
 
     // TODO(https://github.com/AztecProtocol/barretenberg/issues/714) Try out std::valarray?
-    std::array<Fr, _length> evaluations;
+    std::array<Fr, LENGTH> evaluations;
 
     Univariate() = default;
 
-    explicit Univariate(std::array<Fr, _length> evaluations)
+    explicit Univariate(std::array<Fr, LENGTH> evaluations)
         : evaluations(evaluations)
     {}
     ~Univariate() = default;
@@ -40,12 +42,12 @@ template <class Fr, size_t _length> class Univariate {
     explicit Univariate(Fr value)
         : evaluations{}
     {
-        for (size_t i = 0; i < _length; ++i) {
+        for (size_t i = 0; i < LENGTH; ++i) {
             evaluations[i] = value;
         }
     }
     // Construct Univariate from UnivariateView
-    explicit Univariate(UnivariateView<Fr, _length> in)
+    explicit Univariate(UnivariateView<Fr, domain_end, domain_start> in)
         : evaluations{}
     {
         for (size_t i = 0; i < in.evaluations.size(); ++i) {
@@ -53,8 +55,8 @@ template <class Fr, size_t _length> class Univariate {
         }
     }
 
-    Fr& value_at(size_t i) { return evaluations[i]; };
-    const Fr& value_at(size_t i) const { return evaluations[i]; };
+    Fr& value_at(size_t i) { return evaluations[i - domain_start]; };
+    const Fr& value_at(size_t i) const { return evaluations[i - domain_start]; };
     size_t size() { return evaluations.size(); };
 
     // Write the Univariate evaluations to a buffer
@@ -72,8 +74,8 @@ template <class Fr, size_t _length> class Univariate {
 
     static Univariate get_random()
     {
-        auto output = Univariate<Fr, _length>();
-        for (size_t i = 0; i != _length; ++i) {
+        auto output = Univariate<Fr, domain_end, domain_start>();
+        for (size_t i = 0; i != LENGTH; ++i) {
             output.value_at(i) = Fr::random_element();
         }
         return output;
@@ -84,21 +86,21 @@ template <class Fr, size_t _length> class Univariate {
 
     Univariate& operator+=(const Univariate& other)
     {
-        for (size_t i = 0; i < _length; ++i) {
+        for (size_t i = 0; i < LENGTH; ++i) {
             evaluations[i] += other.evaluations[i];
         }
         return *this;
     }
     Univariate& operator-=(const Univariate& other)
     {
-        for (size_t i = 0; i < _length; ++i) {
+        for (size_t i = 0; i < LENGTH; ++i) {
             evaluations[i] -= other.evaluations[i];
         }
         return *this;
     }
     Univariate& operator*=(const Univariate& other)
     {
-        for (size_t i = 0; i < _length; ++i) {
+        for (size_t i = 0; i < LENGTH; ++i) {
             evaluations[i] *= other.evaluations[i];
         }
         return *this;
@@ -178,45 +180,45 @@ template <class Fr, size_t _length> class Univariate {
     }
 
     // Operations between Univariate and UnivariateView
-    Univariate& operator+=(const UnivariateView<Fr, _length>& view)
+    Univariate& operator+=(const UnivariateView<Fr, domain_end, domain_start>& view)
     {
-        for (size_t i = 0; i < _length; ++i) {
+        for (size_t i = 0; i < LENGTH; ++i) {
             evaluations[i] += view.evaluations[i];
         }
         return *this;
     }
 
-    Univariate& operator-=(const UnivariateView<Fr, _length>& view)
+    Univariate& operator-=(const UnivariateView<Fr, domain_end, domain_start>& view)
     {
-        for (size_t i = 0; i < _length; ++i) {
+        for (size_t i = 0; i < LENGTH; ++i) {
             evaluations[i] -= view.evaluations[i];
         }
         return *this;
     }
 
-    Univariate& operator*=(const UnivariateView<Fr, _length>& view)
+    Univariate& operator*=(const UnivariateView<Fr, domain_end, domain_start>& view)
     {
-        for (size_t i = 0; i < _length; ++i) {
+        for (size_t i = 0; i < LENGTH; ++i) {
             evaluations[i] *= view.evaluations[i];
         }
         return *this;
     }
 
-    Univariate operator+(const UnivariateView<Fr, _length>& view) const
+    Univariate operator+(const UnivariateView<Fr, domain_end, domain_start>& view) const
     {
         Univariate res(*this);
         res += view;
         return res;
     }
 
-    Univariate operator-(const UnivariateView<Fr, _length>& view) const
+    Univariate operator-(const UnivariateView<Fr, domain_end, domain_start>& view) const
     {
         Univariate res(*this);
         res -= view;
         return res;
     }
 
-    Univariate operator*(const UnivariateView<Fr, _length>& view) const
+    Univariate operator*(const UnivariateView<Fr, domain_end, domain_start>& view) const
     {
         Univariate res(*this);
         res *= view;
@@ -240,23 +242,25 @@ template <class Fr, size_t _length> class Univariate {
     }
 
     /**
-     * @brief Given a univariate f represented by {f(0), ..., f(t-1)}, compute {f(t), ..., f(u-1)}
-     * and return the Univariate represented by {f(0), ..., f(u-1)}.
+     * @brief Given a univariate f represented by {f(domain_start), ..., f(domain_end - 1)}, compute the evaluations
+     * {f(domain_end),..., f(extended_domain_end -1)} and return the Univariate represented by  {f(domain_start),...,
+     * f(extended_domain_end -1)}
      *
-     * @details Write v_i = f(x_i) on a the domain {x_0, ..., x_{t-1}}. To efficiently compute the needed values of f,
-     * we use the barycentric formula
-     *      - f(x) = B(x) Σ_{i=0}^{t-1} v_i / (d_i*(x-x_i))
+     * @details Write v_i = f(x_i) on a the domain {x_{domain_start}, ..., x_{domain_end-1}}. To efficiently compute the
+     * needed values of f, we use the barycentric formula
+     *      - f(x) = B(x) Σ_{i=domain_start}^{domain_end-1} v_i / (d_i*(x-x_i))
      * where
-     *      - B(x) = Π_{i=0}^{t-1} (x-x_i)
-     *      - d_i  = Π_{j ∈ {0, ..., t-1}, j≠i} (x_i-x_j) for i ∈ {0, ..., t-1}
+     *      - B(x) = Π_{i=domain_start}^{domain_end-1} (x-x_i)
+     *      - d_i  = Π_{j ∈ {domain_start, ..., domain_end-1}, j≠i} (x_i-x_j) for i ∈ {domain_start, ..., domain_end-1}
      *
      * When the domain size is two, extending f = v0(1-X) + v1X to a new value involves just one addition and a
      * subtraction: setting Δ = v1-v0, the values of f(X) are f(0)=v0, f(1)= v0 + Δ, v2 = f(1) + Δ, v3 = f(2) + Δ...
      *
      */
-    template <size_t EXTENDED_LENGTH> Univariate<Fr, EXTENDED_LENGTH> extend_to() const
+    template <size_t EXTENDED_DOMAIN_END> Univariate<Fr, EXTENDED_DOMAIN_END> extend_to() const
     {
-        using Data = BarycentricData<Fr, LENGTH, EXTENDED_LENGTH>;
+        const size_t EXTENDED_LENGTH = EXTENDED_DOMAIN_END - domain_start;
+        using Data = BarycentricData<Fr, LENGTH, EXTENDED_LENGTH>; // ?
         static_assert(EXTENDED_LENGTH >= LENGTH);
 
         Univariate<Fr, EXTENDED_LENGTH> result;
@@ -266,15 +270,15 @@ template <class Fr, size_t _length> class Univariate {
         if constexpr (LENGTH == 2) {
             Fr delta = value_at(1) - value_at(0);
             static_assert(EXTENDED_LENGTH != 0);
-            for (size_t idx = 1; idx < EXTENDED_LENGTH - 1; idx++) {
+            for (size_t idx = domain_start; idx < EXTENDED_DOMAIN_END - 1; idx++) {
                 result.value_at(idx + 1) = result.value_at(idx) + delta;
             }
             return result;
         } else {
-            for (size_t k = LENGTH; k != EXTENDED_LENGTH; ++k) {
+            for (size_t k = domain_end; k != EXTENDED_DOMAIN_END; ++k) {
                 result.value_at(k) = 0;
                 // compute each term v_j / (d_j*(x-x_j)) of the sum
-                for (size_t j = 0; j != LENGTH; ++j) {
+                for (size_t j = domain_start; j != domain_end; ++j) {
                     Fr term = value_at(j);
                     term *= Data::precomputed_denominator_inverses[LENGTH * k + j];
                     result.value_at(k) += term;
@@ -294,9 +298,9 @@ template <class Fr, size_t _length> class Univariate {
      */
     Fr evaluate(const Fr& u)
     {
-        using Data = BarycentricData<Fr, LENGTH, LENGTH>;
+        using Data = BarycentricData<Fr, domain_end, LENGTH, domain_start>;
         Fr full_numerator_value = 1;
-        for (size_t i = 0; i != LENGTH; ++i) {
+        for (size_t i = domain_start; i != domain_end; ++i) {
             full_numerator_value *= u - i;
         }
 
@@ -312,9 +316,9 @@ template <class Fr, size_t _length> class Univariate {
 
         Fr result = 0;
         // compute each term v_j / (d_j*(x-x_j)) of the sum
-        for (size_t i = 0; i != LENGTH; ++i) {
+        for (size_t i = domain_start; i != domain_end; ++i) {
             Fr term = value_at(i);
-            term *= denominator_inverses[i];
+            term *= denominator_inverses[i - domain_start];
             result += term;
         }
         // scale the sum by the the value of of B(x)
@@ -323,98 +327,101 @@ template <class Fr, size_t _length> class Univariate {
     };
 };
 
-template <typename B, class Fr, size_t _length> inline void read(B& it, Univariate<Fr, _length>& univariate)
+template <typename B, class Fr, size_t domain_end, size_t domain_start = 0>
+inline void read(B& it, Univariate<Fr, domain_end, domain_start>& univariate)
 {
     using serialize::read;
     read(it, univariate.evaluations);
 }
 
-template <typename B, class Fr, size_t _length> inline void write(B& it, Univariate<Fr, _length> const& univariate)
+template <typename B, class Fr, size_t domain_end, size_t domain_start = 0>
+inline void write(B& it, Univariate<Fr, domain_end, domain_start> const& univariate)
 {
     using serialize::write;
     write(it, univariate.evaluations);
 }
 
-template <class Fr, size_t view_length> class UnivariateView {
+template <class Fr, size_t domain_end, size_t domain_start = 0> class UnivariateView {
   public:
-    std::span<const Fr, view_length> evaluations;
+    static constexpr size_t LENGTH = domain_end - domain_start;
+    std::span<const Fr, LENGTH> evaluations;
 
     UnivariateView() = default;
 
     const Fr& value_at(size_t i) const { return evaluations[i]; };
 
-    template <size_t full_length>
-    explicit UnivariateView(const Univariate<Fr, full_length>& univariate_in)
-        : evaluations(std::span<const Fr>(univariate_in.evaluations.data(), view_length)){};
+    template <size_t full_domain_end, size_t full_domain_start = 0>
+    explicit UnivariateView(const Univariate<Fr, full_domain_end, full_domain_start>& univariate_in)
+        : evaluations(std::span<const Fr>(univariate_in.evaluations.data(), LENGTH)){};
 
-    Univariate<Fr, view_length> operator+(const UnivariateView& other) const
+    Univariate<Fr, domain_end, domain_start> operator+(const UnivariateView& other) const
     {
-        Univariate<Fr, view_length> res(*this);
+        Univariate<Fr, domain_end, domain_start> res(*this);
         res += other;
         return res;
     }
 
-    Univariate<Fr, view_length> operator-(const UnivariateView& other) const
+    Univariate<Fr, domain_end, domain_start> operator-(const UnivariateView& other) const
     {
-        Univariate<Fr, view_length> res(*this);
+        Univariate<Fr, domain_end, domain_start> res(*this);
         res -= other;
         return res;
     }
 
-    Univariate<Fr, view_length> operator-() const
+    Univariate<Fr, domain_end, domain_start> operator-() const
     {
-        Univariate res(*this);
+        Univariate<Fr, domain_end, domain_start> res(*this);
         for (auto& eval : res.evaluations) {
             eval = -eval;
         }
         return res;
     }
 
-    Univariate<Fr, view_length> operator*(const UnivariateView& other) const
+    Univariate<Fr, domain_end, domain_start> operator*(const UnivariateView& other) const
     {
-        Univariate<Fr, view_length> res(*this);
+        Univariate<Fr, domain_end, domain_start> res(*this);
         res *= other;
         return res;
     }
 
-    Univariate<Fr, view_length> operator*(const Univariate<Fr, view_length>& other) const
+    Univariate<Fr, domain_end, domain_start> operator*(const Univariate<Fr, domain_end, domain_start>& other) const
     {
-        Univariate<Fr, view_length> res(*this);
+        Univariate<Fr, domain_end, domain_start> res(*this);
         res *= other;
         return res;
     }
 
-    Univariate<Fr, view_length> operator+(const Univariate<Fr, view_length>& other) const
+    Univariate<Fr, domain_end, domain_start> operator+(const Univariate<Fr, domain_end, domain_start>& other) const
     {
-        Univariate<Fr, view_length> res(*this);
+        Univariate<Fr, domain_end, domain_start> res(*this);
         res += other;
         return res;
     }
 
-    Univariate<Fr, view_length> operator+(const Fr& other) const
+    Univariate<Fr, domain_end, domain_start> operator+(const Fr& other) const
     {
-        Univariate<Fr, view_length> res(*this);
+        Univariate<Fr, domain_end, domain_start> res(*this);
         res += other;
         return res;
     }
 
-    Univariate<Fr, view_length> operator-(const Fr& other) const
+    Univariate<Fr, domain_end, domain_start> operator-(const Fr& other) const
     {
-        Univariate<Fr, view_length> res(*this);
+        Univariate<Fr, domain_end, domain_start> res(*this);
         res -= other;
         return res;
     }
 
-    Univariate<Fr, view_length> operator*(const Fr& other) const
+    Univariate<Fr, domain_end, domain_start> operator*(const Fr& other) const
     {
-        Univariate<Fr, view_length> res(*this);
+        Univariate<Fr, domain_end, domain_start> res(*this);
         res *= other;
         return res;
     }
 
-    Univariate<Fr, view_length> operator-(const Univariate<Fr, view_length>& other) const
+    Univariate<Fr, domain_end, domain_start> operator-(const Univariate<Fr, domain_end, domain_start>& other) const
     {
-        Univariate<Fr, view_length> res(*this);
+        Univariate<Fr, domain_end, domain_start> res(*this);
         res -= other;
         return res;
     }
diff --git a/barretenberg/cpp/src/barretenberg/polynomials/univariate.test.cpp b/barretenberg/cpp/src/barretenberg/polynomials/univariate.test.cpp
index 5138a51167a..01382baf50d 100644
--- a/barretenberg/cpp/src/barretenberg/polynomials/univariate.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/polynomials/univariate.test.cpp
@@ -158,4 +158,12 @@ TYPED_TEST(UnivariateTest, Serialization)
     }
 }
 
+TYPED_TEST(UnivariateTest, EvaluationCustomDomain)
+{
+    UNIVARIATE_TESTS_ALIASES
+
+    auto poly = Univariate<FF, 3, 1>(std::array<FF, 2>{ 1, 2 });
+    EXPECT_EQ(poly.evaluate(FF(5)), FF(5));
+}
+
 } // namespace barretenberg::test_univariate
diff --git a/barretenberg/cpp/src/barretenberg/proof_system/relations/nested_containers.hpp b/barretenberg/cpp/src/barretenberg/proof_system/relations/nested_containers.hpp
index 601d4f35a83..c8cf2c7f315 100644
--- a/barretenberg/cpp/src/barretenberg/proof_system/relations/nested_containers.hpp
+++ b/barretenberg/cpp/src/barretenberg/proof_system/relations/nested_containers.hpp
@@ -10,29 +10,35 @@ namespace proof_system {
  *
  * @details Credit: https://stackoverflow.com/a/60440611
  */
-template <template <typename, size_t> typename InnerContainer,
+template <template <typename, size_t, size_t> typename InnerContainer,
           typename ValueType,
           auto LENGTHS,
+          size_t sth = 0,
           typename IS = decltype(std::make_index_sequence<LENGTHS.size()>())>
 struct TupleOfContainersOverArray;
-template <template <typename, size_t> typename InnerContainer, typename ValueType, auto LENGTHS, std::size_t... I>
-struct TupleOfContainersOverArray<InnerContainer, ValueType, LENGTHS, std::index_sequence<I...>> {
-    using type = std::tuple<InnerContainer<ValueType, LENGTHS[I]>...>;
+template <template <typename, size_t, size_t> typename InnerContainer,
+          typename ValueType,
+          auto LENGTHS,
+          size_t sth,
+          std::size_t... I>
+struct TupleOfContainersOverArray<InnerContainer, ValueType, LENGTHS, sth, std::index_sequence<I...>> {
+    using type = std::tuple<InnerContainer<ValueType, LENGTHS[I], sth>...>;
 };
 
 // Helpers
-template <typename ValueType, size_t> using ExtractValueType = ValueType;
+template <typename ValueType, size_t, size_t> using ExtractValueType = ValueType;
 
 template <typename Tuple>
 using HomogeneousTupleToArray = std::array<std::tuple_element_t<0, Tuple>, std::tuple_size_v<Tuple>>;
 
 // Types needed for sumcheck and folding.
-template <typename FF, auto LENGTHS>
-using TupleOfUnivariates = typename TupleOfContainersOverArray<barretenberg::Univariate, FF, LENGTHS>::type;
+template <typename FF, auto LENGTHS, size_t sth = 0>
+using TupleOfUnivariates = typename TupleOfContainersOverArray<barretenberg::Univariate, FF, LENGTHS, sth>::type;
 
-template <typename FF, auto LENGTHS>
-using TupleOfValues = typename TupleOfContainersOverArray<ExtractValueType, FF, LENGTHS>::type;
+template <typename FF, auto LENGTHS, size_t sth = 0>
+using TupleOfValues = typename TupleOfContainersOverArray<ExtractValueType, FF, LENGTHS, sth>::type;
 
-template <typename FF, auto LENGTHS> using ArrayOfValues = HomogeneousTupleToArray<TupleOfValues<FF, LENGTHS>>;
+template <typename FF, auto LENGTHS, size_t sth = 0>
+using ArrayOfValues = HomogeneousTupleToArray<TupleOfValues<FF, LENGTHS, sth>>;
 
 } // namespace proof_system
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/transcript/transcript.test.cpp b/barretenberg/cpp/src/barretenberg/transcript/transcript.test.cpp
index 6489f90d014..847b0669903 100644
--- a/barretenberg/cpp/src/barretenberg/transcript/transcript.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/transcript/transcript.test.cpp
@@ -136,7 +136,7 @@ TEST(transcript, univariate_serialization)
     constexpr size_t LENGTH = 8;
 
     using Fr = barretenberg::fr;
-    using Univariate = barretenberg::Univariate<Fr, LENGTH>;
+    using Univariate = barretenberg::Univariate<Fr, 8>;
     using Transcript = transcript::StandardTranscript;
 
     std::vector<uint8_t> g1_vector(64);

From 7ffc3642f6b119dfe8ecccfd26189edb5fad31f8 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Mon, 6 Nov 2023 15:46:18 +0000
Subject: [PATCH 07/86] cleanup

---
 .../honk/composer/ultra_transcript.test.cpp   | 35 --------
 .../honk/proof_system/combiner.test.cpp       |  2 -
 .../honk/proof_system/combiner_example_gen.py | 10 +--
 .../honk/proof_system/protogalaxy.test.cpp    | 84 -------------------
 .../honk/proof_system/protogalaxy_prover.cpp  | 27 ++----
 .../honk/proof_system/protogalaxy_prover.hpp  |  9 --
 .../proof_system/protogalaxy_verifier.cpp     | 24 +-----
 7 files changed, 11 insertions(+), 180 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/honk/composer/ultra_transcript.test.cpp b/barretenberg/cpp/src/barretenberg/honk/composer/ultra_transcript.test.cpp
index d41f0fd61d9..5d2c143aa6c 100644
--- a/barretenberg/cpp/src/barretenberg/honk/composer/ultra_transcript.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/composer/ultra_transcript.test.cpp
@@ -220,38 +220,3 @@ TEST_F(UltraTranscriptTests, StructureTest)
     prover.transcript.deserialize_full_transcript();
     EXPECT_EQ(static_cast<Flavor::Commitment>(prover.transcript.sorted_accum_comm), one_group_val * rand_val);
 }
-
-// TEST_F(UltraTranscriptTests, FoldingManifestTest)
-// {
-//     using Flavor = flavor::Ultra;
-//     auto composer = UltraComposer();
-
-//     std::vector<std::shared_ptr<ProverInstance_<Flavor>>> insts(2);
-//     std::generate(insts.begin(), insts.end(), [&]() {
-//         auto builder = proof_system::UltraCircuitBuilder();
-//         generate_random_test_circuit(builder);
-//         return composer.create_instance(builder);
-//     });
-
-//     // artificially make first instance relaxed
-//     auto log_instance_size = static_cast<size_t>(numeric::get_msb(insts[0]->proving_key->circuit_size));
-//     std::vector<FF> betas(log_instance_size);
-//     for (size_t idx = 0; idx < log_instance_size; idx++) {
-//         betas[idx] = FF::random_element();
-//     }
-//     insts[0]->folding_parameters = { betas, FF(1) };
-
-//     auto prover = composer.create_folding_prover(insts);
-//     auto verifier = composer.create_folding_verifier(insts);
-
-//     auto prover_res = prover.fold_instances();
-//     verifier.fold_public_parameters(prover_res.folding_data);
-
-//     // Check consistency between the manifests generated by the prover and verifier
-//     auto prover_manifest = prover.transcript.get_manifest();
-//     auto verifier_manifest = verifier.transcript.get_manifest();
-//     for (size_t round = 0; round < prover_manifest.size(); ++round) {
-//         ASSERT_EQ(prover_manifest[round], verifier_manifest[round])
-//             << "Prover/Verifier manifest discrepency in round " << round;
-//     }
-// }
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner.test.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner.test.cpp
index 81f01873264..801f856296f 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner.test.cpp
@@ -35,7 +35,6 @@ TEST(Protogalaxy, CombinerOn2Instances)
             std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
             std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
             ProtoGalaxyProver prover;
-            // auto pow_univariate = PowUnivariate<FF>(/*zeta_pow=*/2);
             std::vector<FF> pow_betas = { FF(1), FF(2) };
             auto alpha = FF(0); // focus on the arithmetic relation only
 
@@ -71,7 +70,6 @@ TEST(Protogalaxy, CombinerOn2Instances)
             std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
             std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
             ProtoGalaxyProver prover;
-            // auto pow_univariate = PowUnivariate<FF>(/*zeta_pow=*/2);
             std::vector<FF> pow_betas = { FF(1), FF(2) };
             auto alpha = FF(0); // focus on the arithmetic relation only
 
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner_example_gen.py b/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner_example_gen.py
index 1828e2fac58..4ee35072af2 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner_example_gen.py
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner_example_gen.py
@@ -110,9 +110,6 @@ def compute_first_example():
         accumulator += zeta_pow * relation_value
         zeta_pow *= zeta
 
-
-
-    # accumulator *= extend_one_entity([1, 2])
     return accumulator
 
 
@@ -142,12 +139,9 @@ def compute_second_example():
         result += rel(w_l, w_r, w_o, q_m, q_l, q_r, q_o, q_c)
         result *= 2
 
-        # result *= extend_one_entity([1, 2])
         return result
 
 if __name__ == "__main__":
-    # print(f"First example: \n  {compute_first_example()}")
-    # print(f"Second example:\n  {compute_second_example()}")
-    print(compute_first_example())
-    print(compute_second_example())
+    print(f"First example: \n  {compute_first_example()}")
+    print(f"Second example:\n  {compute_second_example()}")
 
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp
index ca81d65d92e..148c6344dde 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp
@@ -200,68 +200,6 @@ TEST_F(ProtoGalaxyTests, CombinerQuotient)
     }
 }
 
-// TEST_F(ProtoGalaxyTests, Galaxify)
-// {
-//     auto composer = UltraComposer();
-
-//     // Create instance from actual circuit
-//     std::vector<std::shared_ptr<ProverInstance_<Flavor>>> insts(2);
-//     auto builder = proof_system::UltraCircuitBuilder();
-//     auto a = FF::random_element();
-//     auto b = FF::random_element();
-//     builder.add_variable(a);
-//     builder.add_public_variable(a);
-//     builder.add_public_variable(b);
-//     insts[1] = composer.create_instance(builder);
-
-//     auto instance_size = insts[1]->proving_key->circuit_size;
-//     auto log_instance_size = static_cast<size_t>(numeric::get_msb(instance_size));
-
-//     // Create fake accumulator
-//     std::array<barretenberg::Polynomial<FF>, NUM_POLYNOMIALS> random_polynomials;
-//     for (auto& poly : random_polynomials) {
-//         poly = get_random_polynomial(instance_size);
-//     }
-//     auto full_polynomials = construct_ultra_full_polynomials(random_polynomials);
-//     auto relation_parameters = proof_system::RelationParameters<FF>::get_random();
-//     auto alpha = FF::random_element();
-
-//     auto full_honk_evals =
-//         ProtoGalaxyProver::compute_full_honk_evaluations(full_polynomials, alpha, relation_parameters);
-//     std::vector<FF> betas(log_instance_size);
-//     for (size_t idx = 0; idx < log_instance_size; idx++) {
-//         betas[idx] = FF::random_element();
-//     }
-
-//     // Construct pow(\vec{betas}) as in the paper
-//     auto pow_beta = ProtoGalaxyProver::compute_pow_polynomial_at_values(betas, instance_size);
-
-//     // Compute the corresponding target sum and create a dummy accumulator
-//     auto target_sum = FF(0);
-//     for (size_t i = 0; i < instance_size; i++) {
-//         target_sum += full_honk_evals[i] * pow_beta[i];
-//     }
-
-//     insts[0] = std::make_shared<Instance>(
-//         FoldingResult<Flavor>{ .folded_prover_polynomials = full_polynomials,
-//                                .folded_public_inputs = std::vector<FF>{},
-//                                .verification_key = std::make_shared<Flavor::VerificationKey>(),
-//                                .folding_parameters = { betas, target_sum } });
-//     insts[0]->relation_parameters = relation_parameters;
-
-//     // artificially make first instance relaxed
-
-//     auto prover = composer.create_folding_prover(insts);
-//     auto verifier = composer.create_folding_verifier(insts);
-//     prover.prepare_for_folding();
-//     auto new_target_sum_prover = prover.galaxify(alpha)[0];
-
-//     verifier.prepare_for_folding(prover.transcript.proof_data);
-//     auto new_target_sum_verifier = verifier.galaxify(alpha)[0];
-//     info(new_target_sum_prover);
-//     info(new_target_sum_verifier);
-// }
-
 TEST_F(ProtoGalaxyTests, FoldChallenges)
 {
     using Instances = ProverInstances_<Flavor, 2>;
@@ -283,26 +221,4 @@ TEST_F(ProtoGalaxyTests, FoldChallenges)
     EXPECT_EQ(instances.relation_parameters.eta, expected_eta);
 }
 
-// namespace proof_system::honk::instance_tests {
-
-// template <class Flavor> class InstancesTests : public testing::Test {
-//     using FF = typename Flavor::FF;
-//     using Builder = typename Flavor::CircuitBuilder;
-
-//   public:
-//     static void test_parameters_to_univariates()
-//     {
-
-//     };
-// };
-
-// using FlavorTypes = testing::Types<flavor::Ultra>;
-// TYPED_TEST_SUITE(InstancesTests, FlavorTypes);
-
-// TYPED_TEST(InstancesTests, ParametersToUnivariates)
-// {
-//     TestFixture::test_parameters_to_univariates();
-// }
-
-// } // namespace proof_system::honk::instance_tests
 } // namespace protogalaxy_tests
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
index 0e94d29277d..f89252590a8 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
@@ -23,8 +23,6 @@ template <class ProverInstances> void ProtoGalaxyProver_<ProverInstances>::prepa
             transcript.send_to_verifier(domain_separator + "_public_input_" + std::to_string(i), public_input_i);
         }
 
-        // TODO(https://github.com/AztecProtocol/barretenberg/issues/752): establish whether we can use the same
-        // grand product parameters for all instances securely
         auto [eta, beta, gamma] = transcript.get_challenges(
             domain_separator + "_eta", domain_separator + "_beta", domain_separator + "_gamma");
         instance->compute_sorted_accumulator_polynomials(eta);
@@ -41,12 +39,13 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
     prepare_for_folding();
     // TODO(#https://github.com/AztecProtocol/barretenberg/issues/740): Handle the case where we are folding for the
     // first time and accumulator is 0
+    // TODO(#https://github.com/AztecProtocol/barretenberg/issues/763): Fold alpha
     auto [alpha, delta] = transcript.get_challenges("alpha", "delta");
     auto accumulator = get_accumulator();
     auto instance_size = accumulator->prover_polynomials[0].size();
     const auto log_instance_size = static_cast<size_t>(numeric::get_msb(instance_size));
     auto deltas = compute_round_challenge_pows(log_instance_size, delta);
-    // This is not OK here
+
     auto perturbator = compute_perturbator(accumulator, deltas, alpha);
     for (size_t idx = 0; idx <= log_instance_size; idx++) {
         transcript.send_to_verifier("perturbator_" + std::to_string(idx), perturbator[idx]);
@@ -70,31 +69,17 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
     }
     auto combiner_challenge = transcript.get_challenge("combiner_qoutient_challenge");
     auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge);
+
+    // TODO(https://github.com/AztecProtocol/barretenberg/issues/764): Generalise these formulas as well as computation
+    // of Lagrange basis
     auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
     auto lagrange_0_at_challenge = FF(1) - combiner_challenge;
-    // auto lagrange_1_at_challenge = combiner_challenge;
+
     auto new_target_sum = compressed_perturbator * lagrange_0_at_challenge +
                           vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
 
-    // for (size_t idx = 0; idx < accumulator->prover_polynomials.size(); idx++) {
-    //     auto accumulator_poly = accumulator->prover_polynomials[idx];
-    //     auto instance_poly = accumulator->prover_polynomials[idx];
-    //     assert(accumulator_poly.size() == instance_poly.size());
-    //     for (size_t j = 0; j < accumulator_poly.size(); j++) {
-    //         accumulator_poly[j] =
-    //             accumulator_poly[j] * lagrange_0_at_challenge + instance_poly[j] * lagrange_1_at_challenge;
-    //     }
-    // }
-
-    // i have dk evaluations out of which the first k are not good so we evaluate the rest of the polynomials from
-    // k+1
-
-    // we do barycentric evaluation on the combiner
-
     ProverFoldingResult<Flavor> res;
-    // res.folded_prover_polynomials = accumulator->prover_polynomials;
     res.params.target_sum = new_target_sum;
-    // res.params.gate_separation_challenges = betas_star;
     res.folding_data = transcript.proof_data;
     return res;
 }
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
index e7c069ce61a..3645b92966e 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
@@ -240,14 +240,6 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
                                                          const FF alpha)
     {
         size_t common_circuit_size = instances[0]->prover_polynomials._data[0].size();
-        // Precompute the vector of required powers of zeta
-        // TODO(https://github.com/AztecProtocol/barretenberg/issues/751): Parallelize this.
-        // NB: there is a similar TODO in the sumcheck function `compute_univariate`.
-        // std::vector<FF> pow_challenges(common_circuit_size);
-        // pow_challenges[0] = pow_univariate.partial_evaluation_constant;
-        // for (size_t i = 1; i < common_circuit_size; ++i) {
-        //     pow_challenges[i] = pow_challenges[i - 1] * pow_univariate.zeta_pow;
-        // }
 
         // Determine number of threads for multithreading.
         // Note: Multithreading is "on" for every round but we reduce the number of threads from the max available based
@@ -304,7 +296,6 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
                       ProverInstances::NUM>
     compute_combiner_quotient(FF compressed_perturbator, ExtendedUnivariateWithRandomization combiner)
     {
-        // degree dk - k - 1  = d(k - 1) - 1
         std::array<FF, (Flavor::BATCHED_RELATION_TOTAL_LENGTH - 2) * (ProverInstances::NUM - 1)>
             combiner_quotient_evals = {};
 
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp
index 85816023387..24de4b9cd90 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp
@@ -60,34 +60,16 @@ VerifierFoldingResult<typename VerifierInstances::Flavor> ProtoGalaxyVerifier_<
                VerifierInstances::NUM>
         combiner_quotient(combiner_quotient_evals);
     auto combiner_challenge = transcript.get_challenge("combiner_quotient_challenge");
-    auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge); // K(\gamma)
-    // WORKTODO make functions for bigger Z and more lagrange basis
+    auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge);
+
     auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
     auto lagrange_0_at_challenge = FF(1) - combiner_challenge;
-    // auto lagrange_1_at_challenge = combiner_challenge;
+
     auto new_target_sum = perturbator_at_challenge * lagrange_0_at_challenge +
                           vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
 
-    // auto accumulator_public_inputs = accumulator->public_inputs;
-    // auto other_public_inputs = verifier_instances[1]->public_inputs;
-    // std::vector<FF> folded_public_inputs(accumulator->public_input_size);
-    // // extend either with 0s if the sizes are different
-    // for (size_t idx = 0; idx < folded_public_inputs.size(); idx++) {
-    //     folded_public_inputs[idx] = lagrange_0_at_challenge * accumulator->public_inputs[idx] +
-    //                                 lagrange_1_at_challenge * other_public_inputs[idx];
-    // }
-    // // all verification keys have the same size
-    // auto folded_verification_key =
-    //     std::make_shared<VerificationKey>(accumulator->instance_size, accumulator->public_input_size);
-
-    // for (size_t idx = 0; idx < (*folded_verification_key).size(); idx++) {
-    //     (*folded_verification_key)[idx] = (*accumulator->verification_key)[idx] * lagrange_0_at_challenge +
-    //                                       (*verifier_instances[1]->verification_key)[idx] * lagrange_1_at_challenge;
-    // }
     VerifierFoldingResult<Flavor> res;
     res.parameters.target_sum = new_target_sum;
-    // res.folded_verification_key = folded_verification_key;
-    // res.folded_public_inputs = folded_public_inputs;
     return res;
 }
 

From 9d1b5c6ac4f4daef180e8fbc62d8c20c53dfa4cd Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Mon, 6 Nov 2023 16:31:09 +0000
Subject: [PATCH 08/86] comment test

---
 .../src/barretenberg/honk/proof_system/protogalaxy.test.cpp   | 4 +++-
 1 file changed, 3 insertions(+), 1 deletion(-)

diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp
index 148c6344dde..ce6b9cbd620 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp
@@ -176,11 +176,13 @@ TEST_F(ProtoGalaxyTests, PerturbatorPolynomial)
 
 TEST_F(ProtoGalaxyTests, CombinerQuotient)
 {
-    auto compressed_perturbator = FF(2);
+    auto compressed_perturbator = FF(2); // F(\alpha) in the paper
     auto combiner =
         barretenberg::Univariate<FF, 13>(std::array<FF, 13>{ 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32 });
     auto combiner_quotient = ProtoGalaxyProver::compute_combiner_quotient(compressed_perturbator, combiner);
 
+    // K(i) = (G(i) - ( L_0(i) * F(\alpha)) / Z(i), i = {2,.., 13} for ProverInstances::NUM = 2
+    // K(i) = (G(i) - (1 - i) * F(\alpha)) / i * (i - 1)
     auto expected_evals = barretenberg::Univariate<FF, 13, 2>(std::array<FF, 11>{
         (FF(22) - (FF(1) - FF(2)) * compressed_perturbator) / (FF(2) * FF(2 - 1)),
         (FF(23) - (FF(1) - FF(3)) * compressed_perturbator) / (FF(3) * FF(3 - 1)),

From 42ac7e2f654a5b416d1645d60bc5dae8bc370105 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Mon, 6 Nov 2023 16:31:37 +0000
Subject: [PATCH 09/86] comment test

---
 .../src/barretenberg/honk/proof_system/protogalaxy_prover.hpp   | 2 --
 1 file changed, 2 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
index 3645b92966e..dde0b76a386 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
@@ -51,8 +51,6 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      */
     void prepare_for_folding();
 
-    std::array<FF, 3> galaxify(FF);
-
     static std::vector<FF> compute_pow_polynomial_at_values(const std::vector<FF> betas, const size_t instance_size)
     {
         std::vector<FF> pow_betas(instance_size);

From bc63909d980c19861250ba57aed06388d5e87141 Mon Sep 17 00:00:00 2001
From: Cody Gunton <codygunton@gmail.com>
Date: Wed, 8 Nov 2023 02:52:58 -0500
Subject: [PATCH 10/86] Add weirder test

---
 .../barretenberg/honk/proof_system/barycentric.py    | 12 ++++++++++++
 .../src/barretenberg/polynomials/univariate.test.cpp | 11 +++++++++--
 2 files changed, 21 insertions(+), 2 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/barycentric.py b/barretenberg/cpp/src/barretenberg/honk/proof_system/barycentric.py
index af6555e3760..b4f7d3e6c5b 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/barycentric.py
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/barycentric.py
@@ -29,4 +29,16 @@ def get_A_deriv(i, xs):
 result *= s
 print(result)
 
+points = [32, 33, 34, 35, 36]
+evals = [1,11,111,1111,11111]
+        
+z = 2
+ 
+result = get_A_at_z(z, points)
+s = 0
+for i in range(len(evals)):
+    s += evals[i] / ((z - points[i])* get_A_deriv(i, points))
+result *= s
+print(result)
+
     
diff --git a/barretenberg/cpp/src/barretenberg/polynomials/univariate.test.cpp b/barretenberg/cpp/src/barretenberg/polynomials/univariate.test.cpp
index 01382baf50d..238470370a3 100644
--- a/barretenberg/cpp/src/barretenberg/polynomials/univariate.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/polynomials/univariate.test.cpp
@@ -162,8 +162,15 @@ TYPED_TEST(UnivariateTest, EvaluationCustomDomain)
 {
     UNIVARIATE_TESTS_ALIASES
 
-    auto poly = Univariate<FF, 3, 1>(std::array<FF, 2>{ 1, 2 });
-    EXPECT_EQ(poly.evaluate(FF(5)), FF(5));
+    []() {
+        auto poly = Univariate<FF, 3, 1>(std::array<FF, 2>{ 1, 2 });
+        EXPECT_EQ(poly.evaluate(FF(5)), FF(5));
+    }();
+
+    []() {
+        auto poly = Univariate<FF, 37, 32>(std::array<FF, 5>{ 1, 11, 111, 1111, 11111 });
+        EXPECT_EQ(poly.evaluate(FF(2)), FF(294330751));
+    }();
 }
 
 } // namespace barretenberg::test_univariate

From 7d3e01bd1525c6ccf607266e1c79a1b61e131bfa Mon Sep 17 00:00:00 2001
From: Cody Gunton <codygunton@gmail.com>
Date: Wed, 8 Nov 2023 04:49:03 -0500
Subject: [PATCH 11/86] Use clearer type names

---
 .../relations/nested_containers.hpp           | 25 +++++++++----------
 1 file changed, 12 insertions(+), 13 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/proof_system/relations/nested_containers.hpp b/barretenberg/cpp/src/barretenberg/proof_system/relations/nested_containers.hpp
index c8cf2c7f315..90ca8238ce1 100644
--- a/barretenberg/cpp/src/barretenberg/proof_system/relations/nested_containers.hpp
+++ b/barretenberg/cpp/src/barretenberg/proof_system/relations/nested_containers.hpp
@@ -12,17 +12,17 @@ namespace proof_system {
  */
 template <template <typename, size_t, size_t> typename InnerContainer,
           typename ValueType,
-          auto LENGTHS,
-          size_t sth = 0,
-          typename IS = decltype(std::make_index_sequence<LENGTHS.size()>())>
+          auto domain_end,
+          size_t domain_start = 0,
+          typename IS = decltype(std::make_index_sequence<domain_end.size()>())>
 struct TupleOfContainersOverArray;
 template <template <typename, size_t, size_t> typename InnerContainer,
           typename ValueType,
-          auto LENGTHS,
-          size_t sth,
+          auto domain_end,
+          size_t domain_start,
           std::size_t... I>
-struct TupleOfContainersOverArray<InnerContainer, ValueType, LENGTHS, sth, std::index_sequence<I...>> {
-    using type = std::tuple<InnerContainer<ValueType, LENGTHS[I], sth>...>;
+struct TupleOfContainersOverArray<InnerContainer, ValueType, domain_end, domain_start, std::index_sequence<I...>> {
+    using type = std::tuple<InnerContainer<ValueType, domain_end[I], domain_start>...>;
 };
 
 // Helpers
@@ -32,13 +32,12 @@ template <typename Tuple>
 using HomogeneousTupleToArray = std::array<std::tuple_element_t<0, Tuple>, std::tuple_size_v<Tuple>>;
 
 // Types needed for sumcheck and folding.
-template <typename FF, auto LENGTHS, size_t sth = 0>
-using TupleOfUnivariates = typename TupleOfContainersOverArray<barretenberg::Univariate, FF, LENGTHS, sth>::type;
+template <typename FF, auto LENGTHS>
+using TupleOfUnivariates = typename TupleOfContainersOverArray<barretenberg::Univariate, FF, LENGTHS, 0>::type;
 
-template <typename FF, auto LENGTHS, size_t sth = 0>
-using TupleOfValues = typename TupleOfContainersOverArray<ExtractValueType, FF, LENGTHS, sth>::type;
+template <typename FF, auto LENGTHS>
+using TupleOfValues = typename TupleOfContainersOverArray<ExtractValueType, FF, LENGTHS>::type;
 
-template <typename FF, auto LENGTHS, size_t sth = 0>
-using ArrayOfValues = HomogeneousTupleToArray<TupleOfValues<FF, LENGTHS, sth>>;
+template <typename FF, auto LENGTHS> using ArrayOfValues = HomogeneousTupleToArray<TupleOfValues<FF, LENGTHS>>;
 
 } // namespace proof_system
\ No newline at end of file

From 2287a7696b24243cc8e1179b36cddd86c4d4d224 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Wed, 8 Nov 2023 15:41:11 +0000
Subject: [PATCH 12/86] wip

---
 .../honk/composer/eccvm_transcript.test.cpp   |   5 +-
 .../composer/goblin_ultra_transcript.test.cpp |   5 +-
 .../honk/composer/sumcheck.test.cpp           |   8 +-
 .../honk/composer/ultra_transcript.test.cpp   |   5 +-
 .../honk/proof_system/combiner.test.cpp       | 386 +++++++++---------
 .../honk/proof_system/eccvm_prover.cpp        |   4 +-
 .../honk/proof_system/eccvm_verifier.cpp      |   4 +-
 .../honk/proof_system/protogalaxy.test.cpp    |   5 +-
 .../honk/proof_system/protogalaxy_prover.cpp  |  10 +-
 .../honk/proof_system/protogalaxy_prover.hpp  |  40 +-
 .../proof_system/protogalaxy_verifier.cpp     |   3 +-
 .../honk/proof_system/ultra_prover.cpp        |   2 +-
 .../honk/proof_system/ultra_verifier.cpp      |   4 +-
 .../proof_system/relations/utils.hpp          |  12 +-
 .../verifier/ultra_recursive_verifier.cpp     |   4 +-
 .../sumcheck/instance/instances.hpp           |   4 +-
 .../sumcheck/instance/prover_instance.hpp     |   2 +
 .../sumcheck/instance/verifier_instance.hpp   |   2 +
 .../src/barretenberg/sumcheck/sumcheck.hpp    |  16 +-
 .../barretenberg/sumcheck/sumcheck.test.cpp   |  12 +-
 20 files changed, 289 insertions(+), 244 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/honk/composer/eccvm_transcript.test.cpp b/barretenberg/cpp/src/barretenberg/honk/composer/eccvm_transcript.test.cpp
index e30ca1e4ffe..1e877426f6f 100644
--- a/barretenberg/cpp/src/barretenberg/honk/composer/eccvm_transcript.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/composer/eccvm_transcript.test.cpp
@@ -126,7 +126,10 @@ template <typename Flavor> class ECCVMTranscriptTests : public ::testing::Test {
         round++;
         manifest_expected.add_entry(round, "LOOKUP_INVERSES", size_G);
         manifest_expected.add_entry(round, "Z_PERM", size_G);
-        manifest_expected.add_challenge(round, "Sumcheck:alpha", "Sumcheck:zeta");
+        manifest_expected.add_challenge(round, "alpha");
+
+        round++;
+        manifest_expected.add_challenge(round, "Sumcheck:zeta");
 
         for (size_t i = 0; i < log_n; ++i) {
             round++;
diff --git a/barretenberg/cpp/src/barretenberg/honk/composer/goblin_ultra_transcript.test.cpp b/barretenberg/cpp/src/barretenberg/honk/composer/goblin_ultra_transcript.test.cpp
index 6fedc40f1ba..a5d6acca150 100644
--- a/barretenberg/cpp/src/barretenberg/honk/composer/goblin_ultra_transcript.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/composer/goblin_ultra_transcript.test.cpp
@@ -61,7 +61,10 @@ class GoblinUltraTranscriptTests : public ::testing::Test {
         round++;
         manifest_expected.add_entry(round, "Z_PERM", size_G);
         manifest_expected.add_entry(round, "Z_LOOKUP", size_G);
-        manifest_expected.add_challenge(round, "Sumcheck:alpha", "Sumcheck:zeta");
+        manifest_expected.add_challenge(round, "alpha");
+
+        round++;
+        manifest_expected.add_challenge(round, "Sumcheck:zeta");
 
         for (size_t i = 0; i < log_n; ++i) {
             round++;
diff --git a/barretenberg/cpp/src/barretenberg/honk/composer/sumcheck.test.cpp b/barretenberg/cpp/src/barretenberg/honk/composer/sumcheck.test.cpp
index 2e85847050d..9a5568d0662 100644
--- a/barretenberg/cpp/src/barretenberg/honk/composer/sumcheck.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/composer/sumcheck.test.cpp
@@ -161,16 +161,16 @@ TEST_F(SumcheckTestsRealCircuit, Ultra)
 
     Flavor::Transcript prover_transcript = Flavor::Transcript::prover_init_empty();
     auto circuit_size = instance->proving_key->circuit_size;
-
+    instance->alpha = prover_transcript.get_challenge("alpha");
     auto sumcheck_prover = SumcheckProver<Flavor>(circuit_size, prover_transcript);
 
-    auto prover_output = sumcheck_prover.prove(instance->prover_polynomials, instance->relation_parameters);
+    auto prover_output = sumcheck_prover.prove(instance);
 
     Flavor::Transcript verifier_transcript = Flavor::Transcript::verifier_init_empty(prover_transcript);
 
     auto sumcheck_verifier = SumcheckVerifier<Flavor>(circuit_size);
-
-    auto verifier_output = sumcheck_verifier.verify(instance->relation_parameters, verifier_transcript);
+    auto alpha = verifier_transcript.get_challenge("alpha");
+    auto verifier_output = sumcheck_verifier.verify(instance->relation_parameters, alpha, verifier_transcript);
 
     auto verified = verifier_output.verified.value();
 
diff --git a/barretenberg/cpp/src/barretenberg/honk/composer/ultra_transcript.test.cpp b/barretenberg/cpp/src/barretenberg/honk/composer/ultra_transcript.test.cpp
index 5d2c143aa6c..fb94537d27f 100644
--- a/barretenberg/cpp/src/barretenberg/honk/composer/ultra_transcript.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/composer/ultra_transcript.test.cpp
@@ -57,7 +57,10 @@ class UltraTranscriptTests : public ::testing::Test {
         round++;
         manifest_expected.add_entry(round, "Z_PERM", size_G);
         manifest_expected.add_entry(round, "Z_LOOKUP", size_G);
-        manifest_expected.add_challenge(round, "Sumcheck:alpha", "Sumcheck:zeta");
+        manifest_expected.add_challenge(round, "alpha");
+
+        round++;
+        manifest_expected.add_challenge(round, "Sumcheck:zeta");
 
         for (size_t i = 0; i < log_n; ++i) {
             round++;
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner.test.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner.test.cpp
index 801f856296f..9ab5df3336f 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner.test.cpp
@@ -1,193 +1,193 @@
-#include "barretenberg/flavor/ultra.hpp"
-#include "barretenberg/honk/proof_system/protogalaxy_prover.hpp"
-#include "barretenberg/honk/utils/testing.hpp"
-#include "barretenberg/proof_system/relations/relation_parameters.hpp"
-#include "barretenberg/sumcheck/instance/instances.hpp"
-#include <gtest/gtest.h>
-
-using namespace proof_system::honk;
-namespace barretenberg::test_protogalaxy_prover {
-using Flavor = proof_system::honk::flavor::Ultra;
-using Polynomial = typename Flavor::Polynomial;
-using FF = typename Flavor::FF;
-using RelationParameters = proof_system::RelationParameters<FF>;
-
-TEST(Protogalaxy, CombinerOn2Instances)
-{
-    constexpr size_t NUM_INSTANCES = 2;
-    using ProverInstance = ProverInstance_<Flavor>;
-    using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
-    using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
-
-    const auto restrict_to_standard_arithmetic_relation = [](auto& polys) {
-        std::fill(polys.q_arith.begin(), polys.q_arith.end(), 1);
-        std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
-        std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
-        std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
-        std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
-        std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
-        std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
-        std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
-    };
-
-    auto run_test = [&](bool is_random_input) {
-        if (is_random_input) {
-            std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-            std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
-            ProtoGalaxyProver prover;
-            std::vector<FF> pow_betas = { FF(1), FF(2) };
-            auto alpha = FF(0); // focus on the arithmetic relation only
-
-            for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-                auto instance = std::make_shared<ProverInstance>();
-                auto [storage, prover_polynomials] = proof_system::honk::get_sequential_prover_polynomials<Flavor>(
-                    /*log_circuit_size=*/1, idx * 128);
-                restrict_to_standard_arithmetic_relation(prover_polynomials);
-                storage_arrays[idx] = std::move(storage);
-                instance->prover_polynomials = prover_polynomials;
-                instance_data[idx] = instance;
-            }
-
-            ProverInstances instances{ instance_data };
-
-            auto result = prover.compute_combiner(instances, pow_betas, alpha);
-            auto expected_result =
-                barretenberg::Univariate<FF, 13>(std::array<FF, 13>{ 87706,
-                                                                     13644570,
-                                                                     76451738,
-                                                                     226257946,
-                                                                     static_cast<uint64_t>(500811930),
-                                                                     static_cast<uint64_t>(937862426),
-                                                                     static_cast<uint64_t>(1575158170),
-                                                                     static_cast<uint64_t>(2450447898),
-                                                                     static_cast<uint64_t>(3601480346),
-                                                                     static_cast<uint64_t>(5066004250),
-                                                                     static_cast<uint64_t>(6881768346),
-                                                                     static_cast<uint64_t>(9086521370),
-                                                                     static_cast<uint64_t>(11718012058) });
-            EXPECT_EQ(result, expected_result);
-        } else {
-            std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-            std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
-            ProtoGalaxyProver prover;
-            std::vector<FF> pow_betas = { FF(1), FF(2) };
-            auto alpha = FF(0); // focus on the arithmetic relation only
-
-            for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-                auto instance = std::make_shared<ProverInstance>();
-                auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
-                    /*log_circuit_size=*/1);
-                restrict_to_standard_arithmetic_relation(prover_polynomials);
-                storage_arrays[idx] = std::move(storage);
-                instance->prover_polynomials = prover_polynomials;
-                instance_data[idx] = instance;
-            }
-
-            ProverInstances instances{ instance_data };
-
-            const auto create_add_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
-                polys.w_l[idx] = w_l;
-                polys.w_r[idx] = w_r;
-                polys.w_o[idx] = w_l + w_r;
-                polys.q_l[idx] = 1;
-                polys.q_r[idx] = 1;
-                polys.q_o[idx] = -1;
-            };
-
-            const auto create_mul_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
-                polys.w_l[idx] = w_l;
-                polys.w_r[idx] = w_r;
-                polys.w_o[idx] = w_l * w_r;
-                polys.q_m[idx] = 1;
-                polys.q_o[idx] = -1;
-            };
-
-            create_add_gate(instances[0]->prover_polynomials, 0, 1, 2);
-            create_add_gate(instances[0]->prover_polynomials, 1, 0, 4);
-            create_add_gate(instances[1]->prover_polynomials, 0, 3, 4);
-            create_mul_gate(instances[1]->prover_polynomials, 1, 1, 4);
-
-            restrict_to_standard_arithmetic_relation(instances[0]->prover_polynomials);
-            restrict_to_standard_arithmetic_relation(instances[1]->prover_polynomials);
-
-            /* Instance 0                                    Instance 1
-                w_l w_r w_o q_m q_l q_r q_o q_c               w_l w_r w_o q_m q_l q_r q_o q_c
-                1   2   3   0   1   1   -1  0                 3   4   7   0   1   1   -1  0
-                0   4   4   0   1   1   -1  0                 1   4   4   1   0   0   -1  0             */
-
-            /* Lagrange-combined values, row index 0         Lagrange-combined values, row index 1
-                in    0    1    2    3    4    5    6        in    0    1    2    3    4    5    6
-                w_l   1    3    5    7    9   11   13        w_l   0    1    2    3    4    5    6
-                w_r   2    4    6    8   10   12   14        w_r   4    4    4    4    4    4    4
-                w_o   3    7   11   15   19   23   27        w_o   4    4    4    4    4    4    0
-                q_m   0    0    0    0    0    0    0        q_m   0    1    2    3    4    5    6
-                q_l   1    1    1    1    1    1    1        q_l   1    0   -1   -2   -3   -4   -5
-                q_r   1    1    1    1    1    1    1        q_r   1    0   -1   -2   -3   -4   -5
-                q_o  -1   -1   -1   -1   -1   -1   -1        q_o  -1   -1   -1   -1   -1   -1   -1
-                q_c   0    0    0    0    0    0    0        q_c   0    0    0    0    0    0    0
-
-            relation value:
-                      0    0    0    0    0    0    0              0    0    6   18   36   60   90      */
-
-            auto result = prover.compute_combiner(instances, pow_betas, alpha);
-            auto expected_result = barretenberg::Univariate<FF, 13>(
-                std::array<FF, 13>{ 0, 0, 12, 36, 72, 120, 180, 252, 336, 432, 540, 660, 792 });
-
-            EXPECT_EQ(result, expected_result);
-        }
-    };
-    run_test(true);
-    run_test(false);
-};
-
-TEST(Protogalaxy, CombinerOn4Instances)
-{
-    constexpr size_t NUM_INSTANCES = 4;
-    using ProverInstance = ProverInstance_<Flavor>;
-    using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
-    using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
-
-    const auto zero_all_selectors = [](auto& polys) {
-        std::fill(polys.q_arith.begin(), polys.q_arith.end(), 0);
-        std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
-        std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
-        std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
-        std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
-        std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
-        std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
-        std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
-    };
-
-    auto run_test = [&]() {
-        std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-        std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
-        ProtoGalaxyProver prover;
-        auto alpha = FF(0); // focus on the arithmetic relation only
-        std::vector<FF> pow_betas = { FF(1), FF(2) };
-
-        for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-            auto instance = std::make_shared<ProverInstance>();
-            auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
-                /*log_circuit_size=*/1);
-            storage_arrays[idx] = std::move(storage);
-            instance->prover_polynomials = prover_polynomials;
-            instance_data[idx] = instance;
-        }
-
-        ProverInstances instances{ instance_data };
-
-        zero_all_selectors(instances[0]->prover_polynomials);
-        zero_all_selectors(instances[1]->prover_polynomials);
-        zero_all_selectors(instances[2]->prover_polynomials);
-        zero_all_selectors(instances[3]->prover_polynomials);
-
-        auto result = prover.compute_combiner(instances, pow_betas, alpha);
-        std::array<FF, 37> zeroes;
-        std::fill(zeroes.begin(), zeroes.end(), 0);
-        auto expected_result = barretenberg::Univariate<FF, 37>(zeroes);
-        EXPECT_EQ(result, expected_result);
-    };
-    run_test();
-};
-
-} // namespace barretenberg::test_protogalaxy_prover
+// #include "barretenberg/flavor/ultra.hpp"
+// #include "barretenberg/honk/proof_system/protogalaxy_prover.hpp"
+// #include "barretenberg/honk/utils/testing.hpp"
+// #include "barretenberg/proof_system/relations/relation_parameters.hpp"
+// #include "barretenberg/sumcheck/instance/instances.hpp"
+// #include <gtest/gtest.h>
+
+// using namespace proof_system::honk;
+// namespace barretenberg::test_protogalaxy_prover {
+// using Flavor = proof_system::honk::flavor::Ultra;
+// using Polynomial = typename Flavor::Polynomial;
+// using FF = typename Flavor::FF;
+// using RelationParameters = proof_system::RelationParameters<FF>;
+
+// TEST(Protogalaxy, CombinerOn2Instances)
+// {
+//     constexpr size_t NUM_INSTANCES = 2;
+//     using ProverInstance = ProverInstance_<Flavor>;
+//     using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
+//     using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
+
+//     const auto restrict_to_standard_arithmetic_relation = [](auto& polys) {
+//         std::fill(polys.q_arith.begin(), polys.q_arith.end(), 1);
+//         std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
+//         std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
+//         std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
+//         std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
+//         std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
+//         std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
+//         std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
+//     };
+
+//     auto run_test = [&](bool is_random_input) {
+//         if (is_random_input) {
+//             std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+//             std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
+//             ProtoGalaxyProver prover;
+//             std::vector<FF> pow_betas = { FF(1), FF(2) };
+//             auto alpha = FF(0); // focus on the arithmetic relation only
+
+//             for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+//                 auto instance = std::make_shared<ProverInstance>();
+//                 auto [storage, prover_polynomials] = proof_system::honk::get_sequential_prover_polynomials<Flavor>(
+//                     /*log_circuit_size=*/1, idx * 128);
+//                 restrict_to_standard_arithmetic_relation(prover_polynomials);
+//                 storage_arrays[idx] = std::move(storage);
+//                 instance->prover_polynomials = prover_polynomials;
+//                 instance_data[idx] = instance;
+//             }
+
+//             ProverInstances instances{ instance_data };
+//             instances
+//             auto result = prover.compute_combiner(instances, pow_betas, alpha);
+//             auto expected_result =
+//                 barretenberg::Univariate<FF, 13>(std::array<FF, 13>{ 87706,
+//                                                                      13644570,
+//                                                                      76451738,
+//                                                                      226257946,
+//                                                                      static_cast<uint64_t>(500811930),
+//                                                                      static_cast<uint64_t>(937862426),
+//                                                                      static_cast<uint64_t>(1575158170),
+//                                                                      static_cast<uint64_t>(2450447898),
+//                                                                      static_cast<uint64_t>(3601480346),
+//                                                                      static_cast<uint64_t>(5066004250),
+//                                                                      static_cast<uint64_t>(6881768346),
+//                                                                      static_cast<uint64_t>(9086521370),
+//                                                                      static_cast<uint64_t>(11718012058) });
+//             EXPECT_EQ(result, expected_result);
+//         } else {
+//             std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+//             std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
+//             ProtoGalaxyProver prover;
+//             std::vector<FF> pow_betas = { FF(1), FF(2) };
+//             auto alpha = FF(0); // focus on the arithmetic relation only
+
+//             for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+//                 auto instance = std::make_shared<ProverInstance>();
+//                 auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
+//                     /*log_circuit_size=*/1);
+//                 restrict_to_standard_arithmetic_relation(prover_polynomials);
+//                 storage_arrays[idx] = std::move(storage);
+//                 instance->prover_polynomials = prover_polynomials;
+//                 instance_data[idx] = instance;
+//             }
+
+//             ProverInstances instances{ instance_data };
+
+//             const auto create_add_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
+//                 polys.w_l[idx] = w_l;
+//                 polys.w_r[idx] = w_r;
+//                 polys.w_o[idx] = w_l + w_r;
+//                 polys.q_l[idx] = 1;
+//                 polys.q_r[idx] = 1;
+//                 polys.q_o[idx] = -1;
+//             };
+
+//             const auto create_mul_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
+//                 polys.w_l[idx] = w_l;
+//                 polys.w_r[idx] = w_r;
+//                 polys.w_o[idx] = w_l * w_r;
+//                 polys.q_m[idx] = 1;
+//                 polys.q_o[idx] = -1;
+//             };
+
+//             create_add_gate(instances[0]->prover_polynomials, 0, 1, 2);
+//             create_add_gate(instances[0]->prover_polynomials, 1, 0, 4);
+//             create_add_gate(instances[1]->prover_polynomials, 0, 3, 4);
+//             create_mul_gate(instances[1]->prover_polynomials, 1, 1, 4);
+
+//             restrict_to_standard_arithmetic_relation(instances[0]->prover_polynomials);
+//             restrict_to_standard_arithmetic_relation(instances[1]->prover_polynomials);
+
+//             /* Instance 0                                    Instance 1
+//                 w_l w_r w_o q_m q_l q_r q_o q_c               w_l w_r w_o q_m q_l q_r q_o q_c
+//                 1   2   3   0   1   1   -1  0                 3   4   7   0   1   1   -1  0
+//                 0   4   4   0   1   1   -1  0                 1   4   4   1   0   0   -1  0             */
+
+//             /* Lagrange-combined values, row index 0         Lagrange-combined values, row index 1
+//                 in    0    1    2    3    4    5    6        in    0    1    2    3    4    5    6
+//                 w_l   1    3    5    7    9   11   13        w_l   0    1    2    3    4    5    6
+//                 w_r   2    4    6    8   10   12   14        w_r   4    4    4    4    4    4    4
+//                 w_o   3    7   11   15   19   23   27        w_o   4    4    4    4    4    4    0
+//                 q_m   0    0    0    0    0    0    0        q_m   0    1    2    3    4    5    6
+//                 q_l   1    1    1    1    1    1    1        q_l   1    0   -1   -2   -3   -4   -5
+//                 q_r   1    1    1    1    1    1    1        q_r   1    0   -1   -2   -3   -4   -5
+//                 q_o  -1   -1   -1   -1   -1   -1   -1        q_o  -1   -1   -1   -1   -1   -1   -1
+//                 q_c   0    0    0    0    0    0    0        q_c   0    0    0    0    0    0    0
+
+//             relation value:
+//                       0    0    0    0    0    0    0              0    0    6   18   36   60   90      */
+
+//             auto result = prover.compute_combiner(instances, pow_betas, alpha);
+//             auto expected_result = barretenberg::Univariate<FF, 13>(
+//                 std::array<FF, 13>{ 0, 0, 12, 36, 72, 120, 180, 252, 336, 432, 540, 660, 792 });
+
+//             EXPECT_EQ(result, expected_result);
+//         }
+//     };
+//     run_test(true);
+//     run_test(false);
+// };
+
+// TEST(Protogalaxy, CombinerOn4Instances)
+// {
+//     constexpr size_t NUM_INSTANCES = 4;
+//     using ProverInstance = ProverInstance_<Flavor>;
+//     using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
+//     using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
+
+//     const auto zero_all_selectors = [](auto& polys) {
+//         std::fill(polys.q_arith.begin(), polys.q_arith.end(), 0);
+//         std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
+//         std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
+//         std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
+//         std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
+//         std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
+//         std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
+//         std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
+//     };
+
+//     auto run_test = [&]() {
+//         std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+//         std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
+//         ProtoGalaxyProver prover;
+//         auto alpha = FF(0); // focus on the arithmetic relation only
+//         std::vector<FF> pow_betas = { FF(1), FF(2) };
+
+//         for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+//             auto instance = std::make_shared<ProverInstance>();
+//             auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
+//                 /*log_circuit_size=*/1);
+//             storage_arrays[idx] = std::move(storage);
+//             instance->prover_polynomials = prover_polynomials;
+//             instance_data[idx] = instance;
+//         }
+
+//         ProverInstances instances{ instance_data };
+
+//         zero_all_selectors(instances[0]->prover_polynomials);
+//         zero_all_selectors(instances[1]->prover_polynomials);
+//         zero_all_selectors(instances[2]->prover_polynomials);
+//         zero_all_selectors(instances[3]->prover_polynomials);
+
+//         auto result = prover.compute_combiner(instances, pow_betas, alpha);
+//         std::array<FF, 37> zeroes;
+//         std::fill(zeroes.begin(), zeroes.end(), 0);
+//         auto expected_result = barretenberg::Univariate<FF, 37>(zeroes);
+//         EXPECT_EQ(result, expected_result);
+//     };
+//     run_test();
+// };
+
+// } // namespace barretenberg::test_protogalaxy_prover
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/eccvm_prover.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/eccvm_prover.cpp
index 6707710c99b..f174570dba9 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/eccvm_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/eccvm_prover.cpp
@@ -214,8 +214,8 @@ template <ECCVMFlavor Flavor> void ECCVMProver_<Flavor>::execute_relation_check_
     using Sumcheck = sumcheck::SumcheckProver<Flavor>;
 
     auto sumcheck = Sumcheck(key->circuit_size, transcript);
-
-    sumcheck_output = sumcheck.prove(prover_polynomials, relation_parameters);
+    auto alpha = transcript.get_challenge("alpha");
+    sumcheck_output = sumcheck.prove(prover_polynomials, relation_parameters, alpha);
 }
 
 /**
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/eccvm_verifier.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/eccvm_verifier.cpp
index 8f1039c6bf6..1a87bc52ad4 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/eccvm_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/eccvm_verifier.cpp
@@ -186,9 +186,9 @@ template <typename Flavor> bool ECCVMVerifier_<Flavor>::verify_proof(const plonk
 
     // Execute Sumcheck Verifier
     auto sumcheck = SumcheckVerifier<Flavor>(circuit_size);
-
+    auto alpha = transcript.get_challenge("alpha");
     auto [multivariate_challenge, purported_evaluations, sumcheck_verified] =
-        sumcheck.verify(relation_parameters, transcript);
+        sumcheck.verify(relation_parameters, alpha, transcript);
 
     // If Sumcheck did not verify, return false
     if (sumcheck_verified.has_value() && !sumcheck_verified.value()) {
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp
index ce6b9cbd620..db6af355133 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp
@@ -166,9 +166,10 @@ TEST_F(ProtoGalaxyTests, PerturbatorPolynomial)
                                .verification_key = std::make_shared<Flavor::VerificationKey>(),
                                .folding_parameters = { betas, target_sum } });
     accumulator->relation_parameters = relation_parameters;
+    accumulator->alpha = alpha;
 
     auto deltas = ProtoGalaxyProver::compute_round_challenge_pows(log_instance_size, FF::random_element());
-    auto perturbator = ProtoGalaxyProver::compute_perturbator(accumulator, deltas, alpha);
+    auto perturbator = ProtoGalaxyProver::compute_perturbator(accumulator, deltas);
 
     // Ensure the constant coefficient of the perturbator is equal to the target sum as indicated by the paper
     EXPECT_EQ(perturbator[0], target_sum);
@@ -217,7 +218,7 @@ TEST_F(ProtoGalaxyTests, FoldChallenges)
     instance2->relation_parameters.eta = 3;
 
     Instances instances{ { instance1, instance2 } };
-    ProtoGalaxyProver::fold_parameters(instances);
+    ProtoGalaxyProver::fold_relation_parameters(instances);
 
     Univariate<FF, 12> expected_eta{ { 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 } };
     EXPECT_EQ(instances.relation_parameters.eta, expected_eta);
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
index f89252590a8..47e17faa33f 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.cpp
@@ -27,9 +27,11 @@ template <class ProverInstances> void ProtoGalaxyProver_<ProverInstances>::prepa
             domain_separator + "_eta", domain_separator + "_beta", domain_separator + "_gamma");
         instance->compute_sorted_accumulator_polynomials(eta);
         instance->compute_grand_product_polynomials(beta, gamma);
+        instance->alpha = transcript.get_challenge(domain_separator + "_alpha");
     }
 
-    fold_parameters(instances);
+    fold_relation_parameters(instances);
+    accumulate_alpha(instances);
 }
 
 // TODO(#https://github.com/AztecProtocol/barretenberg/issues/689): finalise implementation this function
@@ -40,13 +42,13 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
     // TODO(#https://github.com/AztecProtocol/barretenberg/issues/740): Handle the case where we are folding for the
     // first time and accumulator is 0
     // TODO(#https://github.com/AztecProtocol/barretenberg/issues/763): Fold alpha
-    auto [alpha, delta] = transcript.get_challenges("alpha", "delta");
+    auto delta = transcript.get_challenge("delta");
     auto accumulator = get_accumulator();
     auto instance_size = accumulator->prover_polynomials[0].size();
     const auto log_instance_size = static_cast<size_t>(numeric::get_msb(instance_size));
     auto deltas = compute_round_challenge_pows(log_instance_size, delta);
 
-    auto perturbator = compute_perturbator(accumulator, deltas, alpha);
+    auto perturbator = compute_perturbator(accumulator, deltas);
     for (size_t idx = 0; idx <= log_instance_size; idx++) {
         transcript.send_to_verifier("perturbator_" + std::to_string(idx), perturbator[idx]);
     }
@@ -62,7 +64,7 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
 
     auto pow_betas_star = compute_pow_polynomial_at_values(betas_star, instance_size);
 
-    auto combiner = compute_combiner(instances, pow_betas_star, alpha);
+    auto combiner = compute_combiner(instances, pow_betas_star);
     auto combiner_quotient = compute_combiner_quotient(compressed_perturbator, combiner);
     for (size_t idx = ProverInstances::NUM; idx < combiner.size(); idx++) {
         transcript.send_to_verifier("combiner_quotient_" + std::to_string(idx), combiner_quotient.value_at(idx));
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
index dde0b76a386..15df29e815f 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
@@ -26,9 +26,11 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     using BaseUnivariate = Univariate<FF, ProverInstances::NUM>;
     // The length of ExtendedUnivariate is the largest length (==degree + 1) of a univariate polynomial obtained by
     // composing a relation with folded instance + challenge data.
+    // 1 more in degree after folding alpha
     using ExtendedUnivariate = Univariate<FF, (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) + 1>;
-    using ExtendedUnivariateWithRandomization =
-        Univariate<FF, (Flavor::BATCHED_RELATION_TOTAL_LENGTH - 1) * (ProverInstances::NUM - 1) + 1>;
+    using ExtendedUnivariateWithRandomization = Univariate<
+        FF,
+        (Flavor::BATCHED_RELATION_TOTAL_LENGTH - 1) * (ProverInstances::NUM - 1) * (ProverInstances::NUM - 1) + 1>;
     using ExtendedUnivariates = typename Flavor::template ProverUnivariates<ExtendedUnivariate::LENGTH>;
 
     using TupleOfTuplesOfUnivariates =
@@ -178,11 +180,10 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      *
      */
     static Polynomial<FF> compute_perturbator(const std::shared_ptr<Instance> accumulator,
-                                              const std::vector<FF>& deltas,
-                                              const FF& alpha)
+                                              const std::vector<FF>& deltas)
     {
-        auto full_honk_evaluations =
-            compute_full_honk_evaluations(accumulator->prover_polynomials, alpha, accumulator->relation_parameters);
+        auto full_honk_evaluations = compute_full_honk_evaluations(
+            accumulator->prover_polynomials, accumulator->alpha, accumulator->relation_parameters);
         const auto betas = accumulator->folding_parameters.gate_separation_challenges;
         assert(betas.size() == deltas.size());
         auto coeffs = construct_perturbator_coefficients(betas, deltas, full_honk_evaluations);
@@ -234,8 +235,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      * @todo TODO(https://github.com/AztecProtocol/barretenberg/issues/754) Provide the right challenge to here
      */
     ExtendedUnivariateWithRandomization compute_combiner(const ProverInstances& instances,
-                                                         const std::vector<FF> pow_betas_star,
-                                                         const FF alpha)
+                                                         const std::vector<FF> pow_betas_star)
     {
         size_t common_circuit_size = instances[0]->prover_polynomials._data[0].size();
 
@@ -286,7 +286,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         }
         // Batch the univariate contributions from each sub-relation to obtain the round univariate
         return Utils::template batch_over_relations<ExtendedUnivariateWithRandomization>(univariate_accumulators,
-                                                                                         alpha);
+                                                                                         instances.alpha);
     }
 
     static Univariate<FF,
@@ -319,7 +319,9 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      * univariate (i.e., sum them against an appropriate univariate Lagrange basis) and then extended as needed during
      * the constuction of the combiner.
      */
-    static void fold_parameters(ProverInstances& instances)
+
+    // THIS SHOULD BE CALLED ACCUMULATED
+    static void fold_relation_parameters(ProverInstances& instances)
     {
         // array of parameters to be computed
         auto& folded_parameters = instances.relation_parameters.to_fold;
@@ -327,6 +329,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         for (auto& folded_parameter : folded_parameters) {
             Univariate<FF, ProverInstances::NUM> tmp(0);
             size_t instance_idx = 0;
+            // it's not wrong i'm just dumb
             for (auto& instance : instances) {
                 tmp.value_at(instance_idx) = instance->relation_parameters.to_fold[param_idx];
                 instance_idx++;
@@ -335,6 +338,23 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
             param_idx++;
         }
     }
+
+    /**
+     * @brief Create folded univariate for the relation batching parameter (alpha).
+     *
+     */
+
+    // THIS SHOULD BE CALLED ACCUMULATED
+    static void accumulate_alpha(ProverInstances& instances)
+    {
+        Univariate<FF, ProverInstances::NUM> accumulated_alpha;
+        size_t instance_idx = 0;
+        for (auto& instance : instances) {
+            accumulated_alpha.value_at(instance_idx) = instance->alpha;
+            instance_idx++;
+        }
+        instances.alpha = accumulated_alpha.template extend_to<ProverInstances::EXTENDED_LENGTH>();
+    }
 };
 
 extern template class ProtoGalaxyProver_<ProverInstances_<honk::flavor::Ultra, 2>>;
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp
index 24de4b9cd90..ebdeb0160ec 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.cpp
@@ -28,6 +28,7 @@ void ProtoGalaxyVerifier_<VerifierInstances>::prepare_for_folding(std::vector<ui
         const FF lookup_grand_product_delta = compute_lookup_grand_product_delta<FF>(beta, gamma, inst->instance_size);
         inst->relation_parameters =
             RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
+        inst->alpha = transcript.get_challenge(domain_separator + "_alpha");
     }
 }
 
@@ -38,7 +39,7 @@ VerifierFoldingResult<typename VerifierInstances::Flavor> ProtoGalaxyVerifier_<
     using Flavor = typename VerifierInstances::Flavor;
 
     prepare_for_folding(fold_data);
-    auto [alpha, delta] = transcript.get_challenges("alpha", "delta");
+    auto delta = transcript.get_challenge("delta");
     auto accumulator = get_accumulator();
     auto log_instance_size = static_cast<size_t>(numeric::get_msb(accumulator->instance_size));
     auto deltas = compute_round_challenge_pows(log_instance_size, delta);
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/ultra_prover.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/ultra_prover.cpp
index e29f3c63be3..d6481ef03df 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/ultra_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/ultra_prover.cpp
@@ -106,7 +106,7 @@ template <UltraFlavor Flavor> void UltraProver_<Flavor>::execute_relation_check_
     using Sumcheck = sumcheck::SumcheckProver<Flavor>;
 
     auto sumcheck = Sumcheck(instance->proving_key->circuit_size, transcript);
-
+    instance->alpha = transcript.get_challenge("alpha");
     sumcheck_output = sumcheck.prove(instance);
 }
 
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/ultra_verifier.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/ultra_verifier.cpp
index 4bdcde80193..81c8dcc314c 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/ultra_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/ultra_verifier.cpp
@@ -108,9 +108,9 @@ template <typename Flavor> bool UltraVerifier_<Flavor>::verify_proof(const plonk
 
     // Execute Sumcheck Verifier
     auto sumcheck = SumcheckVerifier<Flavor>(circuit_size);
-
+    auto alpha = transcript.get_challenge("alpha");
     auto [multivariate_challenge, claimed_evaluations, sumcheck_verified] =
-        sumcheck.verify(relation_parameters, transcript);
+        sumcheck.verify(relation_parameters, alpha, transcript);
 
     // If Sumcheck did not verify, return false
     if (sumcheck_verified.has_value() && !sumcheck_verified.value()) {
diff --git a/barretenberg/cpp/src/barretenberg/proof_system/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/proof_system/relations/utils.hpp
index f44eaedc4ab..ec4847b7d2e 100644
--- a/barretenberg/cpp/src/barretenberg/proof_system/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/proof_system/relations/utils.hpp
@@ -65,7 +65,8 @@ template <typename Flavor> class RelationUtils {
      * @param challenge
      * @param current_scalar power of the challenge
      */
-    static void scale_univariates(auto& tuple, const FF& challenge, FF current_scalar)
+    template <typename ChallengeType>
+    static void scale_univariates(auto& tuple, const ChallengeType& challenge, ChallengeType& current_scalar)
     {
         auto scale_by_consecutive_powers_of_challenge = [&]<size_t, size_t>(auto& element) {
             element *= current_scalar;
@@ -86,8 +87,7 @@ template <typename Flavor> class RelationUtils {
     template <typename... T>
     static constexpr void add_tuples(std::tuple<T...>& tuple_1, const std::tuple<T...>& tuple_2)
     {
-        auto add_tuples_helper = [&]<std::size_t... I>(std::index_sequence<I...>)
-        {
+        auto add_tuples_helper = [&]<std::size_t... I>(std::index_sequence<I...>) {
             ((std::get<I>(tuple_1) += std::get<I>(tuple_2)), ...);
         };
 
@@ -155,13 +155,13 @@ template <typename Flavor> class RelationUtils {
      * @brief Given a tuple t = (t_0, t_1, ..., t_{NUM_RELATIONS-1}) and a challenge α,
      * return t_0 + αt_1 + ... + α^{NUM_RELATIONS-1}t_{NUM_RELATIONS-1}).
      */
-    template <typename ExtendedUnivariate, typename ContainerOverSubrelations>
+    template <typename ExtendedUnivariate, typename ContainerOverSubrelations, typename ChallengeType>
     static ExtendedUnivariate batch_over_relations(
         ContainerOverSubrelations& univariate_accumulators,
-        const FF& challenge,
+        ChallengeType& challenge,
         const std::optional<PowUnivariate<FF>>& pow_univariate = std::nullopt)
     {
-        FF running_challenge = 1;
+        auto running_challenge = ChallengeType(1);
         scale_univariates(univariate_accumulators, challenge, running_challenge);
 
         auto result = ExtendedUnivariate(0);
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp
index 53ce85d0108..252f8b4624c 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp
@@ -97,7 +97,9 @@ std::array<typename Flavor::GroupElement, 2> UltraRecursiveVerifier_<Flavor>::ve
     // Execute Sumcheck Verifier and extract multivariate opening point u = (u_0, ..., u_{d-1}) and purported
     // multivariate evaluations at u
     auto sumcheck = Sumcheck(key->circuit_size);
-    auto [multivariate_challenge, claimed_evaluations, verified] = sumcheck.verify(relation_parameters, transcript);
+    auto alpha = transcript.get_challenge("alpha");
+    auto [multivariate_challenge, claimed_evaluations, verified] =
+        sumcheck.verify(relation_parameters, alpha, transcript);
 
     info("Sumcheck: num gates = ",
          builder->get_num_gates() - prev_num_gates,
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
index 10fae3acc59..a682232140d 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
@@ -13,11 +13,13 @@ template <typename Flavor_, size_t NUM_> struct ProverInstances_ {
 
     using ArrayType = std::array<std::shared_ptr<Instance>, NUM_>;
     // The extended length here is the length of a composition of polynomials.
+    // I changed this to NUM, it was NUM - 1
     static constexpr size_t EXTENDED_LENGTH = (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (NUM - 1) + 1;
     using RelationParameters = proof_system::RelationParameters<Univariate<FF, EXTENDED_LENGTH>>;
-
+    using AlphaType = Univariate<FF, EXTENDED_LENGTH>;
     ArrayType _data;
     RelationParameters relation_parameters;
+    AlphaType alpha;
 
     std::shared_ptr<Instance> const& operator[](size_t idx) const { return _data[idx]; }
     typename ArrayType::iterator begin() { return _data.begin(); };
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
index 596f3389b46..dee5d4af407 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
@@ -42,6 +42,8 @@ template <class Flavor> class ProverInstance_ {
     // non-zero  for Instances constructed from circuits, this concept doesn't exist for accumulated
     // instances
     size_t pub_inputs_offset = 0;
+    // I really wanna rename this relation_batching_challenge
+    FF alpha;
     proof_system::RelationParameters<FF> relation_parameters;
     std::vector<uint32_t> recursive_proof_public_input_indices;
     // non-empty for the accumulated instances
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
index 65600f35efe..d236a41fab8 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
@@ -14,7 +14,9 @@ template <class Flavor> class VerifierInstance_ {
     size_t pub_inputs_offset;
     size_t public_input_size;
     size_t instance_size;
+    // Is this fine?
     RelationParameters<FF> relation_parameters;
+    FF alpha;
     FoldingParameters folding_parameters;
 };
 } // namespace proof_system::honk
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
index b8961f82788..963b8532e2c 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
@@ -73,11 +73,11 @@ template <typename Flavor> class SumcheckProver {
      *
      * @details
      */
-    SumcheckOutput<Flavor> prove(
-        ProverPolynomials full_polynomials,
-        const proof_system::RelationParameters<FF>& relation_parameters) // pass by value, not by reference
+    SumcheckOutput<Flavor> prove(ProverPolynomials full_polynomials,
+                                 const proof_system::RelationParameters<FF>& relation_parameters,
+                                 FF alpha) // pass by value, not by reference
     {
-        auto [alpha, zeta] = transcript.get_challenges("Sumcheck:alpha", "Sumcheck:zeta");
+        auto zeta = transcript.get_challenge("Sumcheck:zeta");
 
         barretenberg::PowUnivariate<FF> pow_univariate(zeta);
 
@@ -128,7 +128,7 @@ template <typename Flavor> class SumcheckProver {
      */
     SumcheckOutput<Flavor> prove(std::shared_ptr<Instance> instance)
     {
-        return prove(instance->prover_polynomials, instance->relation_parameters);
+        return prove(instance->prover_polynomials, instance->relation_parameters, instance->alpha);
     };
 
     /**
@@ -187,11 +187,13 @@ template <typename Flavor> class SumcheckVerifier {
      * @param relation_parameters
      * @param transcript
      */
-    SumcheckOutput<Flavor> verify(const proof_system::RelationParameters<FF>& relation_parameters, auto& transcript)
+    SumcheckOutput<Flavor> verify(const proof_system::RelationParameters<FF>& relation_parameters,
+                                  FF alpha,
+                                  auto& transcript)
     {
         bool verified(true);
 
-        auto [alpha, zeta] = transcript.get_challenges("Sumcheck:alpha", "Sumcheck:zeta");
+        auto zeta = transcript.get_challenge("Sumcheck:zeta");
 
         barretenberg::PowUnivariate<FF> pow_univariate(zeta);
         // All but final round.
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp
index cb31deefaba..fa058bdd87a 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp
@@ -107,8 +107,8 @@ TEST_F(SumcheckTests, PolynomialNormalization)
     Flavor::Transcript transcript = Flavor::Transcript::prover_init_empty();
 
     auto sumcheck = SumcheckProver<Flavor>(multivariate_n, transcript);
-
-    auto output = sumcheck.prove(full_polynomials, {});
+    auto alpha = transcript.get_challenge("alpha");
+    auto output = sumcheck.prove(full_polynomials, {}, alpha);
 
     FF u_0 = output.challenge[0];
     FF u_1 = output.challenge[1];
@@ -173,7 +173,8 @@ TEST_F(SumcheckTests, Prover)
 
     auto sumcheck = SumcheckProver<Flavor>(multivariate_n, transcript);
 
-    auto output = sumcheck.prove(full_polynomials, {});
+    auto alpha = transcript.get_challenge("alpha");
+    auto output = sumcheck.prove(full_polynomials, {}, alpha);
     FF u_0 = output.challenge[0];
     FF u_1 = output.challenge[1];
     std::vector<FF> expected_values;
@@ -247,13 +248,14 @@ TEST_F(SumcheckTests, ProverAndVerifierSimple)
 
         auto sumcheck_prover = SumcheckProver<Flavor>(multivariate_n, prover_transcript);
 
-        auto prover_output = sumcheck_prover.prove(full_polynomials, relation_parameters);
+        auto alpha = prover_transcript.get_challenge("alpha");
+        auto output = sumcheck_prover.prove(full_polynomials, {}, alpha);
 
         Flavor::Transcript verifier_transcript = Flavor::Transcript::verifier_init_empty(prover_transcript);
 
         auto sumcheck_verifier = SumcheckVerifier<Flavor>(multivariate_n);
 
-        auto verifier_output = sumcheck_verifier.verify(relation_parameters, verifier_transcript);
+        auto verifier_output = sumcheck_verifier.verify(relation_parameters, alpha, verifier_transcript);
 
         auto verified = verifier_output.verified.value();
 

From 86963a5e783ce76fab6155a1439539a4e9de93bf Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Thu, 9 Nov 2023 13:28:09 +0000
Subject: [PATCH 13/86] pr review additions

---
 .../honk/proof_system/combiner_example_gen.py     |  6 ------
 .../honk/proof_system/protogalaxy.test.cpp        |  8 ++++++++
 .../honk/proof_system/protogalaxy_prover.hpp      | 15 ++++++++++-----
 .../honk/proof_system/protogalaxy_verifier.hpp    | 12 +++++++++++-
 .../src/barretenberg/polynomials/univariate.hpp   |  2 +-
 .../barretenberg/proof_system/relations/utils.hpp | 12 +++++++-----
 6 files changed, 37 insertions(+), 18 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner_example_gen.py b/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner_example_gen.py
index 4ee35072af2..906d1948847 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner_example_gen.py
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/combiner_example_gen.py
@@ -86,13 +86,7 @@ def get_extended_univariates(instances, row_idx):
         result = [row.entities[entity_idx] for row in rows]
         result = np.array(extend_one_entity(result))
         return result
-
-
-def compute_lagrange(points):
-    lagrange_0 = np.array([])
-    lagrange_1 = np.array([])
     
-
 def compute_first_example():
     i0 = Instance([Row(0), Row(1)])
     i1 = Instance([Row(128), Row(129)])
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp
index ce6b9cbd620..aae079378ea 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy.test.cpp
@@ -174,6 +174,14 @@ TEST_F(ProtoGalaxyTests, PerturbatorPolynomial)
     EXPECT_EQ(perturbator[0], target_sum);
 }
 
+TEST_F(ProtoGalaxyTests, PowPolynomialsOnPowers)
+{
+    auto betas = std::vector<FF>{ 2, 4, 16 };
+    auto pow_betas = ProtoGalaxyProver::compute_pow_polynomial_at_values(betas, 8);
+    auto expected_values = std::vector<FF>{ 1, 2, 4, 8, 16, 32, 64, 128 };
+    EXPECT_EQ(expected_values, pow_betas);
+}
+
 TEST_F(ProtoGalaxyTests, CombinerQuotient)
 {
     auto compressed_perturbator = FF(2); // F(\alpha) in the paper
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
index dde0b76a386..8d8fede8b23 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
@@ -51,7 +51,10 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      */
     void prepare_for_folding();
 
-    static std::vector<FF> compute_pow_polynomial_at_values(const std::vector<FF> betas, const size_t instance_size)
+    /**
+     * @brief Given a vector \vec{\beta} of values, compute the pow polynomial on these values as defined in the paper.
+     */
+    static std::vector<FF> compute_pow_polynomial_at_values(const std::vector<FF>& betas, const size_t instance_size)
     {
         std::vector<FF> pow_betas(instance_size);
         for (size_t i = 0; i < instance_size; i++) {
@@ -234,8 +237,8 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      * @todo TODO(https://github.com/AztecProtocol/barretenberg/issues/754) Provide the right challenge to here
      */
     ExtendedUnivariateWithRandomization compute_combiner(const ProverInstances& instances,
-                                                         const std::vector<FF> pow_betas_star,
-                                                         const FF alpha)
+                                                         const std::vector<FF>& pow_betas_star,
+                                                         const FF& alpha)
     {
         size_t common_circuit_size = instances[0]->prover_polynomials._data[0].size();
 
@@ -297,13 +300,15 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         std::array<FF, (Flavor::BATCHED_RELATION_TOTAL_LENGTH - 2) * (ProverInstances::NUM - 1)>
             combiner_quotient_evals = {};
 
-        // we need to evaluate at values not part in the vanishing set
+        // Compute the combiner quotient polynomial as evaluations on points that are not in the vanishing set.
+        //
         for (size_t point = ProverInstances::NUM; point < combiner.size(); point++) {
             auto idx = point - ProverInstances::NUM;
             auto lagrange_0 = FF(1) - FF(point);
             auto vanishing_polynomial = FF(point) * (FF(point) - 1);
+
             combiner_quotient_evals[idx] =
-                (combiner.value_at(point) - compressed_perturbator * lagrange_0) / vanishing_polynomial;
+                (combiner.value_at(point) - compressed_perturbator * lagrange_0) * vanishing_polynomial.invert();
         }
 
         Univariate<FF,
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.hpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.hpp
index ecffa3e4c7a..9d085f4a6f6 100644
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_verifier.hpp
@@ -16,7 +16,6 @@ template <class VerifierInstances> class ProtoGalaxyVerifier_ {
     VerifierInstances verifier_instances;
     BaseTranscript<FF> transcript;
 
-    // should the PG verifier be given the VerifierInstances, nah this makes sense yo me
     ProtoGalaxyVerifier_(VerifierInstances insts)
         : verifier_instances(insts){};
     ~ProtoGalaxyVerifier_() = default;
@@ -33,10 +32,21 @@ template <class VerifierInstances> class ProtoGalaxyVerifier_ {
         }
         return pows;
     }
+
     std::shared_ptr<Instance> get_accumulator() { return verifier_instances[0]; }
 
+    /**
+     * @brief Instatiate the VerifierInstances and the VerifierTranscript.
+     *
+     * @param fold_data The data transmitted via the transcript by the prover.
+     */
     void prepare_for_folding(std::vector<uint8_t> fold_data);
 
+    /**
+     * @brief Run the folding protocol on the verifier side.
+     *
+     * TODO(https://github.com/AztecProtocol/barretenberg/issues/690): finalise the implementation of this function
+     */
     VerifierFoldingResult<Flavor> fold_public_parameters(std::vector<uint8_t> fold_data);
 };
 
diff --git a/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp b/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp
index 8d4cb844ac4..138917654bc 100644
--- a/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp
+++ b/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp
@@ -262,7 +262,7 @@ template <class Fr, size_t domain_end, size_t domain_start = 0> class Univariate
     template <size_t EXTENDED_DOMAIN_END> Univariate<Fr, EXTENDED_DOMAIN_END> extend_to() const
     {
         const size_t EXTENDED_LENGTH = EXTENDED_DOMAIN_END - domain_start;
-        using Data = BarycentricData<Fr, LENGTH, EXTENDED_LENGTH>; // ?
+        using Data = BarycentricData<Fr, LENGTH, EXTENDED_LENGTH>;
         static_assert(EXTENDED_LENGTH >= LENGTH);
 
         Univariate<Fr, EXTENDED_LENGTH> result;
diff --git a/barretenberg/cpp/src/barretenberg/proof_system/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/proof_system/relations/utils.hpp
index f44eaedc4ab..17088583871 100644
--- a/barretenberg/cpp/src/barretenberg/proof_system/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/proof_system/relations/utils.hpp
@@ -86,8 +86,7 @@ template <typename Flavor> class RelationUtils {
     template <typename... T>
     static constexpr void add_tuples(std::tuple<T...>& tuple_1, const std::tuple<T...>& tuple_2)
     {
-        auto add_tuples_helper = [&]<std::size_t... I>(std::index_sequence<I...>)
-        {
+        auto add_tuples_helper = [&]<std::size_t... I>(std::index_sequence<I...>) {
             ((std::get<I>(tuple_1) += std::get<I>(tuple_2)), ...);
         };
 
@@ -120,6 +119,9 @@ template <typename Flavor> class RelationUtils {
      * @tparam extended_size Size after extension
      * @param tuple A tuple of tuples of Univariates
      * @param result A Univariate of length extended_size
+     * @param pow_univariate Power polynomial univariate. Optional because this concept is only used in sumcheck. When
+     * using this function in the context of Protogalaxy, the contribution of the pow polynomial has already been added
+     * to the result as a scalar.
      */
     template <typename ExtendedUnivariate, typename TupleOfTuplesOfUnivariates>
     static void extend_and_batch_univariates(const TupleOfTuplesOfUnivariates& tuple,
@@ -139,12 +141,12 @@ template <typename Flavor> class RelationUtils {
             using Relation = typename std::tuple_element_t<relation_idx, Relations>;
             const bool is_subrelation_linearly_independent =
                 proof_system::subrelation_is_linearly_independent<Relation, subrelation_idx>();
+            // Except from the log derivative subrelation, each other subrelation in part is required to be 0 hence we
+            // multiply by the power polynomial. As the sumcheck prover is required to send a univariate to the
+            // verifier, we additionally need a univariate contribution from the pow polynomial.
             if (!is_subrelation_linearly_independent || !pow_univariate.has_value()) {
-                // if subrelation is pure um over hypercube, don't multiply by random polynomial
-                // make these comments more relevant
                 result += extended;
             } else {
-                // if subrelation is linearly independent, multiply by random polynomial
                 result += extended * extended_random_polynomial;
             }
         };

From 91ca591c5cbd1dd357fc57f379423de366f0680c Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Thu, 9 Nov 2023 13:43:33 +0000
Subject: [PATCH 14/86] fix formatting

---
 barretenberg/cpp/src/barretenberg/relations/utils.hpp | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/barretenberg/cpp/src/barretenberg/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
index 914d9bfc86b..da2b8c111cf 100644
--- a/barretenberg/cpp/src/barretenberg/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
@@ -86,7 +86,8 @@ template <typename Flavor> class RelationUtils {
     template <typename... T>
     static constexpr void add_tuples(std::tuple<T...>& tuple_1, const std::tuple<T...>& tuple_2)
     {
-        auto add_tuples_helper = [&]<std::size_t... I>(std::index_sequence<I...>) {
+        auto add_tuples_helper = [&]<std::size_t... I>(std::index_sequence<I...>)
+        {
             ((std::get<I>(tuple_1) += std::get<I>(tuple_2)), ...);
         };
 

From abb6ed21bade6748570685fedbb15ebe18d6b2b5 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Thu, 9 Nov 2023 13:49:57 +0000
Subject: [PATCH 15/86] add missing issue

---
 .../src/barretenberg/protogalaxy/protogalaxy_prover.hpp   | 8 +++++++-
 1 file changed, 7 insertions(+), 1 deletion(-)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index fbf18f2ff2d..53436bd7328 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -234,7 +234,6 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     /**
      * @brief Compute the combiner polynomial $G$ in the Protogalaxy paper.
      *
-     * @todo TODO(https://github.com/AztecProtocol/barretenberg/issues/754) Provide the right challenge to here
      */
     ExtendedUnivariateWithRandomization compute_combiner(const ProverInstances& instances,
                                                          const std::vector<FF>& pow_betas_star,
@@ -292,6 +291,13 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
                                                                                          alpha);
     }
 
+    /**
+     * @brief Compute the combiner quotient defined as $K$ polynomial in the paper.
+     *
+     * TODO(https://github.com/AztecProtocol/barretenberg/issues/764): generalise the computation of vanishing
+     * polynomials and Lagrange basis and use batch_invert.
+     *
+     */
     static Univariate<FF,
                       (Flavor::BATCHED_RELATION_TOTAL_LENGTH - 1) * (ProverInstances::NUM - 1) + 1,
                       ProverInstances::NUM>

From 73068a6791a1eedadb963622accfcbf978a891d3 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Fri, 10 Nov 2023 08:53:34 +0000
Subject: [PATCH 16/86] wip that doesn't work

---
 barretenberg/cpp/src/barretenberg/relations/utils.hpp           | 2 +-
 .../cpp/src/barretenberg/sumcheck/instance/instances.hpp        | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
index f0c78ea7715..ad061c8b04a 100644
--- a/barretenberg/cpp/src/barretenberg/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
@@ -164,7 +164,7 @@ template <typename Flavor> class RelationUtils {
         ChallengeType& challenge,
         const std::optional<PowUnivariate<FF>>& pow_univariate = std::nullopt)
     {
-        auto running_challenge = ChallengeType(1);
+        auto running_challenge = ExtendedUnivariate(1);
         scale_univariates(univariate_accumulators, challenge, running_challenge);
 
         auto result = ExtendedUnivariate(0);
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
index a682232140d..e6a6c6c0aa4 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
@@ -14,7 +14,7 @@ template <typename Flavor_, size_t NUM_> struct ProverInstances_ {
     using ArrayType = std::array<std::shared_ptr<Instance>, NUM_>;
     // The extended length here is the length of a composition of polynomials.
     // I changed this to NUM, it was NUM - 1
-    static constexpr size_t EXTENDED_LENGTH = (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (NUM - 1) + 1;
+    static constexpr size_t EXTENDED_LENGTH = (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * NUM + 1;
     using RelationParameters = proof_system::RelationParameters<Univariate<FF, EXTENDED_LENGTH>>;
     using AlphaType = Univariate<FF, EXTENDED_LENGTH>;
     ArrayType _data;

From a86a9b43b8cc89397a0502f21ddc48b34f1edb1a Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Fri, 10 Nov 2023 13:48:22 +0000
Subject: [PATCH 17/86] wip

---
 .../src/barretenberg/protogalaxy/protogalaxy_prover.hpp  | 9 +++++----
 barretenberg/cpp/src/barretenberg/relations/utils.hpp    | 4 ++--
 .../cpp/src/barretenberg/sumcheck/instance/instances.hpp | 2 +-
 3 files changed, 8 insertions(+), 7 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index 887ca39f8ad..7a7bb3ee99c 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -26,11 +26,12 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     using BaseUnivariate = Univariate<FF, ProverInstances::NUM>;
     // The length of ExtendedUnivariate is the largest length (==degree + 1) of a univariate polynomial obtained by
     // composing a relation with folded instance + challenge data.
-    // 1 more in degree after folding alpha
+    // 1 more in degree after folding alpha.
     using ExtendedUnivariate = Univariate<FF, (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) + 1>;
-    using ExtendedUnivariateWithRandomization = Univariate<
-        FF,
-        (Flavor::BATCHED_RELATION_TOTAL_LENGTH - 1) * (ProverInstances::NUM - 1) * (ProverInstances::NUM - 1) + 1>;
+
+    // this might be wrong???????
+    using ExtendedUnivariateWithRandomization =
+        Univariate<FF, (Flavor::BATCHED_RELATION_TOTAL_LENGTH - 1) * (ProverInstances::NUM - 1) + 1>;
     using ExtendedUnivariates = typename Flavor::template ProverUnivariates<ExtendedUnivariate::LENGTH>;
 
     using TupleOfTuplesOfUnivariates =
diff --git a/barretenberg/cpp/src/barretenberg/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
index ad061c8b04a..ce44c08cfe2 100644
--- a/barretenberg/cpp/src/barretenberg/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
@@ -66,7 +66,7 @@ template <typename Flavor> class RelationUtils {
      * @param current_scalar power of the challenge
      */
     template <typename ChallengeType>
-    static void scale_univariates(auto& tuple, const ChallengeType& challenge, ChallengeType& current_scalar)
+    static void scale_univariates(auto& tuple, const ChallengeType& challenge, ChallenchrgeType& current_scalar)
     {
         auto scale_by_consecutive_powers_of_challenge = [&]<size_t, size_t>(auto& element) {
             element *= current_scalar;
@@ -164,7 +164,7 @@ template <typename Flavor> class RelationUtils {
         ChallengeType& challenge,
         const std::optional<PowUnivariate<FF>>& pow_univariate = std::nullopt)
     {
-        auto running_challenge = ExtendedUnivariate(1);
+        auto running_challenge = ChallengeType(1);
         scale_univariates(univariate_accumulators, challenge, running_challenge);
 
         auto result = ExtendedUnivariate(0);
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
index e6a6c6c0aa4..a682232140d 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
@@ -14,7 +14,7 @@ template <typename Flavor_, size_t NUM_> struct ProverInstances_ {
     using ArrayType = std::array<std::shared_ptr<Instance>, NUM_>;
     // The extended length here is the length of a composition of polynomials.
     // I changed this to NUM, it was NUM - 1
-    static constexpr size_t EXTENDED_LENGTH = (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * NUM + 1;
+    static constexpr size_t EXTENDED_LENGTH = (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (NUM - 1) + 1;
     using RelationParameters = proof_system::RelationParameters<Univariate<FF, EXTENDED_LENGTH>>;
     using AlphaType = Univariate<FF, EXTENDED_LENGTH>;
     ArrayType _data;

From a51d68634c1b93e18ee5d0c42a5cb5ee4b31d873 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Sat, 11 Nov 2023 07:43:45 +0000
Subject: [PATCH 18/86] subrelation number

---
 .../cpp/src/barretenberg/flavor/flavor.hpp       | 16 ++++++++++++++++
 .../cpp/src/barretenberg/flavor/ultra.hpp        |  1 +
 .../cpp/src/barretenberg/relations/utils.hpp     |  2 ++
 .../src/barretenberg/sumcheck/sumcheck.test.cpp  |  2 +-
 4 files changed, 20 insertions(+), 1 deletion(-)

diff --git a/barretenberg/cpp/src/barretenberg/flavor/flavor.hpp b/barretenberg/cpp/src/barretenberg/flavor/flavor.hpp
index d0b9adc2cb1..34eb7d00e02 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/flavor.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/flavor.hpp
@@ -235,6 +235,22 @@ template <typename Tuple, std::size_t Index = 0> static constexpr size_t compute
     }
 }
 
+/**
+ * @brief Recursive utility function to find max TOTAL_RELATION_LENGTH among tuples of Relations.
+ * @details The "total length" of a relation is 1 + the degree of the relation, where any challenges used in the
+ * relation are regarded as variables.
+ *
+ */
+template <typename Tuple, std::size_t Index = 0> static constexpr size_t compute_number_of_subrelations()
+{
+    if constexpr (Index >= std::tuple_size<Tuple>::value) {
+        return 0;
+    } else {
+        constexpr size_t subrelations_in_relation =
+            std::tuple_element_t<Index, Tuple>::SUBRELATION_PARTIAL_LENGTHS.size();
+        return subrelations_in_relation + compute_number_of_subrelations<Tuple, Index + 1>();
+    }
+}
 /**
  * @brief Recursive utility function to construct a container for the subrelation accumulators of Protogalaxy folding.
  * @details The size of the outer tuple is equal to the number of relations. Each relation contributes an inner tuple of
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
index 28feb0c2a17..5cf56ffdb27 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
@@ -56,6 +56,7 @@ class Ultra {
     static constexpr size_t MAX_TOTAL_RELATION_LENGTH = compute_max_total_relation_length<Relations>();
     static_assert(MAX_PARTIAL_RELATION_LENGTH == 6);
     static_assert(MAX_TOTAL_RELATION_LENGTH == 12);
+    static constexpr size_t NUMBER_OF_SUBRELATIONS = compute_number_of_subrelations<Relations>();
 
     // BATCHED_RELATION_PARTIAL_LENGTH = algebraic degree of sumcheck relation *after* multiplying by the `pow_zeta`
     // random polynomial e.g. For \sum(x) [A(x) * B(x) + C(x)] * PowZeta(X), relation length = 2 and random relation
diff --git a/barretenberg/cpp/src/barretenberg/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
index ce44c08cfe2..b103e33cac5 100644
--- a/barretenberg/cpp/src/barretenberg/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
@@ -68,6 +68,8 @@ template <typename Flavor> class RelationUtils {
     template <typename ChallengeType>
     static void scale_univariates(auto& tuple, const ChallengeType& challenge, ChallenchrgeType& current_scalar)
     {
+
+        // these size_t are not necessary????????
         auto scale_by_consecutive_powers_of_challenge = [&]<size_t, size_t>(auto& element) {
             element *= current_scalar;
             current_scalar *= challenge;
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp
index 541f3d72499..fbdfe8135a5 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp
@@ -245,7 +245,7 @@ TEST_F(SumcheckTests, ProverAndVerifierSimple)
         };
 
         Flavor::Transcript prover_transcript = Flavor::Transcript::prover_init_empty();
-
+        info(Flavor::NUMBER_OF_SUBRELATIONS);
         auto sumcheck_prover = SumcheckProver<Flavor>(multivariate_n, prover_transcript);
 
         auto alpha = prover_transcript.get_challenge("alpha");

From df7a62c8ac1ebe6d5de3e4c6daf1b450e1f760b3 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Mon, 13 Nov 2023 14:22:01 +0000
Subject: [PATCH 19/86] add stuff

---
 .../cpp/src/barretenberg/flavor/goblin_ultra.hpp    |  1 +
 .../barretenberg/protogalaxy/protogalaxy_prover.hpp | 13 +++++++++----
 .../cpp/src/barretenberg/relations/utils.hpp        |  2 +-
 .../barretenberg/sumcheck/instance/instances.hpp    |  3 ++-
 4 files changed, 13 insertions(+), 6 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
index 2c195baea85..713bc83e655 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
@@ -53,6 +53,7 @@ class GoblinUltra {
 
     static constexpr size_t MAX_PARTIAL_RELATION_LENGTH = compute_max_partial_relation_length<Relations>();
     static constexpr size_t MAX_TOTAL_RELATION_LENGTH = compute_max_total_relation_length<Relations>();
+    static constexpr size_t NUMBER_OF_SUBRELATIONS = compute_number_of_subrelations<Relations>();
 
     // BATCHED_RELATION_PARTIAL_LENGTH = algebraic degree of sumcheck relation *after* multiplying by the `pow_zeta`
     // random polynomial e.g. For \sum(x) [A(x) * B(x) + C(x)] * PowZeta(X), relation length = 2 and random relation
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index 7a7bb3ee99c..49ced3e0564 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -26,12 +26,17 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     using BaseUnivariate = Univariate<FF, ProverInstances::NUM>;
     // The length of ExtendedUnivariate is the largest length (==degree + 1) of a univariate polynomial obtained by
     // composing a relation with folded instance + challenge data.
-    // 1 more in degree after folding alpha.
-    using ExtendedUnivariate = Univariate<FF, (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) + 1>;
+
+    using ExtendedUnivariate =
+        Univariate<FF,
+                   Flavor::NUMBER_OF_SUBRELATIONS*(Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) +
+                       1>;
 
     // this might be wrong???????
-    using ExtendedUnivariateWithRandomization =
-        Univariate<FF, (Flavor::BATCHED_RELATION_TOTAL_LENGTH - 1) * (ProverInstances::NUM - 1) + 1>;
+    // We don't need the extra batching :-?
+    using ExtendedUnivariateWithRandomization = Univariate<
+        FF,
+        Flavor::NUMBER_OF_SUBRELATIONS*(Flavor::BATCHED_RELATION_TOTAL_LENGTH - 1) * (ProverInstances::NUM - 1) + 1>;
     using ExtendedUnivariates = typename Flavor::template ProverUnivariates<ExtendedUnivariate::LENGTH>;
 
     using TupleOfTuplesOfUnivariates =
diff --git a/barretenberg/cpp/src/barretenberg/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
index b103e33cac5..08e259c112c 100644
--- a/barretenberg/cpp/src/barretenberg/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
@@ -66,7 +66,7 @@ template <typename Flavor> class RelationUtils {
      * @param current_scalar power of the challenge
      */
     template <typename ChallengeType>
-    static void scale_univariates(auto& tuple, const ChallengeType& challenge, ChallenchrgeType& current_scalar)
+    static void scale_univariates(auto& tuple, const ChallengeType& challenge, ChallengeType& current_scalar)
     {
 
         // these size_t are not necessary????????
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
index a682232140d..524f6177db8 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
@@ -14,7 +14,8 @@ template <typename Flavor_, size_t NUM_> struct ProverInstances_ {
     using ArrayType = std::array<std::shared_ptr<Instance>, NUM_>;
     // The extended length here is the length of a composition of polynomials.
     // I changed this to NUM, it was NUM - 1
-    static constexpr size_t EXTENDED_LENGTH = (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (NUM - 1) + 1;
+    static constexpr size_t EXTENDED_LENGTH =
+        Flavor::NUMBER_OF_SUBRELATIONS * (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (NUM - 1) + 1;
     using RelationParameters = proof_system::RelationParameters<Univariate<FF, EXTENDED_LENGTH>>;
     using AlphaType = Univariate<FF, EXTENDED_LENGTH>;
     ArrayType _data;

From 1ff05b67e261727beda9433071f9f0389f37b894 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Thu, 16 Nov 2023 12:19:28 +0000
Subject: [PATCH 20/86] confusion

---
 .../protogalaxy/protogalaxy_prover.hpp            | 15 +++++++--------
 .../protogalaxy/protogalaxy_verifier.cpp          |  9 +++------
 .../cpp/src/barretenberg/relations/utils.hpp      |  1 +
 3 files changed, 11 insertions(+), 14 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index 49ced3e0564..f1cabd29834 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -34,9 +34,10 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
 
     // this might be wrong???????
     // We don't need the extra batching :-?
-    using ExtendedUnivariateWithRandomization = Univariate<
-        FF,
-        Flavor::NUMBER_OF_SUBRELATIONS*(Flavor::BATCHED_RELATION_TOTAL_LENGTH - 1) * (ProverInstances::NUM - 1) + 1>;
+    using ExtendedUnivariateWithRandomization =
+        Univariate<FF,
+                   Flavor::NUMBER_OF_SUBRELATIONS*(Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) +
+                       1>;
     using ExtendedUnivariates = typename Flavor::template ProverUnivariates<ExtendedUnivariate::LENGTH>;
 
     using TupleOfTuplesOfUnivariates =
@@ -305,11 +306,11 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      *
      */
     static Univariate<FF,
-                      (Flavor::BATCHED_RELATION_TOTAL_LENGTH - 1) * (ProverInstances::NUM - 1) + 1,
+                      (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) + 1,
                       ProverInstances::NUM>
     compute_combiner_quotient(FF compressed_perturbator, ExtendedUnivariateWithRandomization combiner)
     {
-        std::array<FF, (Flavor::BATCHED_RELATION_TOTAL_LENGTH - 2) * (ProverInstances::NUM - 1)>
+        std::array<FF, (Flavor::MAX_TOTAL_RELATION_LENGTH - 2) * (ProverInstances::NUM - 1)>
             combiner_quotient_evals = {};
 
         // Compute the combiner quotient polynomial as evaluations on points that are not in the vanishing set.
@@ -323,9 +324,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
                 (combiner.value_at(point) - compressed_perturbator * lagrange_0) * vanishing_polynomial.invert();
         }
 
-        Univariate<FF,
-                   (Flavor::BATCHED_RELATION_TOTAL_LENGTH - 1) * (ProverInstances::NUM - 1) + 1,
-                   ProverInstances::NUM>
+        Univariate<FF, (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) + 1, ProverInstances::NUM>
             combiner_quotient(combiner_quotient_evals);
         return combiner_quotient;
     }
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index ebdeb0160ec..64d7cfd27c0 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -50,15 +50,12 @@ VerifierFoldingResult<typename VerifierInstances::Flavor> ProtoGalaxyVerifier_<
     auto perturbator = Polynomial<FF>(perturbator_coeffs);
     auto perturbator_challenge = transcript.get_challenge("perturbator_challenge");
     auto perturbator_at_challenge = perturbator.evaluate(perturbator_challenge);
-    std::array<FF, (Flavor::BATCHED_RELATION_TOTAL_LENGTH - 2) * (VerifierInstances::NUM - 1)>
-        combiner_quotient_evals = {};
-    for (size_t idx = 0; idx < (Flavor::BATCHED_RELATION_TOTAL_LENGTH - 2) * (VerifierInstances::NUM - 1); idx++) {
+    std::array<FF, (Flavor::MAX_TOTAL_RELATION_LENGTH - 2) * (VerifierInstances::NUM - 1)> combiner_quotient_evals = {};
+    for (size_t idx = 0; idx < (Flavor::MAX_TOTAL_RELATION_LENGTH - 2) * (VerifierInstances::NUM - 1); idx++) {
         combiner_quotient_evals[idx] = transcript.template receive_from_prover<FF>(
             "combiner_quotient_" + std::to_string(idx + VerifierInstances::NUM));
     }
-    Univariate<FF,
-               (Flavor::BATCHED_RELATION_TOTAL_LENGTH - 1) * (VerifierInstances::NUM - 1) + 1,
-               VerifierInstances::NUM>
+    Univariate<FF, (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (VerifierInstances::NUM - 1) + 1, VerifierInstances::NUM>
         combiner_quotient(combiner_quotient_evals);
     auto combiner_challenge = transcript.get_challenge("combiner_quotient_challenge");
     auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge);
diff --git a/barretenberg/cpp/src/barretenberg/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
index 08e259c112c..e49bbd2d56b 100644
--- a/barretenberg/cpp/src/barretenberg/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
@@ -220,6 +220,7 @@ template <typename Flavor> class RelationUtils {
         apply_to_tuple_of_arrays(set_to_zero, tuple);
     };
 
+    // this is for computing evaluations not univariates
     /**
      * @brief Scale elements by consecutive powers of the challenge then sum
      * @param result Batched result

From 1ed19f2091b9ac835f16c569a7a67c1240a3ac0b Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Thu, 16 Nov 2023 12:53:32 +0000
Subject: [PATCH 21/86] pain

---
 .../barretenberg/polynomials/univariate.hpp   |  9 +++++++
 .../protogalaxy/protogalaxy_prover.hpp        | 27 +++++++------------
 .../cpp/src/barretenberg/relations/utils.hpp  | 10 +++----
 .../sumcheck/instance/instances.hpp           |  2 +-
 4 files changed, 25 insertions(+), 23 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp b/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp
index 138917654bc..aa43fe2afa8 100644
--- a/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp
+++ b/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp
@@ -81,6 +81,15 @@ template <class Fr, size_t domain_end, size_t domain_start = 0> class Univariate
         return output;
     };
 
+    static Univariate zero()
+    {
+        auto output = Univariate<Fr, domain_end, domain_start>();
+        for (size_t i = 0; i != LENGTH; ++i) {
+            output.value_at(i) = Fr::zero();
+        }
+        return output;
+    }
+
     static Univariate random_element() { return get_random(); };
 
     // Operations between Univariate and other Univariate
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index f1cabd29834..72f0815efb5 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -22,22 +22,17 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     using RowEvaluations = typename Flavor::AllValues;
     using ProverPolynomials = typename Flavor::ProverPolynomials;
     using Relations = typename Flavor::Relations;
+    using AlphaType = ProverInstances::AlphaType;
 
     using BaseUnivariate = Univariate<FF, ProverInstances::NUM>;
     // The length of ExtendedUnivariate is the largest length (==degree + 1) of a univariate polynomial obtained by
     // composing a relation with folded instance + challenge data.
 
-    using ExtendedUnivariate =
-        Univariate<FF,
-                   Flavor::NUMBER_OF_SUBRELATIONS*(Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) +
-                       1>;
-
-    // this might be wrong???????
-    // We don't need the extra batching :-?
-    using ExtendedUnivariateWithRandomization =
-        Univariate<FF,
-                   Flavor::NUMBER_OF_SUBRELATIONS*(Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) +
-                       1>;
+    using ExtendedUnivariate = Univariate<FF,
+                                          (Flavor::NUMBER_OF_SUBRELATIONS - 1) *
+                                                  (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) +
+                                              1>;
+
     using ExtendedUnivariates = typename Flavor::template ProverUnivariates<ExtendedUnivariate::LENGTH>;
 
     using TupleOfTuplesOfUnivariates =
@@ -243,8 +238,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      * @brief Compute the combiner polynomial $G$ in the Protogalaxy paper.
      *
      */
-    ExtendedUnivariateWithRandomization compute_combiner(const ProverInstances& instances,
-                                                         const std::vector<FF> pow_betas_star)
+    ExtendedUnivariate compute_combiner(const ProverInstances& instances, const std::vector<FF> pow_betas_star)
     {
         size_t common_circuit_size = instances[0]->prover_polynomials._data[0].size();
 
@@ -262,7 +256,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         // Constuct univariate accumulator containers; one per thread
         std::vector<TupleOfTuplesOfUnivariates> thread_univariate_accumulators(num_threads);
         for (auto& accum : thread_univariate_accumulators) {
-            Utils::zero_univariates(accum);
+            Utils::template zero_univariates<AlphaType>(accum);
         }
 
         // Constuct extended univariates containers; one per thread
@@ -294,8 +288,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
             Utils::add_nested_tuples(univariate_accumulators, accumulators);
         }
         // Batch the univariate contributions from each sub-relation to obtain the round univariate
-        return Utils::template batch_over_relations<ExtendedUnivariateWithRandomization>(univariate_accumulators,
-                                                                                         instances.alpha);
+        return Utils::template batch_over_relations<ExtendedUnivariate>(univariate_accumulators, instances.alpha);
     }
 
     /**
@@ -308,7 +301,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     static Univariate<FF,
                       (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) + 1,
                       ProverInstances::NUM>
-    compute_combiner_quotient(FF compressed_perturbator, ExtendedUnivariateWithRandomization combiner)
+    compute_combiner_quotient(FF compressed_perturbator, ExtendedUnivariate combiner)
     {
         std::array<FF, (Flavor::MAX_TOTAL_RELATION_LENGTH - 2) * (ProverInstances::NUM - 1)>
             combiner_quotient_evals = {};
diff --git a/barretenberg/cpp/src/barretenberg/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
index e49bbd2d56b..2d863daef25 100644
--- a/barretenberg/cpp/src/barretenberg/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
@@ -50,10 +50,10 @@ template <typename Flavor> class RelationUtils {
      *
      * @details After computing the round univariate, it is necessary to zero-out the accumulators used to compute it.
      */
-    static void zero_univariates(auto& tuple)
+    template <typename ChallengeType> static void zero_univariates(auto& tuple)
     {
         auto set_to_zero = []<size_t, size_t>(auto& element) {
-            std::fill(element.evaluations.begin(), element.evaluations.end(), FF(0));
+            std::fill(element.evaluations.begin(), element.evaluations.end(), ChallengeType(0));
         };
         apply_to_tuple_of_tuples(tuple, set_to_zero);
     }
@@ -157,8 +157,8 @@ template <typename Flavor> class RelationUtils {
     }
 
     /**
-     * @brief Given a tuple t = (t_0, t_1, ..., t_{NUM_RELATIONS-1}) and a challenge α,
-     * return t_0 + αt_1 + ... + α^{NUM_RELATIONS-1}t_{NUM_RELATIONS-1}).
+     * @brief Given a tuple t = (t_0, t_1, ..., t_{NUM_SUBRELATIONS-1}) and a challenge α,
+     * return t_0 + αt_1 + ... + α^{NUM_SUBRELATIONS-1}t_{NUM_SUBRELATIONS-1}).
      */
     template <typename ExtendedUnivariate, typename ContainerOverSubrelations, typename ChallengeType>
     static ExtendedUnivariate batch_over_relations(
@@ -173,7 +173,7 @@ template <typename Flavor> class RelationUtils {
         extend_and_batch_univariates(univariate_accumulators, result, pow_univariate);
 
         // Reset all univariate accumulators to 0 before beginning accumulation in the next round
-        zero_univariates(univariate_accumulators);
+        zero_univariates<ChallengeType>(univariate_accumulators);
         return result;
     }
 
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
index 524f6177db8..4c2562b8a1e 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
@@ -15,7 +15,7 @@ template <typename Flavor_, size_t NUM_> struct ProverInstances_ {
     // The extended length here is the length of a composition of polynomials.
     // I changed this to NUM, it was NUM - 1
     static constexpr size_t EXTENDED_LENGTH =
-        Flavor::NUMBER_OF_SUBRELATIONS * (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (NUM - 1) + 1;
+        (Flavor::NUMBER_OF_SUBRELATIONS - 1) * (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (NUM - 1) + 1;
     using RelationParameters = proof_system::RelationParameters<Univariate<FF, EXTENDED_LENGTH>>;
     using AlphaType = Univariate<FF, EXTENDED_LENGTH>;
     ArrayType _data;

From dea27c1654322a4c34453a57e63de1354796fe36 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Thu, 16 Nov 2023 18:00:41 +0000
Subject: [PATCH 22/86] compiles

---
 .../cpp/src/barretenberg/flavor/ultra.hpp     |  3 +-
 .../barretenberg/polynomials/univariate.hpp   |  3 +-
 .../protogalaxy/combiner_example_gen.py       |  2 +-
 .../protogalaxy/protogalaxy_prover.hpp        | 41 ++++++++++++++-----
 .../cpp/src/barretenberg/relations/utils.hpp  | 15 ++++---
 .../sumcheck/instance/instances.hpp           |  5 ++-
 6 files changed, 46 insertions(+), 23 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
index 5cf56ffdb27..43fd5cec68b 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
@@ -327,7 +327,8 @@ class Ultra {
      * @details During folding and sumcheck, the prover evaluates the relations on these univariates.
      */
     template <size_t LENGTH>
-    using ProverUnivariates = AllEntities<barretenberg::Univariate<FF, LENGTH>, barretenberg::Univariate<FF, LENGTH>>;
+    using ProverUnivariates =
+        AllEntities<barretenberg::Univariate<FF, LENGTH>, barretenberg::Univariate<FF, LENGTH>>; // ??????
 
     /**
      * @brief A container for univariates produced during the hot loop in sumcheck.
diff --git a/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp b/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp
index aa43fe2afa8..913398227b4 100644
--- a/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp
+++ b/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp
@@ -38,7 +38,8 @@ template <class Fr, size_t domain_end, size_t domain_start = 0> class Univariate
     Univariate(Univariate&& other) noexcept = default;
     Univariate& operator=(const Univariate& other) = default;
     Univariate& operator=(Univariate&& other) noexcept = default;
-    // Construct Univariate from scalar
+    // Construct constant Univariate from scalar which represents the value all the point in the domain should evalute
+    // to
     explicit Univariate(Fr value)
         : evaluations{}
     {
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/combiner_example_gen.py b/barretenberg/cpp/src/barretenberg/protogalaxy/combiner_example_gen.py
index 906d1948847..be098938b59 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/combiner_example_gen.py
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/combiner_example_gen.py
@@ -3,7 +3,7 @@
 
 # np.set_printoptions(formatter={'int': hex})
 
-EXTENDED_RELATION_LENGTH = 13
+EXTENDED_RELATION_LENGTH = 12
 
 class Row:
     def __init__(self, start):
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index 72f0815efb5..f460cd459b4 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -27,11 +27,12 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     using BaseUnivariate = Univariate<FF, ProverInstances::NUM>;
     // The length of ExtendedUnivariate is the largest length (==degree + 1) of a univariate polynomial obtained by
     // composing a relation with folded instance + challenge data.
-
-    using ExtendedUnivariate = Univariate<FF,
-                                          (Flavor::NUMBER_OF_SUBRELATIONS - 1) *
-                                                  (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) +
-                                              1>;
+    using ExtendedUnivariate = Univariate<FF, (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) + 1>;
+    using ExtendedUnivariateWithRandomization =
+        Univariate<FF,
+                   (Flavor::NUMBER_OF_SUBRELATIONS - 1) * (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) *
+                           (ProverInstances::NUM - 1) +
+                       1>;
 
     using ExtendedUnivariates = typename Flavor::template ProverUnivariates<ExtendedUnivariate::LENGTH>;
 
@@ -238,7 +239,8 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      * @brief Compute the combiner polynomial $G$ in the Protogalaxy paper.
      *
      */
-    ExtendedUnivariate compute_combiner(const ProverInstances& instances, const std::vector<FF> pow_betas_star)
+    ExtendedUnivariateWithRandomization compute_combiner(const ProverInstances& instances,
+                                                         const std::vector<FF> pow_betas_star)
     {
         size_t common_circuit_size = instances[0]->prover_polynomials._data[0].size();
 
@@ -256,7 +258,8 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         // Constuct univariate accumulator containers; one per thread
         std::vector<TupleOfTuplesOfUnivariates> thread_univariate_accumulators(num_threads);
         for (auto& accum : thread_univariate_accumulators) {
-            Utils::template zero_univariates<AlphaType>(accum);
+            // just normal relation lengths
+            Utils::zero_univariates(accum);
         }
 
         // Constuct extended univariates containers; one per thread
@@ -288,7 +291,25 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
             Utils::add_nested_tuples(univariate_accumulators, accumulators);
         }
         // Batch the univariate contributions from each sub-relation to obtain the round univariate
-        return Utils::template batch_over_relations<ExtendedUnivariate>(univariate_accumulators, instances.alpha);
+        return batch_over_relations(univariate_accumulators, instances.alpha);
+    }
+    static ExtendedUnivariateWithRandomization batch_over_relations(TupleOfTuplesOfUnivariates univariate_accumulators,
+                                                                    AlphaType alpha)
+    {
+        auto running_challenge = AlphaType(1);
+
+        auto result = ExtendedUnivariateWithRandomization(0);
+        auto scale_and_sum = [&]<size_t, size_t>(auto& element) {
+            auto extended = element.template extend_to<ProverInstances::BATCHED_EXTENDED_LENGTH>();
+            extended *= running_challenge;
+            running_challenge *= alpha;
+            result += extended;
+        };
+
+        Utils::apply_to_tuple_of_tuples(univariate_accumulators, scale_and_sum);
+        Utils::zero_univariates(univariate_accumulators);
+
+        return result;
     }
 
     /**
@@ -301,7 +322,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     static Univariate<FF,
                       (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) + 1,
                       ProverInstances::NUM>
-    compute_combiner_quotient(FF compressed_perturbator, ExtendedUnivariate combiner)
+    compute_combiner_quotient(FF compressed_perturbator, ExtendedUnivariateWithRandomization combiner)
     {
         std::array<FF, (Flavor::MAX_TOTAL_RELATION_LENGTH - 2) * (ProverInstances::NUM - 1)>
             combiner_quotient_evals = {};
@@ -362,7 +383,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
             accumulated_alpha.value_at(instance_idx) = instance->alpha;
             instance_idx++;
         }
-        instances.alpha = accumulated_alpha.template extend_to<ProverInstances::EXTENDED_LENGTH>();
+        instances.alpha = accumulated_alpha.template extend_to<ProverInstances::BATCHED_EXTENDED_LENGTH>();
     }
 };
 
diff --git a/barretenberg/cpp/src/barretenberg/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
index 2d863daef25..59f8e1cc622 100644
--- a/barretenberg/cpp/src/barretenberg/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
@@ -50,10 +50,10 @@ template <typename Flavor> class RelationUtils {
      *
      * @details After computing the round univariate, it is necessary to zero-out the accumulators used to compute it.
      */
-    template <typename ChallengeType> static void zero_univariates(auto& tuple)
+    static void zero_univariates(auto& tuple)
     {
         auto set_to_zero = []<size_t, size_t>(auto& element) {
-            std::fill(element.evaluations.begin(), element.evaluations.end(), ChallengeType(0));
+            std::fill(element.evaluations.begin(), element.evaluations.end(), FF(0));
         };
         apply_to_tuple_of_tuples(tuple, set_to_zero);
     }
@@ -65,8 +65,7 @@ template <typename Flavor> class RelationUtils {
      * @param challenge
      * @param current_scalar power of the challenge
      */
-    template <typename ChallengeType>
-    static void scale_univariates(auto& tuple, const ChallengeType& challenge, ChallengeType& current_scalar)
+    static void scale_univariates(auto& tuple, const FF& challenge, FF& current_scalar)
     {
 
         // these size_t are not necessary????????
@@ -160,20 +159,20 @@ template <typename Flavor> class RelationUtils {
      * @brief Given a tuple t = (t_0, t_1, ..., t_{NUM_SUBRELATIONS-1}) and a challenge α,
      * return t_0 + αt_1 + ... + α^{NUM_SUBRELATIONS-1}t_{NUM_SUBRELATIONS-1}).
      */
-    template <typename ExtendedUnivariate, typename ContainerOverSubrelations, typename ChallengeType>
+    template <typename ExtendedUnivariate, typename ContainerOverSubrelations>
     static ExtendedUnivariate batch_over_relations(
         ContainerOverSubrelations& univariate_accumulators,
-        ChallengeType& challenge,
+        FF& challenge,
         const std::optional<PowUnivariate<FF>>& pow_univariate = std::nullopt)
     {
-        auto running_challenge = ChallengeType(1);
+        auto running_challenge = FF(1);
         scale_univariates(univariate_accumulators, challenge, running_challenge);
 
         auto result = ExtendedUnivariate(0);
         extend_and_batch_univariates(univariate_accumulators, result, pow_univariate);
 
         // Reset all univariate accumulators to 0 before beginning accumulation in the next round
-        zero_univariates<ChallengeType>(univariate_accumulators);
+        zero_univariates(univariate_accumulators);
         return result;
     }
 
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
index 4c2562b8a1e..7f8141a0e3d 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
@@ -14,10 +14,11 @@ template <typename Flavor_, size_t NUM_> struct ProverInstances_ {
     using ArrayType = std::array<std::shared_ptr<Instance>, NUM_>;
     // The extended length here is the length of a composition of polynomials.
     // I changed this to NUM, it was NUM - 1
-    static constexpr size_t EXTENDED_LENGTH =
+    static constexpr size_t EXTENDED_LENGTH = (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (NUM - 1) + 1;
+    static constexpr size_t BATCHED_EXTENDED_LENGTH =
         (Flavor::NUMBER_OF_SUBRELATIONS - 1) * (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (NUM - 1) + 1;
     using RelationParameters = proof_system::RelationParameters<Univariate<FF, EXTENDED_LENGTH>>;
-    using AlphaType = Univariate<FF, EXTENDED_LENGTH>;
+    using AlphaType = Univariate<FF, BATCHED_EXTENDED_LENGTH>;
     ArrayType _data;
     RelationParameters relation_parameters;
     AlphaType alpha;

From 073db883c0a2df1c135f4a2ba5723a0829784fc1 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Fri, 17 Nov 2023 22:52:34 +0000
Subject: [PATCH 23/86] works

---
 .../protogalaxy/combiner.test.cpp             | 388 +++++++++---------
 .../protogalaxy/combiner_example_gen.py       |   2 +-
 .../protogalaxy/protogalaxy_prover.cpp        |   2 +-
 .../protogalaxy/protogalaxy_prover.hpp        |  31 +-
 .../cpp/src/barretenberg/relations/utils.hpp  |  69 +---
 .../sumcheck/instance/instances.hpp           |   3 +-
 .../barretenberg/sumcheck/sumcheck_round.hpp  |  58 ++-
 .../sumcheck/sumcheck_round.test.cpp          |   2 +-
 .../ultra_honk/protogalaxy.test.cpp           |  21 +
 9 files changed, 293 insertions(+), 283 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp
index 3416ca67f9e..dce50d0220c 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp
@@ -1,193 +1,195 @@
-// #include "barretenberg/flavor/ultra.hpp"
-// #include "barretenberg/honk/utils/testing.hpp"
-// #include "barretenberg/protogalaxy/protogalaxy_prover.hpp"
-// #include "barretenberg/relations/relation_parameters.hpp"
-// #include "barretenberg/sumcheck/instance/instances.hpp"
-// #include <gtest/gtest.h>
-
-// using namespace proof_system::honk;
-// namespace barretenberg::test_protogalaxy_prover {
-// using Flavor = proof_system::honk::flavor::Ultra;
-// using Polynomial = typename Flavor::Polynomial;
-// using FF = typename Flavor::FF;
-// using RelationParameters = proof_system::RelationParameters<FF>;
-
-// TEST(Protogalaxy, CombinerOn2Instances)
-// {
-//     constexpr size_t NUM_INSTANCES = 2;
-//     using ProverInstance = ProverInstance_<Flavor>;
-//     using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
-//     using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
-
-//     const auto restrict_to_standard_arithmetic_relation = [](auto& polys) {
-//         std::fill(polys.q_arith.begin(), polys.q_arith.end(), 1);
-//         std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
-//         std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
-//         std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
-//         std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
-//         std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
-//         std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
-//         std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
-//     };
-
-//     auto run_test = [&](bool is_random_input) {
-//         if (is_random_input) {
-//             std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-//             std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
-//             ProtoGalaxyProver prover;
-//             std::vector<FF> pow_betas = { FF(1), FF(2) };
-//             // auto alpha = FF(0); // focus on the arithmetic relation only
-
-//             for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-//                 auto instance = std::make_shared<ProverInstance>();
-//                 auto [storage, prover_polynomials] = proof_system::honk::get_sequential_prover_polynomials<Flavor>(
-//                     /*log_circuit_size=*/1, idx * 128);
-//                 restrict_to_standard_arithmetic_relation(prover_polynomials);
-//                 storage_arrays[idx] = std::move(storage);
-//                 instance->prover_polynomials = prover_polynomials;
-//                 instance_data[idx] = instance;
-//             }
-
-//             ProverInstances instances{ instance_data };
-
-//             auto result = prover.compute_combiner(instances, pow_betas);
-//             auto expected_result = barretenberg::Univariate<FF, 12>(std::array<FF, 12>{
-//                 87706,
-//                 13644570,
-//                 76451738,
-//                 226257946,
-//                 static_cast<uint64_t>(500811930),
-//                 static_cast<uint64_t>(937862426),
-//                 static_cast<uint64_t>(1575158170),
-//                 static_cast<uint64_t>(2450447898),
-//                 static_cast<uint64_t>(3601480346),
-//                 static_cast<uint64_t>(5066004250),
-//                 static_cast<uint64_t>(6881768346),
-//                 static_cast<uint64_t>(9086521370),
-//             });
-//             EXPECT_EQ(result, expected_result);
-//         } else {
-//             std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-//             std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
-//             ProtoGalaxyProver prover;
-//             std::vector<FF> pow_betas = { FF(1), FF(2) };
-
-//             for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-//                 auto instance = std::make_shared<ProverInstance>();
-//                 auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
-//                     /*log_circuit_size=*/1);
-//                 restrict_to_standard_arithmetic_relation(prover_polynomials);
-//                 storage_arrays[idx] = std::move(storage);
-//                 instance->prover_polynomials = prover_polynomials;
-//                 instance_data[idx] = instance;
-//             }
-
-//             ProverInstances instances{ instance_data };
-//             // instances->alpha = FF(0); // focus on the arithmetic relation only
-
-//             const auto create_add_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
-//                 polys.w_l[idx] = w_l;
-//                 polys.w_r[idx] = w_r;
-//                 polys.w_o[idx] = w_l + w_r;
-//                 polys.q_l[idx] = 1;
-//                 polys.q_r[idx] = 1;
-//                 polys.q_o[idx] = -1;
-//             };
-
-//             const auto create_mul_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
-//                 polys.w_l[idx] = w_l;
-//                 polys.w_r[idx] = w_r;
-//                 polys.w_o[idx] = w_l * w_r;
-//                 polys.q_m[idx] = 1;
-//                 polys.q_o[idx] = -1;
-//             };
-
-//             create_add_gate(instances[0]->prover_polynomials, 0, 1, 2);
-//             create_add_gate(instances[0]->prover_polynomials, 1, 0, 4);
-//             create_add_gate(instances[1]->prover_polynomials, 0, 3, 4);
-//             create_mul_gate(instances[1]->prover_polynomials, 1, 1, 4);
-
-//             restrict_to_standard_arithmetic_relation(instances[0]->prover_polynomials);
-//             restrict_to_standard_arithmetic_relation(instances[1]->prover_polynomials);
-
-//             /* Instance 0                                    Instance 1
-//                 w_l w_r w_o q_m q_l q_r q_o q_c               w_l w_r w_o q_m q_l q_r q_o q_c
-//                 1   2   3   0   1   1   -1  0                 3   4   7   0   1   1   -1  0
-//                 0   4   4   0   1   1   -1  0                 1   4   4   1   0   0   -1  0             */
-
-//             /* Lagrange-combined values, row index 0         Lagrange-combined values, row index 1
-//                 in    0    1    2    3    4    5    6        in    0    1    2    3    4    5    6
-//                 w_l   1    3    5    7    9   11   13        w_l   0    1    2    3    4    5    6
-//                 w_r   2    4    6    8   10   12   14        w_r   4    4    4    4    4    4    4
-//                 w_o   3    7   11   15   19   23   27        w_o   4    4    4    4    4    4    0
-//                 q_m   0    0    0    0    0    0    0        q_m   0    1    2    3    4    5    6
-//                 q_l   1    1    1    1    1    1    1        q_l   1    0   -1   -2   -3   -4   -5
-//                 q_r   1    1    1    1    1    1    1        q_r   1    0   -1   -2   -3   -4   -5
-//                 q_o  -1   -1   -1   -1   -1   -1   -1        q_o  -1   -1   -1   -1   -1   -1   -1
-//                 q_c   0    0    0    0    0    0    0        q_c   0    0    0    0    0    0    0
-
-//             relation value:
-//                       0    0    0    0    0    0    0              0    0    6   18   36   60   90      */
-
-//             auto result = prover.compute_combiner(instances, pow_betas);
-//             auto expected_result = barretenberg::Univariate<FF, 12>(
-//                 std::array<FF, 12>{ 0, 0, 12, 36, 72, 120, 180, 252, 336, 432, 540, 660 });
-
-//             EXPECT_EQ(result, expected_result);
-//         }
-//     };
-//     run_test(true);
-//     run_test(false);
-// };
-
-// // TEST(Protogalaxy, CombinerOn4Instances)
-// // {
-// //     constexpr size_t NUM_INSTANCES = 4;
-// //     using ProverInstance = ProverInstance_<Flavor>;
-// //     using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
-// //     using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
-
-// //     const auto zero_all_selectors = [](auto& polys) {
-// //         std::fill(polys.q_arith.begin(), polys.q_arith.end(), 0);
-// //         std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
-// //         std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
-// //         std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
-// //         std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
-// //         std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
-// //         std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
-// //         std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
-// //     };
-
-// //     auto run_test = [&]() {
-// //         std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-// //         std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
-// //         ProtoGalaxyProver prover;
-// //         std::vector<FF> pow_betas = { FF(1), FF(2) };
-
-// //         for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-// //             auto instance = std::make_shared<ProverInstance>();
-// //             auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
-// //                 /*log_circuit_size=*/1);
-// //             storage_arrays[idx] = std::move(storage);
-// //             instance->prover_polynomials = prover_polynomials;
-// //             instance_data[idx] = instance;
-// //         }
-
-// //         ProverInstances instances{ instance_data };
-// //         // instances.alpha = Univariate; // focus on the arithmetic relation only
-
-// //         zero_all_selectors(instances[0]->prover_polynomials);
-// //         zero_all_selectors(instances[1]->prover_polynomials);
-// //         zero_all_selectors(instances[2]->prover_polynomials);
-// //         zero_all_selectors(instances[3]->prover_polynomials);
-
-// //         auto result = prover.compute_combiner(instances, pow_betas);
-// //         std::array<FF, 34> zeroes;
-// //         std::fill(zeroes.begin(), zeroes.end(), 0);
-// //         auto expected_result = barretenberg::Univariate<FF, 34>(zeroes);
-// //         EXPECT_EQ(result, expected_result);
-// //     };
-// //     run_test();
-// // };
-
-// } // namespace barretenberg::test_protogalaxy_prover
+#include "barretenberg/flavor/ultra.hpp"
+#include "barretenberg/honk/utils/testing.hpp"
+#include "barretenberg/protogalaxy/protogalaxy_prover.hpp"
+#include "barretenberg/relations/relation_parameters.hpp"
+#include "barretenberg/sumcheck/instance/instances.hpp"
+#include <gtest/gtest.h>
+
+using namespace proof_system::honk;
+namespace barretenberg::test_protogalaxy_prover {
+using Flavor = proof_system::honk::flavor::Ultra;
+using Polynomial = typename Flavor::Polynomial;
+using FF = typename Flavor::FF;
+using RelationParameters = proof_system::RelationParameters<FF>;
+
+// TODO(https://github.com/AztecProtocol/barretenberg/issues/780): Improve combiner tests to check more than the
+// arithmetic relation so we more than unit test folding relation parameters and alpha as well.
+TEST(Protogalaxy, CombinerOn2Instances)
+{
+    constexpr size_t NUM_INSTANCES = 2;
+    using ProverInstance = ProverInstance_<Flavor>;
+    using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
+    using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
+
+    const auto restrict_to_standard_arithmetic_relation = [](auto& polys) {
+        std::fill(polys.q_arith.begin(), polys.q_arith.end(), 1);
+        std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
+        std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
+        std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
+        std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
+        std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
+        std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
+        std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
+    };
+
+    auto run_test = [&](bool is_random_input) {
+        if (is_random_input) {
+            std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+            std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
+            ProtoGalaxyProver prover;
+            std::vector<FF> pow_betas = { FF(1), FF(2) };
+
+            for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+                auto instance = std::make_shared<ProverInstance>();
+                auto [storage, prover_polynomials] = proof_system::honk::get_sequential_prover_polynomials<Flavor>(
+                    /*log_circuit_size=*/1, idx * 128);
+                restrict_to_standard_arithmetic_relation(prover_polynomials);
+                storage_arrays[idx] = std::move(storage);
+                instance->prover_polynomials = prover_polynomials;
+                instance_data[idx] = instance;
+            }
+
+            ProverInstances instances{ instance_data };
+            instances.alpha = Univariate<FF, 13>(FF(0)); // focus on the arithmetic relation only
+
+            auto result = prover.compute_combiner(instances, pow_betas);
+            auto expected_result =
+                barretenberg::Univariate<FF, 13>(std::array<FF, 13>{ 87706,
+                                                                     13644570,
+                                                                     76451738,
+                                                                     226257946,
+                                                                     static_cast<uint64_t>(500811930),
+                                                                     static_cast<uint64_t>(937862426),
+                                                                     static_cast<uint64_t>(1575158170),
+                                                                     static_cast<uint64_t>(2450447898),
+                                                                     static_cast<uint64_t>(3601480346),
+                                                                     static_cast<uint64_t>(5066004250),
+                                                                     static_cast<uint64_t>(6881768346),
+                                                                     static_cast<uint64_t>(9086521370),
+                                                                     static_cast<uint64_t>(11718012058) });
+            EXPECT_EQ(result, expected_result);
+        } else {
+            std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+            std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
+            ProtoGalaxyProver prover;
+            std::vector<FF> pow_betas = { FF(1), FF(2) };
+
+            for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+                auto instance = std::make_shared<ProverInstance>();
+                auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
+                    /*log_circuit_size=*/1);
+                restrict_to_standard_arithmetic_relation(prover_polynomials);
+                storage_arrays[idx] = std::move(storage);
+                instance->prover_polynomials = prover_polynomials;
+                instance_data[idx] = instance;
+            }
+
+            ProverInstances instances{ instance_data };
+            instances.alpha = Univariate<FF, 13>(FF(0)); // focus on the arithmetic relation only
+
+            const auto create_add_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
+                polys.w_l[idx] = w_l;
+                polys.w_r[idx] = w_r;
+                polys.w_o[idx] = w_l + w_r;
+                polys.q_l[idx] = 1;
+                polys.q_r[idx] = 1;
+                polys.q_o[idx] = -1;
+            };
+
+            const auto create_mul_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
+                polys.w_l[idx] = w_l;
+                polys.w_r[idx] = w_r;
+                polys.w_o[idx] = w_l * w_r;
+                polys.q_m[idx] = 1;
+                polys.q_o[idx] = -1;
+            };
+
+            create_add_gate(instances[0]->prover_polynomials, 0, 1, 2);
+            create_add_gate(instances[0]->prover_polynomials, 1, 0, 4);
+            create_add_gate(instances[1]->prover_polynomials, 0, 3, 4);
+            create_mul_gate(instances[1]->prover_polynomials, 1, 1, 4);
+
+            restrict_to_standard_arithmetic_relation(instances[0]->prover_polynomials);
+            restrict_to_standard_arithmetic_relation(instances[1]->prover_polynomials);
+
+            /* Instance 0                                    Instance 1
+                w_l w_r w_o q_m q_l q_r q_o q_c               w_l w_r w_o q_m q_l q_r q_o q_c
+                1   2   3   0   1   1   -1  0                 3   4   7   0   1   1   -1  0
+                0   4   4   0   1   1   -1  0                 1   4   4   1   0   0   -1  0             */
+
+            /* Lagrange-combined values, row index 0         Lagrange-combined values, row index 1
+                in    0    1    2    3    4    5    6        in    0    1    2    3    4    5    6
+                w_l   1    3    5    7    9   11   13        w_l   0    1    2    3    4    5    6
+                w_r   2    4    6    8   10   12   14        w_r   4    4    4    4    4    4    4
+                w_o   3    7   11   15   19   23   27        w_o   4    4    4    4    4    4    0
+                q_m   0    0    0    0    0    0    0        q_m   0    1    2    3    4    5    6
+                q_l   1    1    1    1    1    1    1        q_l   1    0   -1   -2   -3   -4   -5
+                q_r   1    1    1    1    1    1    1        q_r   1    0   -1   -2   -3   -4   -5
+                q_o  -1   -1   -1   -1   -1   -1   -1        q_o  -1   -1   -1   -1   -1   -1   -1
+                q_c   0    0    0    0    0    0    0        q_c   0    0    0    0    0    0    0
+
+            relation value:
+                      0    0    0    0    0    0    0              0    0    6   18   36   60   90      */
+
+            auto result = prover.compute_combiner(instances, pow_betas);
+            auto expected_result = barretenberg::Univariate<FF, 13>(
+                std::array<FF, 13>{ 0, 0, 12, 36, 72, 120, 180, 252, 336, 432, 540, 660, 792 });
+
+            EXPECT_EQ(result, expected_result);
+        }
+    };
+    run_test(true);
+    run_test(false);
+};
+
+TEST(Protogalaxy, CombinerOn4Instances)
+{
+    constexpr size_t NUM_INSTANCES = 4;
+    using ProverInstance = ProverInstance_<Flavor>;
+    using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
+    using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
+
+    const auto zero_all_selectors = [](auto& polys) {
+        std::fill(polys.q_arith.begin(), polys.q_arith.end(), 0);
+        std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
+        std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
+        std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
+        std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
+        std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
+        std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
+        std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
+    };
+
+    auto run_test = [&]() {
+        std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+        std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
+        ProtoGalaxyProver prover;
+        std::vector<FF> pow_betas = { FF(1), FF(2) };
+
+        for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+            auto instance = std::make_shared<ProverInstance>();
+            auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
+                /*log_circuit_size=*/1);
+            storage_arrays[idx] = std::move(storage);
+            instance->prover_polynomials = prover_polynomials;
+            instance_data[idx] = instance;
+        }
+
+        ProverInstances instances{ instance_data };
+        instances.alpha = Univariate<FF, 43>(FF(0)); // focus on the arithmetic relation only
+
+        zero_all_selectors(instances[0]->prover_polynomials);
+        zero_all_selectors(instances[1]->prover_polynomials);
+        zero_all_selectors(instances[2]->prover_polynomials);
+        zero_all_selectors(instances[3]->prover_polynomials);
+
+        auto result = prover.compute_combiner(instances, pow_betas);
+        std::array<FF, 43> zeroes;
+        std::fill(zeroes.begin(), zeroes.end(), 0);
+        auto expected_result = barretenberg::Univariate<FF, 43>(zeroes);
+        EXPECT_EQ(result, expected_result);
+    };
+    run_test();
+};
+
+} // namespace barretenberg::test_protogalaxy_prover
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/combiner_example_gen.py b/barretenberg/cpp/src/barretenberg/protogalaxy/combiner_example_gen.py
index be098938b59..906d1948847 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/combiner_example_gen.py
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/combiner_example_gen.py
@@ -3,7 +3,7 @@
 
 # np.set_printoptions(formatter={'int': hex})
 
-EXTENDED_RELATION_LENGTH = 12
+EXTENDED_RELATION_LENGTH = 13
 
 class Row:
     def __init__(self, start):
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index d505f73bab7..d52af9f2a84 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -31,7 +31,7 @@ template <class ProverInstances> void ProtoGalaxyProver_<ProverInstances>::prepa
     }
 
     fold_relation_parameters(instances);
-    accumulate_alpha(instances);
+    fold_alpha(instances);
 }
 
 // TODO(#https://github.com/AztecProtocol/barretenberg/issues/689): finalise implementation this function
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index c3445915fb1..9d703a3bb07 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -30,9 +30,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     using ExtendedUnivariate = Univariate<FF, (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) + 1>;
     using ExtendedUnivariateWithRandomization =
         Univariate<FF,
-                   (Flavor::NUMBER_OF_SUBRELATIONS - 1) * (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) *
-                           (ProverInstances::NUM - 1) +
-                       1>;
+                   (Flavor::MAX_TOTAL_RELATION_LENGTH - 1 + ProverInstances::NUM - 1) * (ProverInstances::NUM - 1) + 1>;
 
     using ExtendedUnivariates = typename Flavor::template ProverUnivariates<ExtendedUnivariate::LENGTH>;
 
@@ -297,17 +295,17 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     static ExtendedUnivariateWithRandomization batch_over_relations(TupleOfTuplesOfUnivariates univariate_accumulators,
                                                                     AlphaType alpha)
     {
-        auto running_challenge = AlphaType(1);
 
-        auto result = ExtendedUnivariateWithRandomization(0);
-        auto scale_and_sum = [&]<size_t, size_t>(auto& element) {
+        // First relation does not get multiplied by a batching challenge
+        auto result = std::get<0>(std::get<0>(univariate_accumulators))
+                          .template extend_to<ProverInstances::BATCHED_EXTENDED_LENGTH>();
+        auto scale_and_sum = [&]<size_t outer_idx, size_t>(auto& element) {
             auto extended = element.template extend_to<ProverInstances::BATCHED_EXTENDED_LENGTH>();
-            extended *= running_challenge;
-            running_challenge *= alpha;
+            extended *= alpha;
             result += extended;
         };
 
-        Utils::apply_to_tuple_of_tuples(univariate_accumulators, scale_and_sum);
+        Utils::template apply_to_tuple_of_tuples<0, 1>(univariate_accumulators, scale_and_sum);
         Utils::zero_univariates(univariate_accumulators);
 
         return result;
@@ -320,13 +318,10 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      * polynomials and Lagrange basis and use batch_invert.
      *
      */
-    static Univariate<FF,
-                      (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) + 1,
-                      ProverInstances::NUM>
-    compute_combiner_quotient(FF compressed_perturbator, ExtendedUnivariateWithRandomization combiner)
+    static Univariate<FF, ProverInstances::BATCHED_EXTENDED_LENGTH, ProverInstances::NUM> compute_combiner_quotient(
+        FF compressed_perturbator, ExtendedUnivariateWithRandomization combiner)
     {
-        std::array<FF, (Flavor::MAX_TOTAL_RELATION_LENGTH - 2) * (ProverInstances::NUM - 1)>
-            combiner_quotient_evals = {};
+        std::array<FF, ProverInstances::BATCHED_EXTENDED_LENGTH - ProverInstances::NUM> combiner_quotient_evals = {};
 
         // Compute the combiner quotient polynomial as evaluations on points that are not in the vanishing set.
         //
@@ -339,8 +334,8 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
                 (combiner.value_at(point) - compressed_perturbator * lagrange_0) * vanishing_polynomial.invert();
         }
 
-        Univariate<FF, (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (ProverInstances::NUM - 1) + 1, ProverInstances::NUM>
-            combiner_quotient(combiner_quotient_evals);
+        Univariate<FF, ProverInstances::BATCHED_EXTENDED_LENGTH, ProverInstances::NUM> combiner_quotient(
+            combiner_quotient_evals);
         return combiner_quotient;
     }
 
@@ -376,7 +371,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      */
 
     // THIS SHOULD BE CALLED ACCUMULATED
-    static void accumulate_alpha(ProverInstances& instances)
+    static void fold_alpha(ProverInstances& instances)
     {
         Univariate<FF, ProverInstances::NUM> accumulated_alpha;
         size_t instance_idx = 0;
diff --git a/barretenberg/cpp/src/barretenberg/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
index 303b160ce28..e20213f6287 100644
--- a/barretenberg/cpp/src/barretenberg/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
@@ -25,7 +25,7 @@ template <typename Flavor> class RelationUtils {
      * @param tuple A Tuple of tuples of Univariates
      * @param operation Operation to apply to Univariates
      */
-    template <class Operation, size_t outer_idx = 0, size_t inner_idx = 0>
+    template <size_t outer_idx = 0, size_t inner_idx = 0, class Operation>
     static void apply_to_tuple_of_tuples(auto& tuple, Operation&& operation)
     {
         auto& inner_tuple = std::get<outer_idx>(tuple);
@@ -39,9 +39,9 @@ template <typename Flavor> class RelationUtils {
 
         // Recurse over inner and outer tuples
         if constexpr (inner_idx + 1 < inner_size) {
-            apply_to_tuple_of_tuples<Operation, outer_idx, inner_idx + 1>(tuple, std::forward<Operation>(operation));
+            apply_to_tuple_of_tuples<outer_idx, inner_idx + 1, Operation>(tuple, std::forward<Operation>(operation));
         } else if constexpr (outer_idx + 1 < outer_size) {
-            apply_to_tuple_of_tuples<Operation, outer_idx + 1, 0>(tuple, std::forward<Operation>(operation));
+            apply_to_tuple_of_tuples<outer_idx + 1, 0, Operation>(tuple, std::forward<Operation>(operation));
         }
     }
 
@@ -67,8 +67,6 @@ template <typename Flavor> class RelationUtils {
      */
     static void scale_univariates(auto& tuple, const FF& challenge, FF& current_scalar)
     {
-
-        // these size_t are not necessary????????
         auto scale_by_consecutive_powers_of_challenge = [&]<size_t, size_t>(auto& element) {
             element *= current_scalar;
             current_scalar *= challenge;
@@ -115,67 +113,6 @@ template <typename Flavor> class RelationUtils {
         }
     }
 
-    /**
-     * @brief Extend Univariates to specified size then sum them
-     *
-     * @tparam extended_size Size after extension
-     * @param tuple A tuple of tuples of Univariates
-     * @param result A Univariate of length extended_size
-     * @param pow_univariate Power polynomial univariate. Optional because this concept is only used in sumcheck. When
-     * using this function in the context of Protogalaxy, the contribution of the pow polynomial has already been added
-     * to the result as a scalar.
-     */
-    template <typename ExtendedUnivariate, typename TupleOfTuplesOfUnivariates>
-    static void extend_and_batch_univariates(const TupleOfTuplesOfUnivariates& tuple,
-                                             ExtendedUnivariate& result,
-                                             const std::optional<PowUnivariate<FF>>& pow_univariate = std::nullopt)
-    {
-        ExtendedUnivariate extended_random_polynomial;
-        // Random poly R(X) = (1-X) + X.zeta_pow
-        if (pow_univariate.has_value()) {
-            auto random_polynomial = Univariate<FF, 2>({ 1, pow_univariate.value().zeta_pow });
-            extended_random_polynomial = random_polynomial.template extend_to<ExtendedUnivariate::LENGTH>();
-        }
-
-        auto extend_and_sum = [&]<size_t relation_idx, size_t subrelation_idx, typename Element>(Element& element) {
-            auto extended = element.template extend_to<ExtendedUnivariate::LENGTH>();
-
-            using Relation = typename std::tuple_element_t<relation_idx, Relations>;
-            const bool is_subrelation_linearly_independent =
-                proof_system::subrelation_is_linearly_independent<Relation, subrelation_idx>();
-            // Except from the log derivative subrelation, each other subrelation in part is required to be 0 hence we
-            // multiply by the power polynomial. As the sumcheck prover is required to send a univariate to the
-            // verifier, we additionally need a univariate contribution from the pow polynomial.
-            if (!is_subrelation_linearly_independent || !pow_univariate.has_value()) {
-                result += extended;
-            } else {
-                result += extended * extended_random_polynomial;
-            }
-        };
-        apply_to_tuple_of_tuples(tuple, extend_and_sum);
-    }
-
-    /**
-     * @brief Given a tuple t = (t_0, t_1, ..., t_{NUM_SUBRELATIONS-1}) and a challenge α,
-     * return t_0 + αt_1 + ... + α^{NUM_SUBRELATIONS-1}t_{NUM_SUBRELATIONS-1}).
-     */
-    template <typename ExtendedUnivariate, typename ContainerOverSubrelations>
-    static ExtendedUnivariate batch_over_relations(
-        ContainerOverSubrelations& univariate_accumulators,
-        const FF& challenge,
-        const std::optional<PowUnivariate<FF>>& pow_univariate = std::nullopt)
-    {
-        auto running_challenge = FF(1);
-        scale_univariates(univariate_accumulators, challenge, running_challenge);
-
-        auto result = ExtendedUnivariate(0);
-        extend_and_batch_univariates(univariate_accumulators, result, pow_univariate);
-
-        // Reset all univariate accumulators to 0 before beginning accumulation in the next round
-        zero_univariates(univariate_accumulators);
-        return result;
-    }
-
     /**
      * @brief Calculate the contribution of each relation to the expected value of the full Honk relation.
      *
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
index 6bab1c95480..af3df6989fc 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
@@ -16,8 +16,7 @@ template <typename Flavor_, size_t NUM_> struct ProverInstances_ {
     // The extended length here is the length of a composition of polynomials.
     // I changed this to NUM, it was NUM - 1
     static constexpr size_t EXTENDED_LENGTH = (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (NUM - 1) + 1;
-    static constexpr size_t BATCHED_EXTENDED_LENGTH =
-        (Flavor::NUMBER_OF_SUBRELATIONS - 1) * (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (NUM - 1) + 1;
+    static constexpr size_t BATCHED_EXTENDED_LENGTH = (Flavor::MAX_TOTAL_RELATION_LENGTH - 1 + NUM - 1) * (NUM - 1) + 1;
     using RelationParameters = proof_system::RelationParameters<Univariate<FF, EXTENDED_LENGTH>>;
     using AlphaType = Univariate<FF, BATCHED_EXTENDED_LENGTH>;
     ArrayType _data;
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
index 5aafd72a24c..b25956c9f27 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
@@ -164,10 +164,66 @@ template <typename Flavor> class SumcheckProverRound {
             Utils::add_nested_tuples(univariate_accumulators, accumulators);
         }
         // Batch the univariate contributions from each sub-relation to obtain the round univariate
-        return Utils::template batch_over_relations<barretenberg::Univariate<FF, BATCHED_RELATION_PARTIAL_LENGTH>>(
+        return batch_over_relations<barretenberg::Univariate<FF, BATCHED_RELATION_PARTIAL_LENGTH>>(
             univariate_accumulators, alpha, pow_univariate);
     }
 
+    /**
+     * @brief Given a tuple t = (t_0, t_1, ..., t_{NUM_SUBRELATIONS-1}) and a challenge α,
+     * return t_0 + αt_1 + ... + α^{NUM_SUBRELATIONS-1}t_{NUM_SUBRELATIONS-1}).
+     */
+    template <typename ExtendedUnivariate, typename ContainerOverSubrelations>
+    static ExtendedUnivariate batch_over_relations(ContainerOverSubrelations& univariate_accumulators,
+                                                   const FF& challenge,
+                                                   const barretenberg::PowUnivariate<FF>& pow_univariate)
+    {
+        auto running_challenge = FF(1);
+        Utils::scale_univariates(univariate_accumulators, challenge, running_challenge);
+
+        auto result = ExtendedUnivariate(0);
+        extend_and_batch_univariates(univariate_accumulators, result, pow_univariate);
+
+        // Reset all univariate accumulators to 0 before beginning accumulation in the next round
+        Utils::zero_univariates(univariate_accumulators);
+        return result;
+    }
+
+    /**
+     * @brief Extend Univariates to specified size then sum them
+     *
+     * @tparam extended_size Size after extension
+     * @param tuple A tuple of tuples of Univariates
+     * @param result A Univariate of length extended_size
+     * @param pow_univariate Power polynomial univariate
+     */
+    template <typename ExtendedUnivariate, typename TupleOfTuplesOfUnivariates>
+    static void extend_and_batch_univariates(const TupleOfTuplesOfUnivariates& tuple,
+                                             ExtendedUnivariate& result,
+                                             const barretenberg::PowUnivariate<FF>& pow_univariate)
+    {
+        ExtendedUnivariate extended_random_polynomial;
+        // Random poly R(X) = (1-X) + X.zeta_pow
+        auto random_polynomial = barretenberg::Univariate<FF, 2>({ 1, pow_univariate.zeta_pow });
+        extended_random_polynomial = random_polynomial.template extend_to<ExtendedUnivariate::LENGTH>();
+
+        auto extend_and_sum = [&]<size_t relation_idx, size_t subrelation_idx, typename Element>(Element& element) {
+            auto extended = element.template extend_to<ExtendedUnivariate::LENGTH>();
+
+            using Relation = typename std::tuple_element_t<relation_idx, Relations>;
+            const bool is_subrelation_linearly_independent =
+                proof_system::subrelation_is_linearly_independent<Relation, subrelation_idx>();
+            // Except from the log derivative subrelation, each other subrelation in part is required to be 0 hence we
+            // multiply by the power polynomial. As the sumcheck prover is required to send a univariate to the
+            // verifier, we additionally need a univariate contribution from the pow polynomial.
+            if (!is_subrelation_linearly_independent) {
+                result += extended;
+            } else {
+                result += extended * extended_random_polynomial;
+            }
+        };
+        Utils::apply_to_tuple_of_tuples(tuple, extend_and_sum);
+    }
+
   private:
     /**
      * @brief For a given edge, calculate the contribution of each relation to the prover round univariate (S_l in the
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.test.cpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.test.cpp
index 1f4a9021383..2856012b163 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.test.cpp
@@ -43,7 +43,7 @@ TEST(SumcheckRound, SumcheckTupleOfTuplesOfUnivariates)
     // Use extend_and_batch_univariates to extend to MAX_LENGTH then accumulate
     PowUnivariate<FF> pow_univariate(1);
     auto result = Univariate<FF, MAX_LENGTH>();
-    Utils::extend_and_batch_univariates(tuple_of_tuples, result, pow_univariate);
+    SumcheckProverRound<Flavor>::extend_and_batch_univariates(tuple_of_tuples, result, pow_univariate);
 
     // Repeat the batching process manually
     auto result_expected = univariate_1.template extend_to<MAX_LENGTH>() * 1 +
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
index 2bc682db59e..fac96d16cc6 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
@@ -231,4 +231,25 @@ TEST_F(ProtoGalaxyTests, FoldChallenges)
     EXPECT_EQ(instances.relation_parameters.eta, expected_eta);
 }
 
+TEST_F(ProtoGalaxyTests, FoldAlpha)
+{
+    using Instances = ProverInstances_<Flavor, 2>;
+    using Instance = typename Instances::Instance;
+
+    Builder builder1;
+    auto instance1 = std::make_shared<Instance>(builder1);
+    instance1->alpha = 2;
+
+    Builder builder2;
+    builder2.add_variable(3);
+    auto instance2 = std::make_shared<Instance>(builder2);
+    instance2->alpha = 4;
+
+    Instances instances{ { instance1, instance2 } };
+    ProtoGalaxyProver::fold_alpha(instances);
+
+    Univariate<FF, 13> expected_alpha{ { 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26 } };
+    EXPECT_EQ(instances.alpha, expected_alpha);
+}
+
 } // namespace protogalaxy_tests
\ No newline at end of file

From 348f561007e35a1ee2d51463bca3f104431803d2 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Fri, 17 Nov 2023 22:59:48 +0000
Subject: [PATCH 24/86] more naming

---
 barretenberg/cpp/src/barretenberg/flavor/flavor.hpp   |  4 +---
 .../cpp/src/barretenberg/polynomials/univariate.hpp   |  2 +-
 .../barretenberg/protogalaxy/protogalaxy_prover.hpp   | 11 +++++------
 .../barretenberg/protogalaxy/protogalaxy_verifier.cpp |  8 ++++----
 barretenberg/cpp/src/barretenberg/relations/utils.hpp |  3 ++-
 .../src/barretenberg/sumcheck/instance/instances.hpp  |  1 +
 .../sumcheck/instance/prover_instance.hpp             |  1 -
 .../sumcheck/instance/verifier_instance.hpp           |  1 -
 8 files changed, 14 insertions(+), 17 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/flavor/flavor.hpp b/barretenberg/cpp/src/barretenberg/flavor/flavor.hpp
index ec3d302d523..4f9f1d515b9 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/flavor.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/flavor.hpp
@@ -244,9 +244,7 @@ template <typename Tuple, std::size_t Index = 0> static constexpr size_t compute
 }
 
 /**
- * @brief Recursive utility function to find max TOTAL_RELATION_LENGTH among tuples of Relations.
- * @details The "total length" of a relation is 1 + the degree of the relation, where any challenges used in the
- * relation are regarded as variables.
+ * @brief Recursive utility function to find the number of subrelations.
  *
  */
 template <typename Tuple, std::size_t Index = 0> static constexpr size_t compute_number_of_subrelations()
diff --git a/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp b/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp
index 32a45c26966..a3770cad6b9 100644
--- a/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp
+++ b/barretenberg/cpp/src/barretenberg/polynomials/univariate.hpp
@@ -38,7 +38,7 @@ template <class Fr, size_t domain_end, size_t domain_start = 0> class Univariate
     Univariate(Univariate&& other) noexcept = default;
     Univariate& operator=(const Univariate& other) = default;
     Univariate& operator=(Univariate&& other) noexcept = default;
-    // Construct constant Univariate from scalar which represents the value all the point in the domain should evalute
+    // Construct constant Univariate from scalar which represents the value that all the points in the domain evaluate
     // to
     explicit Univariate(Fr value)
         : evaluations{}
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index 9d703a3bb07..e0961905d97 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -22,7 +22,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     using RowEvaluations = typename Flavor::AllValues;
     using ProverPolynomials = typename Flavor::ProverPolynomials;
     using Relations = typename Flavor::Relations;
-    using AlphaType = ProverInstances::AlphaType;
+    using AlphaType = typename ProverInstances::AlphaType;
 
     using BaseUnivariate = Univariate<FF, ProverInstances::NUM>;
     // The length of ExtendedUnivariate is the largest length (==degree + 1) of a univariate polynomial obtained by
@@ -345,8 +345,6 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      * univariate (i.e., sum them against an appropriate univariate Lagrange basis) and then extended as needed during
      * the constuction of the combiner.
      */
-
-    // THIS SHOULD BE CALLED ACCUMULATED
     static void fold_relation_parameters(ProverInstances& instances)
     {
         // array of parameters to be computed
@@ -355,7 +353,6 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         for (auto& folded_parameter : folded_parameters) {
             Univariate<FF, ProverInstances::NUM> tmp(0);
             size_t instance_idx = 0;
-            // it's not wrong i'm just dumb
             for (auto& instance : instances) {
                 tmp.value_at(instance_idx) = instance->relation_parameters.to_fold[param_idx];
                 instance_idx++;
@@ -369,8 +366,10 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      * @brief Create folded univariate for the relation batching parameter (alpha).
      *
      */
-
-    // THIS SHOULD BE CALLED ACCUMULATED
+    // TODO(https://github.com/AztecProtocol/barretenberg/issues/772): At the moment we have a single α per Instance, we
+    // fold them and then we use the unique folded_α for each folded subrelation that is batched in the combiner. This
+    // is obviously insecure. We need to generate α_i for each subrelation_i, fold them and then use folded_α_i when
+    // batching the i-th folded subrelation in the combiner.
     static void fold_alpha(ProverInstances& instances)
     {
         Univariate<FF, ProverInstances::NUM> accumulated_alpha;
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index f26b59028eb..ed6441dae39 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -52,13 +52,13 @@ VerifierFoldingResult<typename VerifierInstances::Flavor> ProtoGalaxyVerifier_<
     auto perturbator_at_challenge = perturbator.evaluate(perturbator_challenge);
 
     // Thed degree of K(X) is dk - k - 1 = k(d - 1) - 1. Hence we need  k(d - 1) evaluations to represent it.
-    std::array<FF, (Flavor::MAX_TOTAL_RELATION_LENGTH - 2) * (VerifierInstances::NUM - 1)> combiner_quotient_evals = {};
-    for (size_t idx = 0; idx < (Flavor::MAX_TOTAL_RELATION_LENGTH - 2) * (VerifierInstances::NUM - 1); idx++) {
+    std::array<FF, VerifierInstances::BATCHED_EXTENDED_LENGTH - VerifierInstances::NUM> combiner_quotient_evals = {};
+    for (size_t idx = 0; idx < VerifierInstances::BATCHED_EXTENDED_LENGTH - VerifierInstances::NUM; idx++) {
         combiner_quotient_evals[idx] = transcript.template receive_from_prover<FF>(
             "combiner_quotient_" + std::to_string(idx + VerifierInstances::NUM));
     }
-    Univariate<FF, (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (VerifierInstances::NUM - 1) + 1, VerifierInstances::NUM>
-        combiner_quotient(combiner_quotient_evals);
+    Univariate<FF, VerifierInstances::BATCHED_EXTENDED_LENGTH, VerifierInstances::NUM> combiner_quotient(
+        combiner_quotient_evals);
     auto combiner_challenge = transcript.get_challenge("combiner_quotient_challenge");
     auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge);
 
diff --git a/barretenberg/cpp/src/barretenberg/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
index e20213f6287..6b52f317db7 100644
--- a/barretenberg/cpp/src/barretenberg/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
@@ -86,7 +86,8 @@ template <typename Flavor> class RelationUtils {
     template <typename... T>
     static constexpr void add_tuples(std::tuple<T...>& tuple_1, const std::tuple<T...>& tuple_2)
     {
-        auto add_tuples_helper = [&]<std::size_t... I>(std::index_sequence<I...>) {
+        auto add_tuples_helper = [&]<std::size_t... I>(std::index_sequence<I...>)
+        {
             ((std::get<I>(tuple_1) += std::get<I>(tuple_2)), ...);
         };
 
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
index af3df6989fc..ab67af98d4a 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
@@ -93,6 +93,7 @@ template <typename Flavor_, size_t NUM_> struct VerifierInstances_ {
 
   public:
     static constexpr size_t NUM = NUM_;
+    static constexpr size_t BATCHED_EXTENDED_LENGTH = (Flavor::MAX_TOTAL_RELATION_LENGTH - 1 + NUM - 1) * (NUM - 1) + 1;
     ArrayType _data;
     std::shared_ptr<Instance> const& operator[](size_t idx) const { return _data[idx]; }
     typename ArrayType::iterator begin() { return _data.begin(); };
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
index 797b983177a..c72e940a206 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
@@ -42,7 +42,6 @@ template <class Flavor> class ProverInstance_ {
     // non-zero  for Instances constructed from circuits, this concept doesn't exist for accumulated
     // instances
     size_t pub_inputs_offset = 0;
-    // I really wanna rename this relation_batching_challenge
     FF alpha;
     proof_system::RelationParameters<FF> relation_parameters;
     std::vector<uint32_t> recursive_proof_public_input_indices;
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
index dc4e6389b90..fb14bd32b8b 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
@@ -14,7 +14,6 @@ template <class Flavor> class VerifierInstance_ {
     size_t pub_inputs_offset;
     size_t public_input_size;
     size_t instance_size;
-    // Is this fine?
     RelationParameters<FF> relation_parameters;
     FF alpha;
     FoldingParameters folding_parameters;

From c380cea9f3f97b3e716686ead61ee4c5cda7f7cd Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Sat, 18 Nov 2023 22:39:44 +0000
Subject: [PATCH 25/86] fix sumcheck test that was failing

---
 .../cpp/src/barretenberg/sumcheck/sumcheck.test.cpp    | 10 ++++------
 1 file changed, 4 insertions(+), 6 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp
index 1784beee2b8..33864bb8442 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp
@@ -249,18 +249,16 @@ TEST_F(SumcheckTests, ProverAndVerifierSimple)
         };
 
         Flavor::Transcript prover_transcript = Flavor::Transcript::prover_init_empty();
-        info(Flavor::NUMBER_OF_SUBRELATIONS);
-        info(Flavor::MAX_TOTAL_RELATION_LENGTH);
         auto sumcheck_prover = SumcheckProver<Flavor>(multivariate_n, prover_transcript);
 
-        auto alpha = prover_transcript.get_challenge("alpha");
-        auto output = sumcheck_prover.prove(full_polynomials, {}, alpha);
+        auto prover_alpha = prover_transcript.get_challenge("alpha");
+        auto output = sumcheck_prover.prove(full_polynomials, {}, prover_alpha);
 
         Flavor::Transcript verifier_transcript = Flavor::Transcript::verifier_init_empty(prover_transcript);
 
         auto sumcheck_verifier = SumcheckVerifier<Flavor>(multivariate_n);
-
-        auto verifier_output = sumcheck_verifier.verify(relation_parameters, alpha, verifier_transcript);
+        auto verifier_alpha = verifier_transcript.get_challenge("alpha");
+        auto verifier_output = sumcheck_verifier.verify(relation_parameters, verifier_alpha, verifier_transcript);
 
         auto verified = verifier_output.verified.value();
 

From 9d16416e1aa9078d0acccca1cf42b0c8b368eab3 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Mon, 20 Nov 2023 13:56:01 +0000
Subject: [PATCH 26/86] stuff

---
 .../protogalaxy/combiner_example_gen.py       |  8 -----
 .../protogalaxy/protogalaxy_prover.cpp        |  9 +++---
 .../protogalaxy/protogalaxy_prover.hpp        |  4 ---
 .../protogalaxy/protogalaxy_verifier.cpp      | 31 +++++++++----------
 4 files changed, 20 insertions(+), 32 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/combiner_example_gen.py b/barretenberg/cpp/src/barretenberg/protogalaxy/combiner_example_gen.py
index f65ab68e97f..ef5a9d61b20 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/combiner_example_gen.py
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/combiner_example_gen.py
@@ -105,10 +105,6 @@ def compute_first_example():
         zeta_pow *= zeta
         print(zeta_pow)
 
-<<<<<<< HEAD:barretenberg/cpp/src/barretenberg/honk/proof_system/combiner_example_gen.py
-    # accumulator *= extend_one_entity([1, 2])
-=======
->>>>>>> origin/mm/pg-fold-alpha:barretenberg/cpp/src/barretenberg/protogalaxy/combiner_example_gen.py
     return accumulator
 
 
@@ -138,10 +134,6 @@ def compute_second_example():
         result += rel(w_l, w_r, w_o, q_m, q_l, q_r, q_o, q_c)
         result *= 2
 
-<<<<<<< HEAD:barretenberg/cpp/src/barretenberg/honk/proof_system/combiner_example_gen.py
-        # result *= extend_one_entity([1, 2])
-=======
->>>>>>> origin/mm/pg-fold-alpha:barretenberg/cpp/src/barretenberg/protogalaxy/combiner_example_gen.py
         return result
 
 if __name__ == "__main__":
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index 9539b1f26c2..651206d9412 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -80,8 +80,9 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
     // auto lagrange_1_at_challenge = combiner_challenge;
     auto new_target_sum = compressed_perturbator * lagrange_0_at_challenge +
                           vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
-
-    // for (size_t idx = 0; idx < accumulator->prover_polynomials.size(); idx++) {
+    // auto accumulator_prover_polynomials_view = accumulator->prover_polynomials.pointer_view();
+    // auto instance_prover_polynomials_view = instances[0]->prover_polynomials.pointer_view();
+    // for (size_t idx = 0; idx < a; idx++) {
     //     auto accumulator_poly = accumulator->prover_polynomials[idx];
     //     auto instance_poly = accumulator->prover_polynomials[idx];
     //     assert(accumulator_poly.size() == instance_poly.size());
@@ -94,8 +95,8 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
     ProverFoldingResult<Flavor> res;
     res.params.target_sum = new_target_sum;
     res.folding_data = transcript.proof_data;
-    // res.folded_prover_polynomials = accumulator->prover_polynomials;
-    // res.params.gate_separation_challenges = betas_star;
+    res.folded_prover_polynomials = accumulator->prover_polynomials;
+    res.params.gate_separation_challenges = betas_star;
     return res;
 }
 template class ProtoGalaxyProver_<ProverInstances_<honk::flavor::Ultra, 2>>;
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index fd605180db0..e0961905d97 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -54,14 +54,10 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      */
     void prepare_for_folding();
 
-<<<<<<< HEAD:barretenberg/cpp/src/barretenberg/honk/proof_system/protogalaxy_prover.hpp
-    static std::vector<FF> compute_pow_polynomial_at_values(const std::vector<FF> betas, const size_t instance_size)
-=======
     /**
      * @brief Given a vector \vec{\beta} of values, compute the pow polynomial on these values as defined in the paper.
      */
     static std::vector<FF> compute_pow_polynomial_at_values(const std::vector<FF>& betas, const size_t instance_size)
->>>>>>> origin/mm/pg-fold-alpha:barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
     {
         std::vector<FF> pow_betas(instance_size);
         for (size_t i = 0; i < instance_size; i++) {
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index 1a603349c1d..5f0709041e2 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -61,29 +61,28 @@ VerifierFoldingResult<typename VerifierInstances::Flavor> ProtoGalaxyVerifier_<
         combiner_quotient_evals);
     auto combiner_challenge = transcript.get_challenge("combiner_quotient_challenge");
     auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge);
-
     auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
     auto lagrange_0_at_challenge = FF(1) - combiner_challenge;
-    // auto lagrange_1_at_challenge = combiner_challenge;
+    auto lagrange_1_at_challenge = combiner_challenge;
 
     auto new_target_sum = perturbator_at_challenge * lagrange_0_at_challenge +
                           vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
 
-    // auto accumulator_public_inputs = accumulator->public_inputs;
-    // auto other_public_inputs = verifier_instances[1]->public_inputs;
-    // std::vector<FF> folded_public_inputs(accumulator->public_input_size);
-    // // extend either with 0s if the sizes are different
-    // for (size_t idx = 0; idx < folded_public_inputs.size(); idx++) {
-    //     folded_public_inputs[idx] = lagrange_0_at_challenge * accumulator->public_inputs[idx] +
-    //                                 lagrange_1_at_challenge * other_public_inputs[idx];
-    // }
-    // // all verification keys have the same size
-    // auto folded_verification_key = VerificationKey(accumulator->instance_size, accumulator->public_input_size);
+    auto accumulator_public_inputs = accumulator->public_inputs;
+    auto other_public_inputs = verifier_instances[1]->public_inputs;
+    std::vector<FF> folded_public_inputs(accumulator->public_input_size);
+    // extend either with 0s if the sizes are different
+    for (size_t idx = 0; idx < folded_public_inputs.size(); idx++) {
+        folded_public_inputs[idx] = lagrange_0_at_challenge * accumulator->public_inputs[idx] +
+                                    lagrange_1_at_challenge * other_public_inputs[idx];
+    }
+    // all verification keys have the same size
+    auto folded_verification_key = VerificationKey(accumulator->instance_size, accumulator->public_input_size);
 
-    // for (size_t idx = 0; idx < folded_verification_key.size(); idx++) {
-    //     folded_verification_key[idx] = (*accumulator->verification_key)[idx] * lagrange_0_at_challenge +
-    //                                    (*verifier_instances[1]->verification_key)[idx] * lagrange_1_at_challenge;
-    // }
+    for (size_t idx = 0; idx < folded_verification_key.size(); idx++) {
+        folded_verification_key[idx] = (*accumulator->verification_key)[idx] * lagrange_0_at_challenge +
+                                       (*verifier_instances[1]->verification_key)[idx] * lagrange_1_at_challenge;
+    }
     VerifierFoldingResult<Flavor> res;
     res.parameters.target_sum = new_target_sum;
     return res;

From 0472c1015e32c9e733ceead267ff494b9a58dacf Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Mon, 20 Nov 2023 16:25:15 +0000
Subject: [PATCH 27/86] confusion

---
 .../protogalaxy/protogalaxy_prover.cpp        | 23 +++++++++----------
 1 file changed, 11 insertions(+), 12 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index 651206d9412..be1dce3edec 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -77,20 +77,19 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
     auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
     auto lagrange_0_at_challenge = FF(1) - combiner_challenge;
 
-    // auto lagrange_1_at_challenge = combiner_challenge;
+    auto lagrange_1_at_challenge = combiner_challenge;
     auto new_target_sum = compressed_perturbator * lagrange_0_at_challenge +
                           vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
-    // auto accumulator_prover_polynomials_view = accumulator->prover_polynomials.pointer_view();
-    // auto instance_prover_polynomials_view = instances[0]->prover_polynomials.pointer_view();
-    // for (size_t idx = 0; idx < a; idx++) {
-    //     auto accumulator_poly = accumulator->prover_polynomials[idx];
-    //     auto instance_poly = accumulator->prover_polynomials[idx];
-    //     assert(accumulator_poly.size() == instance_poly.size());
-    //     for (size_t j = 0; j < accumulator_poly.size(); j++) {
-    //         accumulator_poly[j] =
-    //             accumulator_poly[j] * lagrange_0_at_challenge + instance_poly * lagrange_1_at_challenge;
-    //     }
-    // }
+
+    for (size_t idx = 0; idx < accumulator->prover_polynomials.size(); idx++) {
+        auto accumulator_poly = accumulator->prover_polynomials[idx];
+        auto instance_poly = accumulator->prover_polynomials[idx];
+        assert(accumulator_poly.size() == instance_poly.size());
+        for (size_t j = 0; j < accumulator_poly.size(); j++) {
+            accumulator_poly[j] =
+                accumulator_poly[j] * lagrange_0_at_challenge + instance_poly * lagrange_1_at_challenge;
+        }
+    }
 
     ProverFoldingResult<Flavor> res;
     res.params.target_sum = new_target_sum;

From 2ae95d2ea375976870f06161d9ded7cd46841a11 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Tue, 21 Nov 2023 08:52:14 +0000
Subject: [PATCH 28/86] stuff with pointer view

---
 .../protogalaxy/protogalaxy_prover.cpp         | 12 +++++-------
 .../protogalaxy/protogalaxy_verifier.cpp       | 13 ++++++++-----
 .../sumcheck/instance/instances.hpp            | 18 +++++++++---------
 3 files changed, 22 insertions(+), 21 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index be1dce3edec..d7937f29b22 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -81,13 +81,11 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
     auto new_target_sum = compressed_perturbator * lagrange_0_at_challenge +
                           vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
 
-    for (size_t idx = 0; idx < accumulator->prover_polynomials.size(); idx++) {
-        auto accumulator_poly = accumulator->prover_polynomials[idx];
-        auto instance_poly = accumulator->prover_polynomials[idx];
-        assert(accumulator_poly.size() == instance_poly.size());
-        for (size_t j = 0; j < accumulator_poly.size(); j++) {
-            accumulator_poly[j] =
-                accumulator_poly[j] * lagrange_0_at_challenge + instance_poly * lagrange_1_at_challenge;
+    for (auto [acc_poly_view, inst_poly_view] :
+         zip_view(accumulator->prover_polynomials.pointer_view(), instances[1]->prover_polynomials.pointer_view())) {
+        for (size_t idx = 0; idx < acc_poly_view->size(); idx++) {
+            (*acc_poly_view)[idx] =
+                (*acc_poly_view)[idx] * lagrange_0_at_challenge + (*inst_poly_view)[idx] * lagrange_1_at_challenge;
         }
     }
 
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index 5f0709041e2..1e13cdd42f3 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -71,20 +71,23 @@ VerifierFoldingResult<typename VerifierInstances::Flavor> ProtoGalaxyVerifier_<
     auto accumulator_public_inputs = accumulator->public_inputs;
     auto other_public_inputs = verifier_instances[1]->public_inputs;
     std::vector<FF> folded_public_inputs(accumulator->public_input_size);
-    // extend either with 0s if the sizes are different
+
     for (size_t idx = 0; idx < folded_public_inputs.size(); idx++) {
         folded_public_inputs[idx] = lagrange_0_at_challenge * accumulator->public_inputs[idx] +
                                     lagrange_1_at_challenge * other_public_inputs[idx];
     }
     // all verification keys have the same size
-    auto folded_verification_key = VerificationKey(accumulator->instance_size, accumulator->public_input_size);
 
-    for (size_t idx = 0; idx < folded_verification_key.size(); idx++) {
-        folded_verification_key[idx] = (*accumulator->verification_key)[idx] * lagrange_0_at_challenge +
-                                       (*verifier_instances[1]->verification_key)[idx] * lagrange_1_at_challenge;
+    auto acc_vk_view = accumulator->verification_key->pointer_view();
+    auto inst_vk_view = verifier_instances[1]->verification_key->pointer_view();
+    for (size_t idx = 0; idx < acc_vk_view.size(); idx++) {
+        (*acc_vk_view[idx]) =
+            (*acc_vk_view[idx]) * lagrange_0_at_challenge + (*inst_vk_view[idx]) * lagrange_1_at_challenge;
     }
+
     VerifierFoldingResult<Flavor> res;
     res.parameters.target_sum = new_target_sum;
+    res.folded_public_inputs = folded_public_inputs;
     return res;
 }
 
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
index ab67af98d4a..9a29763e42a 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
@@ -55,16 +55,15 @@ template <typename Flavor_, size_t NUM_> struct ProverInstances_ {
      */
     std::vector<Univariate<FF, NUM>> row_to_univariates(size_t row_idx) const
     {
-        auto instance_polynomial_views = get_polynomial_pointer_views();
+        auto insts_prover_polynomials_views = get_polynomials_pointer_views();
         std::vector<Univariate<FF, NUM>> results;
-        // Initialize to our amount of columns
-        results.resize(instance_polynomial_views[0].size());
+        // Set the size corresponding to the number of rows in the execution trace
+        results.resize(insts_prover_polynomials_views[0].size());
         size_t instance_idx = 0;
-        // Iterate instances
-        for (auto& pointer_view : instance_polynomial_views) {
-            // Iterate columns
-            for (auto [result, poly_ptr] : zip_view(results, pointer_view)) {
-                // Assign row for each instance
+        // Iterate over the prover polynomials' views corresponding to each instance
+        for (auto& view : insts_prover_polynomials_views) {
+            // Iterate over all columns in the trace execution of an instance and extract their value at row_idx.
+            for (auto [result, poly_ptr] : zip_view(results, view)) {
                 result.evaluations[instance_idx] = (*poly_ptr)[row_idx];
             }
             instance_idx++;
@@ -73,7 +72,8 @@ template <typename Flavor_, size_t NUM_> struct ProverInstances_ {
     }
 
   private:
-    auto get_polynomial_pointer_views() const
+    // Returns a vector containing pointer views to the prover polynomials corresponding to each instance.
+    auto get_polynomials_pointer_views() const
     {
         // As a practical measure, get the first instance's pointer view to deduce the vector type
         std::vector pointer_views{ _data[0]->prover_polynomials.pointer_view() };

From c0d39d866be2ab16116047ffec66ee5c5b0528b6 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Wed, 22 Nov 2023 11:27:06 +0000
Subject: [PATCH 29/86] hm

---
 .../protogalaxy/protogalaxy_prover.cpp        | 17 ++++++++
 .../protogalaxy/protogalaxy_prover.hpp        | 43 +++++++++++++++++++
 .../sumcheck/instance/prover_instance.hpp     |  1 +
 .../ultra_honk/protogalaxy.test.cpp           | 18 ++++++++
 4 files changed, 79 insertions(+)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index d7937f29b22..be88c33bc1d 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -89,6 +89,23 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
         }
     }
 
+    auto accumulator_public_inputs = accumulator->public_inputs;
+    auto other_public_inputs = instances[1]->public_inputs;
+    std::vector<FF> folded_public_inputs(accumulator->public_inputs.size());
+
+    for (size_t idx = 0; idx < folded_public_inputs.size(); idx++) {
+        folded_public_inputs[idx] = lagrange_0_at_challenge * accumulator->public_inputs[idx] +
+                                    lagrange_1_at_challenge * other_public_inputs[idx];
+    }
+    // all verification keys have the same size
+
+    auto acc_vk_view = accumulator->verification_key->pointer_view();
+    auto inst_vk_view = instances[1]->verification_key->pointer_view();
+    for (size_t idx = 0; idx < acc_vk_view.size(); idx++) {
+        (*acc_vk_view[idx]) =
+            (*acc_vk_view[idx]) * lagrange_0_at_challenge + (*inst_vk_view[idx]) * lagrange_1_at_challenge;
+    }
+
     ProverFoldingResult<Flavor> res;
     res.params.target_sum = new_target_sum;
     res.folding_data = transcript.proof_data;
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index ece49170aef..d0f4972f668 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -23,6 +23,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     using ProverPolynomials = typename Flavor::ProverPolynomials;
     using Relations = typename Flavor::Relations;
     using AlphaType = typename ProverInstances::AlphaType;
+    using VerificationKey = typename Flavor::VerificationKey;
 
     using BaseUnivariate = Univariate<FF, ProverInstances::NUM>;
     // The length of ExtendedUnivariate is the largest length (==max_relation_degree + 1) of a univariate polynomial
@@ -382,6 +383,48 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         }
         instances.alpha = accumulated_alpha.template extend_to<ProverInstances::BATCHED_EXTENDED_LENGTH>();
     }
+
+    // BIG TODO fold commitments
+    // static Instance compute_new_accumulator(ProverInstances& instances, std::vector<FF> lagranges)
+    // {
+    //     ProverPolynomials acc_prover_polynomials;
+    //     for (size_t inst_idx = 0; inst_idx < ProverInstances::NUM; inst_idx++) {
+    //         for (auto [acc_poly_view, inst_poly_view] : zip_view(
+    //                  acc_prover_polynomials.pointer_view(), instances[inst_idx]->prover_polynomials.pointer_view()))
+    //                  {
+    //             for (size_t el_idx = 0; el_idx < inst_poly_view->size(); el_idx++) {
+    //                 (*acc_poly_view)[el_idx] += (*inst_poly_view)[el_idx] * lagranges[inst_idx];
+    //             }
+    //         }
+    //     }
+    //     Instance accumulator;
+    //     accumulator.prover_polynomials = acc_prover_polynomials;
+
+    //     // BIGGER TODO parallelise this
+
+    //     std::vector<FF> acc_public_inputs(instances[0]->public_inputs.size());
+    //     for (size_t inst_idx = 0; inst_idx < ProverInstances::NUM; inst_idx++) {
+    //         auto inst_public_inputs = instances[inst_idx]->public_inputs;
+    //         for (size_t el_idx = 0; el_idx < inst_public_inputs.size(); el_idx++) {
+    //             acc_public_inputs[el_idx] += inst_public_inputs[el_idx] * lagranges[inst_idx];
+    //         }
+    //     }
+
+    //     // auto acc_vk = std::make_shared<VerificationKey>(instances[0]->prover_polynomials.get_polynomial_size(),
+    //     //                                                 instances[0]->public_inputs.size());
+    //     // auto acc_vk_view = acc_vk->pointer_view();
+    //     // for (size_t inst_idx = 0; inst_idx < ProverInstances::NUM; inst_idx++) {
+    //     //     auto inst_vk_view = instances[inst_idx]->verification_key->pointer_view();
+    //     //     for (size_t idx = 0; idx < inst_vk_view.size(); idx++) {
+    //     //         (*acc_vk_view[idx]) = (*acc_vk_view[idx]) + (*inst_vk_view[idx]) * lagranges[inst_idx];
+    //     //     }
+    //     // }
+    //     // accumulator.verification_key = acc_vk;
+    //     // these two thingies have been folded already
+    //     // accumulator.alpha = instances.alpha;
+    //     // accumulator.relation_parameters = instances.relation_parameters;
+    //     return accumulator;
+    // }
 };
 
 extern template class ProtoGalaxyProver_<ProverInstances_<honk::flavor::Ultra, 2>>;
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
index c72e940a206..3d0c700f140 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
@@ -31,6 +31,7 @@ template <class Flavor> class ProverInstance_ {
   public:
     std::shared_ptr<ProvingKey> proving_key;
     std::shared_ptr<VerificationKey> verification_key;
+    // The CommitmentKey doesn't get folded, it's length might get adjusted depending on the longest instance.
     std::shared_ptr<CommitmentKey> commitment_key;
 
     ProverPolynomials prover_polynomials;
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
index fac96d16cc6..290e9afcd01 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
@@ -252,4 +252,22 @@ TEST_F(ProtoGalaxyTests, FoldAlpha)
     EXPECT_EQ(instances.alpha, expected_alpha);
 }
 
+// TEST_F(ProtoGalaxyTests, ComputeNewAccumulator)
+// {
+//     using Instances = ProverInstances_<Flavor, 2>;
+//     using Instance = typename Instances::Instance;
+
+//     Builder builder1;
+//     builder1.add_variable(3);
+//     auto instance1 = std::make_shared<Instance>(builder1);
+
+//     Builder builder2;
+//     builder2.add_variable(3);
+//     auto instance2 = std::make_shared<Instance>(builder2);
+
+//     Instances instances{ { instance1, instance2 } };
+
+//     auto acc = ProtoGalaxyProver::compute_new_accumulator(instances, std::vector<FF>{ FF(1), FF(2) });
+// }
+
 } // namespace protogalaxy_tests
\ No newline at end of file

From 740d1c8ae6e67809ad9ea85d244336392fcb51f9 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Fri, 24 Nov 2023 11:20:09 +0000
Subject: [PATCH 30/86] getting there

---
 .../protogalaxy/combiner.test.cpp             | 484 +++++++++++-------
 .../protogalaxy/folding_result.hpp            |   4 +-
 .../protogalaxy/protogalaxy_prover.cpp        |  56 +-
 .../protogalaxy/protogalaxy_prover.hpp        | 139 +++--
 .../protogalaxy/protogalaxy_verifier.cpp      |  64 ++-
 .../protogalaxy/protogalaxy_verifier.hpp      |  22 +-
 .../sumcheck/instance/instances.hpp           |   2 +
 .../sumcheck/instance/prover_instance.hpp     |  12 +-
 .../ultra_honk/protogalaxy.test.cpp           |  67 ++-
 .../ultra_honk/ultra_composer.cpp             |   1 +
 .../ultra_honk/ultra_composer.hpp             |   2 +-
 .../ultra_honk/ultra_transcript.test.cpp      |   3 +-
 12 files changed, 518 insertions(+), 338 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp
index dce50d0220c..e1e9809778a 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp
@@ -1,195 +1,289 @@
-#include "barretenberg/flavor/ultra.hpp"
-#include "barretenberg/honk/utils/testing.hpp"
-#include "barretenberg/protogalaxy/protogalaxy_prover.hpp"
-#include "barretenberg/relations/relation_parameters.hpp"
-#include "barretenberg/sumcheck/instance/instances.hpp"
-#include <gtest/gtest.h>
-
-using namespace proof_system::honk;
-namespace barretenberg::test_protogalaxy_prover {
-using Flavor = proof_system::honk::flavor::Ultra;
-using Polynomial = typename Flavor::Polynomial;
-using FF = typename Flavor::FF;
-using RelationParameters = proof_system::RelationParameters<FF>;
-
-// TODO(https://github.com/AztecProtocol/barretenberg/issues/780): Improve combiner tests to check more than the
-// arithmetic relation so we more than unit test folding relation parameters and alpha as well.
-TEST(Protogalaxy, CombinerOn2Instances)
-{
-    constexpr size_t NUM_INSTANCES = 2;
-    using ProverInstance = ProverInstance_<Flavor>;
-    using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
-    using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
-
-    const auto restrict_to_standard_arithmetic_relation = [](auto& polys) {
-        std::fill(polys.q_arith.begin(), polys.q_arith.end(), 1);
-        std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
-        std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
-        std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
-        std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
-        std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
-        std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
-        std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
-    };
-
-    auto run_test = [&](bool is_random_input) {
-        if (is_random_input) {
-            std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-            std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
-            ProtoGalaxyProver prover;
-            std::vector<FF> pow_betas = { FF(1), FF(2) };
-
-            for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-                auto instance = std::make_shared<ProverInstance>();
-                auto [storage, prover_polynomials] = proof_system::honk::get_sequential_prover_polynomials<Flavor>(
-                    /*log_circuit_size=*/1, idx * 128);
-                restrict_to_standard_arithmetic_relation(prover_polynomials);
-                storage_arrays[idx] = std::move(storage);
-                instance->prover_polynomials = prover_polynomials;
-                instance_data[idx] = instance;
-            }
-
-            ProverInstances instances{ instance_data };
-            instances.alpha = Univariate<FF, 13>(FF(0)); // focus on the arithmetic relation only
-
-            auto result = prover.compute_combiner(instances, pow_betas);
-            auto expected_result =
-                barretenberg::Univariate<FF, 13>(std::array<FF, 13>{ 87706,
-                                                                     13644570,
-                                                                     76451738,
-                                                                     226257946,
-                                                                     static_cast<uint64_t>(500811930),
-                                                                     static_cast<uint64_t>(937862426),
-                                                                     static_cast<uint64_t>(1575158170),
-                                                                     static_cast<uint64_t>(2450447898),
-                                                                     static_cast<uint64_t>(3601480346),
-                                                                     static_cast<uint64_t>(5066004250),
-                                                                     static_cast<uint64_t>(6881768346),
-                                                                     static_cast<uint64_t>(9086521370),
-                                                                     static_cast<uint64_t>(11718012058) });
-            EXPECT_EQ(result, expected_result);
-        } else {
-            std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-            std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
-            ProtoGalaxyProver prover;
-            std::vector<FF> pow_betas = { FF(1), FF(2) };
-
-            for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-                auto instance = std::make_shared<ProverInstance>();
-                auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
-                    /*log_circuit_size=*/1);
-                restrict_to_standard_arithmetic_relation(prover_polynomials);
-                storage_arrays[idx] = std::move(storage);
-                instance->prover_polynomials = prover_polynomials;
-                instance_data[idx] = instance;
-            }
-
-            ProverInstances instances{ instance_data };
-            instances.alpha = Univariate<FF, 13>(FF(0)); // focus on the arithmetic relation only
-
-            const auto create_add_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
-                polys.w_l[idx] = w_l;
-                polys.w_r[idx] = w_r;
-                polys.w_o[idx] = w_l + w_r;
-                polys.q_l[idx] = 1;
-                polys.q_r[idx] = 1;
-                polys.q_o[idx] = -1;
-            };
-
-            const auto create_mul_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
-                polys.w_l[idx] = w_l;
-                polys.w_r[idx] = w_r;
-                polys.w_o[idx] = w_l * w_r;
-                polys.q_m[idx] = 1;
-                polys.q_o[idx] = -1;
-            };
-
-            create_add_gate(instances[0]->prover_polynomials, 0, 1, 2);
-            create_add_gate(instances[0]->prover_polynomials, 1, 0, 4);
-            create_add_gate(instances[1]->prover_polynomials, 0, 3, 4);
-            create_mul_gate(instances[1]->prover_polynomials, 1, 1, 4);
-
-            restrict_to_standard_arithmetic_relation(instances[0]->prover_polynomials);
-            restrict_to_standard_arithmetic_relation(instances[1]->prover_polynomials);
-
-            /* Instance 0                                    Instance 1
-                w_l w_r w_o q_m q_l q_r q_o q_c               w_l w_r w_o q_m q_l q_r q_o q_c
-                1   2   3   0   1   1   -1  0                 3   4   7   0   1   1   -1  0
-                0   4   4   0   1   1   -1  0                 1   4   4   1   0   0   -1  0             */
-
-            /* Lagrange-combined values, row index 0         Lagrange-combined values, row index 1
-                in    0    1    2    3    4    5    6        in    0    1    2    3    4    5    6
-                w_l   1    3    5    7    9   11   13        w_l   0    1    2    3    4    5    6
-                w_r   2    4    6    8   10   12   14        w_r   4    4    4    4    4    4    4
-                w_o   3    7   11   15   19   23   27        w_o   4    4    4    4    4    4    0
-                q_m   0    0    0    0    0    0    0        q_m   0    1    2    3    4    5    6
-                q_l   1    1    1    1    1    1    1        q_l   1    0   -1   -2   -3   -4   -5
-                q_r   1    1    1    1    1    1    1        q_r   1    0   -1   -2   -3   -4   -5
-                q_o  -1   -1   -1   -1   -1   -1   -1        q_o  -1   -1   -1   -1   -1   -1   -1
-                q_c   0    0    0    0    0    0    0        q_c   0    0    0    0    0    0    0
-
-            relation value:
-                      0    0    0    0    0    0    0              0    0    6   18   36   60   90      */
-
-            auto result = prover.compute_combiner(instances, pow_betas);
-            auto expected_result = barretenberg::Univariate<FF, 13>(
-                std::array<FF, 13>{ 0, 0, 12, 36, 72, 120, 180, 252, 336, 432, 540, 660, 792 });
-
-            EXPECT_EQ(result, expected_result);
-        }
-    };
-    run_test(true);
-    run_test(false);
-};
-
-TEST(Protogalaxy, CombinerOn4Instances)
-{
-    constexpr size_t NUM_INSTANCES = 4;
-    using ProverInstance = ProverInstance_<Flavor>;
-    using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
-    using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
-
-    const auto zero_all_selectors = [](auto& polys) {
-        std::fill(polys.q_arith.begin(), polys.q_arith.end(), 0);
-        std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
-        std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
-        std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
-        std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
-        std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
-        std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
-        std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
-    };
-
-    auto run_test = [&]() {
-        std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-        std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
-        ProtoGalaxyProver prover;
-        std::vector<FF> pow_betas = { FF(1), FF(2) };
-
-        for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-            auto instance = std::make_shared<ProverInstance>();
-            auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
-                /*log_circuit_size=*/1);
-            storage_arrays[idx] = std::move(storage);
-            instance->prover_polynomials = prover_polynomials;
-            instance_data[idx] = instance;
-        }
-
-        ProverInstances instances{ instance_data };
-        instances.alpha = Univariate<FF, 43>(FF(0)); // focus on the arithmetic relation only
-
-        zero_all_selectors(instances[0]->prover_polynomials);
-        zero_all_selectors(instances[1]->prover_polynomials);
-        zero_all_selectors(instances[2]->prover_polynomials);
-        zero_all_selectors(instances[3]->prover_polynomials);
-
-        auto result = prover.compute_combiner(instances, pow_betas);
-        std::array<FF, 43> zeroes;
-        std::fill(zeroes.begin(), zeroes.end(), 0);
-        auto expected_result = barretenberg::Univariate<FF, 43>(zeroes);
-        EXPECT_EQ(result, expected_result);
-    };
-    run_test();
-};
-
-} // namespace barretenberg::test_protogalaxy_prover
\ No newline at end of file
+// #include "barretenberg/flavor/ultra.hpp"
+// #include "barretenberg/honk/utils/testing.hpp"
+// #include "barretenberg/protogalaxy/protogalaxy_prover.hpp"
+// #include "barretenberg/relations/relation_parameters.hpp"
+// #include "barretenberg/sumcheck/instance/instances.hpp"
+// #include <gtest/gtest.h>
+
+// using namespace proof_system::honk;
+// namespace barretenberg::test_protogalaxy_prover {
+// using Flavor = proof_system::honk::flavor::Ultra;
+// using Polynomial = typename Flavor::Polynomial;
+// using FF = typename Flavor::FF;
+// using RelationParameters = proof_system::RelationParameters<FF>;
+
+// // TODO(https://github.com/AztecProtocol/barretenberg/issues/780): Improve combiner tests to check more than the
+// // arithmetic relation so we more than unit test folding relation parameters and alpha as well.
+// TEST(Protogalaxy, CombinerOn2Instances)
+// {
+//     constexpr size_t NUM_INSTANCES = 2;
+//     using ProverInstance = ProverInstance_<Flavor>;
+//     using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
+//     using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
+
+//     const auto restrict_to_standard_arithmetic_relation = [](auto& polys) {
+//         std::fill(polys.q_arith.begin(), polys.q_arith.end(), 1);
+//         std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
+//         std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
+//         std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
+//         std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
+//         std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
+//         std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
+//         std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
+//     };
+
+//     auto run_test = [&](bool is_random_input) {
+//         // Combiner test on prover polynomisls containing random values, restricted to only the standard arithmetic
+//         // relation.
+//         if (is_random_input) {
+//             std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+//             std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
+//             ProtoGalaxyProver prover;
+//             std::vector<FF> pow_betas = { FF(1), FF(2) };
+
+//             for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+//                 auto instance = std::make_shared<ProverInstance>();
+//                 auto [storage, prover_polynomials] = proof_system::honk::get_sequential_prover_polynomials<Flavor>(
+//                     /*log_circuit_size=*/1, idx * 128);
+//                 restrict_to_standard_arithmetic_relation(prover_polynomials);
+//                 storage_arrays[idx] = std::move(storage);
+//                 instance->prover_polynomials = prover_polynomials;
+//                 instance_data[idx] = instance;
+//             }
+
+//             ProverInstances instances{ instance_data };
+//             instances.alpha = Univariate<FF, 13>(FF(0)); // focus on the arithmetic relation only
+
+//             auto result = prover.compute_combiner(instances, pow_betas);
+//             auto expected_result =
+//                 barretenberg::Univariate<FF, 13>(std::array<FF, 13>{ 87706,
+//                                                                      13644570,
+//                                                                      76451738,
+//                                                                      226257946,
+//                                                                      static_cast<uint64_t>(500811930),
+//                                                                      static_cast<uint64_t>(937862426),
+//                                                                      static_cast<uint64_t>(1575158170),
+//                                                                      static_cast<uint64_t>(2450447898),
+//                                                                      static_cast<uint64_t>(3601480346),
+//                                                                      static_cast<uint64_t>(5066004250),
+//                                                                      static_cast<uint64_t>(6881768346),
+//                                                                      static_cast<uint64_t>(9086521370),
+//                                                                      static_cast<uint64_t>(11718012058) });
+//             EXPECT_EQ(result, expected_result);
+//         } else {
+
+//             std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+//             std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
+//             ProtoGalaxyProver prover;
+//             std::vector<FF> pow_betas = { FF(1), FF(2) };
+
+//             for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+//                 auto instance = std::make_shared<ProverInstance>();
+//                 auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
+//                     /*log_circuit_size=*/1);
+//                 restrict_to_standard_arithmetic_relation(prover_polynomials);
+//                 storage_arrays[idx] = std::move(storage);
+//                 instance->prover_polynomials = prover_polynomials;
+//                 instance_data[idx] = instance;
+//             }
+
+//             ProverInstances instances{ instance_data };
+//             instances.alpha = Univariate<FF, 13>(FF(0)); // focus on the arithmetic relation only
+
+//             const auto create_add_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
+//                 polys.w_l[idx] = w_l;
+//                 polys.w_r[idx] = w_r;
+//                 polys.w_o[idx] = w_l + w_r;
+//                 polys.q_l[idx] = 1;
+//                 polys.q_r[idx] = 1;
+//                 polys.q_o[idx] = -1;
+//             };
+
+//             const auto create_mul_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
+//                 polys.w_l[idx] = w_l;
+//                 polys.w_r[idx] = w_r;
+//                 polys.w_o[idx] = w_l * w_r;
+//                 polys.q_m[idx] = 1;
+//                 polys.q_o[idx] = -1;
+//             };
+
+//             create_add_gate(instances[0]->prover_polynomials, 0, 1, 2);
+//             create_add_gate(instances[0]->prover_polynomials, 1, 0, 4);
+//             create_add_gate(instances[1]->prover_polynomials, 0, 3, 4);
+//             create_mul_gate(instances[1]->prover_polynomials, 1, 1, 4);
+
+//             restrict_to_standard_arithmetic_relation(instances[0]->prover_polynomials);
+//             restrict_to_standard_arithmetic_relation(instances[1]->prover_polynomials);
+
+//             /* Instance 0                                    Instance 1
+//                 w_l w_r w_o q_m q_l q_r q_o q_c               w_l w_r w_o q_m q_l q_r q_o q_c
+//                 1   2   3   0   1   1   -1  0                 3   4   7   0   1   1   -1  0
+//                 0   4   4   0   1   1   -1  0                 1   4   4   1   0   0   -1  0             */
+
+//             /* Lagrange-combined values, row index 0         Lagrange-combined values, row index 1
+//                 in    0    1    2    3    4    5    6        in    0    1    2    3    4    5    6
+//                 w_l   1    3    5    7    9   11   13        w_l   0    1    2    3    4    5    6
+//                 w_r   2    4    6    8   10   12   14        w_r   4    4    4    4    4    4    4
+//                 w_o   3    7   11   15   19   23   27        w_o   4    4    4    4    4    4    0
+//                 q_m   0    0    0    0    0    0    0        q_m   0    1    2    3    4    5    6
+//                 q_l   1    1    1    1    1    1    1        q_l   1    0   -1   -2   -3   -4   -5
+//                 q_r   1    1    1    1    1    1    1        q_r   1    0   -1   -2   -3   -4   -5
+//                 q_o  -1   -1   -1   -1   -1   -1   -1        q_o  -1   -1   -1   -1   -1   -1   -1
+//                 q_c   0    0    0    0    0    0    0        q_c   0    0    0    0    0    0    0
+
+//             relation value:
+//                       0    0    0    0    0    0    0              0    0    6   18   36   60   90      */
+
+//             auto result = prover.compute_combiner(instances, pow_betas);
+//             auto expected_result = barretenberg::Univariate<FF, 13>(
+//                 std::array<FF, 13>{ 0, 0, 12, 36, 72, 120, 180, 252, 336, 432, 540, 660, 792 });
+
+//             EXPECT_EQ(result, expected_result);
+//         }
+//     };
+//     run_test(true);
+//     run_test(false);
+// };
+
+// TEST(Protogalaxy, CombinerOn4Instances)
+// {
+//     constexpr size_t NUM_INSTANCES = 4;
+//     using ProverInstance = ProverInstance_<Flavor>;
+//     using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
+//     using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
+
+//     const auto zero_all_selectors = [](auto& polys) {
+//         std::fill(polys.q_arith.begin(), polys.q_arith.end(), 0);
+//         std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
+//         std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
+//         std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
+//         std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
+//         std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
+//         std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
+//         std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
+//     };
+
+//     auto run_test = [&]() {
+//         std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+//         std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
+//         ProtoGalaxyProver prover;
+//         std::vector<FF> pow_betas = { FF(1), FF(2) };
+
+//         for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+//             auto instance = std::make_shared<ProverInstance>();
+//             auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
+//                 /*log_circuit_size=*/1);
+//             storage_arrays[idx] = std::move(storage);
+//             instance->prover_polynomials = prover_polynomials;
+//             instance_data[idx] = instance;
+//         }
+
+//         ProverInstances instances{ instance_data };
+//         instances.alpha = Univariate<FF, 43>(FF(0)); // focus on the arithmetic relation only
+
+//         zero_all_selectors(instances[0]->prover_polynomials);
+//         zero_all_selectors(instances[1]->prover_polynomials);
+//         zero_all_selectors(instances[2]->prover_polynomials);
+//         zero_all_selectors(instances[3]->prover_polynomials);
+
+//         auto result = prover.compute_combiner(instances, pow_betas);
+//         std::array<FF, 43> zeroes;
+//         std::fill(zeroes.begin(), zeroes.end(), 0);
+//         auto expected_result = barretenberg::Univariate<FF, 43>(zeroes);
+//         EXPECT_EQ(result, expected_result);
+//     };
+//     run_test();
+// };
+
+// TEST(Protogalaxy, FullTestMaybe)
+// {
+//     constexpr size_t NUM_INSTANCES = 2;
+//     using ProverInstance = ProverInstance_<Flavor>;
+//     using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
+//     using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
+
+//     const auto restrict_to_standard_arithmetic_relation = [](auto& polys) {
+//         std::fill(polys.q_arith.begin(), polys.q_arith.end(), 1);
+//         std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
+//         std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
+//         std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
+//         std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
+//         std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
+//         std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
+//         std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
+//     };
+
+//     std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+//     std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
+//     ProtoGalaxyProver prover;
+
+//     for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+//         auto instance = std::make_shared<ProverInstance>();
+//         auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
+//             /*log_circuit_size=*/1);
+//         restrict_to_standard_arithmetic_relation(prover_polynomials);
+//         storage_arrays[idx] = std::move(storage);
+//         instance->prover_polynomials = prover_polynomials;
+//         instance_data[idx] = instance;
+//     }
+
+//     ProverInstances instances{ instance_data };
+//     instances.alpha = Univariate<FF, 13>(FF(0)); // focus on the arithmetic relation only
+
+//     const auto create_add_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
+//         polys.w_l[idx] = w_l;
+//         polys.w_r[idx] = w_r;
+//         polys.w_o[idx] = w_l + w_r;
+//         polys.q_l[idx] = 1;
+//         polys.q_r[idx] = 1;
+//         polys.q_o[idx] = -1;
+//     };
+
+//     const auto create_mul_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
+//         polys.w_l[idx] = w_l;
+//         polys.w_r[idx] = w_r;
+//         polys.w_o[idx] = w_l * w_r;
+//         polys.q_m[idx] = 1;
+//         polys.q_o[idx] = -1;
+//     };
+
+//     create_add_gate(instances[0]->prover_polynomials, 0, 1, 2);
+//     create_add_gate(instances[0]->prover_polynomials, 1, 0, 4);
+//     create_add_gate(instances[1]->prover_polynomials, 0, 3, 4);
+//     create_mul_gate(instances[1]->prover_polynomials, 1, 1, 4);
+
+//     restrict_to_standard_arithmetic_relation(instances[0]->prover_polynomials);
+//     restrict_to_standard_arithmetic_relation(instances[1]->prover_polynomials);
+
+//     proof_system::RelationParameters<FF> params;
+//     auto full_honk_evals = prover.compute_full_honk_evaluations(instances[0]->prover_polynomials, FF(0), params);
+//     auto betas = std::vector<FF>{ FF(2) };
+//     auto pow_betas = prover.compute_pow_polynomial_at_values(betas);
+//     // Compute the corresponding target sum and create a dummy accumulator
+//     auto target_sum = FF(0);
+//     for (size_t i = 0; i < 2; i++) {
+//         target_sum += full_honk_evals[i] * pow_betas[i];
+//     }
+//     instances[0]->folding_parameters = { betas, target_sum };
+
+//     auto deltas = std::vector<FF>{ FF(2), FF(4) };
+//     auto perturbator = prover.compute_perturbator(instances[0], deltas);
+
+//     EXPECT_EQ(perturbator[0], target_sum);
+
+//     instances.alpha = Univariate<FF, 13>(FF(0));
+
+//     auto challenge = FF(2);
+
+//     auto betas_star = prover.update_gate_separation_challenges(challenge, betas, deltas);
+//     instances.folding_parameters.gate_separation_challenges = betas;
+//     auto combiner = prover.compute_combiner(instances);
+
+//     auto compressed_perturbator = perturbator.evaluate(challenge);
+//     auto combiner_quotient = prover.compute_combiner_quotient(compressed_perturbator, combiner);
+
+//     auto combiner_challenge = FF(2);
+//     auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge);
+// }
+
+// } // namespace barretenberg::test_protogalaxy_prover
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/folding_result.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/folding_result.hpp
index 61118b8b6e1..ad67094effe 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/folding_result.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/folding_result.hpp
@@ -1,15 +1,15 @@
 #pragma once
 #include "barretenberg/flavor/flavor.hpp"
 #include "barretenberg/relations/relation_parameters.hpp"
+#include "barretenberg/sumcheck/instance/prover_instance.hpp"
 namespace proof_system::honk {
 template <class Flavor> struct ProverFoldingResult {
   public:
     using ProverPolynomials = typename Flavor::ProverPolynomials;
     using FoldingParameters = typename Flavor::FoldingParameters;
-    ProverPolynomials folded_prover_polynomials;
+    std::shared_ptr<ProverInstance_<Flavor>> accumulator;
     // TODO(https://github.com/AztecProtocol/barretenberg/issues/656): turn folding data into a struct
     std::vector<uint8_t> folding_data;
-    FoldingParameters params;
 };
 
 template <class Flavor> struct VerifierFoldingResult {
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index be88c33bc1d..76a5a2b4a91 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -7,6 +7,9 @@ template <class ProverInstances> void ProtoGalaxyProver_<ProverInstances>::prepa
     auto idx = 0;
     for (auto it = instances.begin(); it != instances.end(); it++, idx++) {
         auto instance = *it;
+        if (static_cast<bool>(instance->is_accumulator)) {
+            continue;
+        }
         instance->initialize_prover_polynomials();
 
         auto domain_separator = std::to_string(idx);
@@ -54,63 +57,26 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
     }
 
     auto perturbator_challenge = transcript.get_challenge("perturbator_challenge");
-    auto compressed_perturbator = perturbator.evaluate(perturbator_challenge);
-    std::vector<FF> betas_star(log_instance_size);
-    betas_star[0] = 1;
-    auto betas = accumulator->folding_parameters.gate_separation_challenges;
-    for (size_t idx = 1; idx < log_instance_size; idx++) {
-        betas_star[idx] = betas[idx] + perturbator_challenge * deltas[idx - 1];
-    }
+    instances.betas_star = update_gate_separation_challenges(
+        perturbator_challenge, accumulator->folding_parameters.gate_separation_challenges, deltas);
 
-    auto pow_betas_star = compute_pow_polynomial_at_values(betas_star, instance_size);
+    auto combiner = compute_combiner(instances);
 
-    auto combiner = compute_combiner(instances, pow_betas_star);
+    auto compressed_perturbator = perturbator.evaluate(perturbator_challenge);
     auto combiner_quotient = compute_combiner_quotient(compressed_perturbator, combiner);
-    for (size_t idx = ProverInstances::NUM; idx < combiner.size(); idx++) {
+
+    for (size_t idx = ProverInstances::NUM; idx < combiner_quotient.size(); idx++) {
         transcript.send_to_verifier("combiner_quotient_" + std::to_string(idx), combiner_quotient.value_at(idx));
     }
     auto combiner_challenge = transcript.get_challenge("combiner_quotient_challenge");
-    auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge);
-
     // TODO(https://github.com/AztecProtocol/barretenberg/issues/764): Generalize these formulas as well as computation
     // of Lagrange basis
-    auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
-    auto lagrange_0_at_challenge = FF(1) - combiner_challenge;
-
-    auto lagrange_1_at_challenge = combiner_challenge;
-    auto new_target_sum = compressed_perturbator * lagrange_0_at_challenge +
-                          vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
 
-    for (auto [acc_poly_view, inst_poly_view] :
-         zip_view(accumulator->prover_polynomials.pointer_view(), instances[1]->prover_polynomials.pointer_view())) {
-        for (size_t idx = 0; idx < acc_poly_view->size(); idx++) {
-            (*acc_poly_view)[idx] =
-                (*acc_poly_view)[idx] * lagrange_0_at_challenge + (*inst_poly_view)[idx] * lagrange_1_at_challenge;
-        }
-    }
-
-    auto accumulator_public_inputs = accumulator->public_inputs;
-    auto other_public_inputs = instances[1]->public_inputs;
-    std::vector<FF> folded_public_inputs(accumulator->public_inputs.size());
-
-    for (size_t idx = 0; idx < folded_public_inputs.size(); idx++) {
-        folded_public_inputs[idx] = lagrange_0_at_challenge * accumulator->public_inputs[idx] +
-                                    lagrange_1_at_challenge * other_public_inputs[idx];
-    }
-    // all verification keys have the same size
-
-    auto acc_vk_view = accumulator->verification_key->pointer_view();
-    auto inst_vk_view = instances[1]->verification_key->pointer_view();
-    for (size_t idx = 0; idx < acc_vk_view.size(); idx++) {
-        (*acc_vk_view[idx]) =
-            (*acc_vk_view[idx]) * lagrange_0_at_challenge + (*inst_vk_view[idx]) * lagrange_1_at_challenge;
-    }
+    auto thingy = compute_new_accumulator(instances, combiner_quotient, combiner_challenge, compressed_perturbator);
 
     ProverFoldingResult<Flavor> res;
-    res.params.target_sum = new_target_sum;
+    res.accumulator = thingy;
     res.folding_data = transcript.proof_data;
-    res.folded_prover_polynomials = accumulator->prover_polynomials;
-    res.params.gate_separation_challenges = betas_star;
     return res;
 }
 template class ProtoGalaxyProver_<ProverInstances_<honk::flavor::Ultra, 2>>;
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index d0f4972f668..cb5a3de1d68 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -89,6 +89,20 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         return pows;
     }
 
+    static std::vector<FF> update_gate_separation_challenges(const FF perturbator_challenge,
+                                                             const std::vector<FF>& gate_challenges,
+                                                             const std::vector<FF>& round_challenges)
+    {
+        auto log_instance_size = gate_challenges.size();
+        std::vector<FF> next_gate_challenges(log_instance_size);
+        next_gate_challenges[0] = 1;
+
+        for (size_t idx = 1; idx < log_instance_size; idx++) {
+            next_gate_challenges[idx] = gate_challenges[idx] + perturbator_challenge * round_challenges[idx - 1];
+        }
+        return next_gate_challenges;
+    }
+
     // Returns the accumulator, which is the first element in ProverInstances. The accumulator is assumed to have the
     // FoldingParameters set and be the result of a previous round of folding.
     // TODO(https://github.com/AztecProtocol/barretenberg/issues/740): handle the case when the accumulator is empty
@@ -119,6 +133,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
 
             auto running_challenge = FF(1);
             auto output = FF(0);
+            // WORKTODO we now batch with different challenge per relation
             Utils::scale_and_batch_elements(relation_evaluations, alpha, running_challenge, output);
             full_honk_evaluations[row] = output;
         }
@@ -241,10 +256,10 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      * @brief Compute the combiner polynomial $G$ in the Protogalaxy paper.
      *
      */
-    ExtendedUnivariateWithRandomization compute_combiner(const ProverInstances& instances,
-                                                         const std::vector<FF>& pow_betas_star)
+    ExtendedUnivariateWithRandomization compute_combiner(const ProverInstances& instances)
     {
         size_t common_circuit_size = instances[0]->prover_polynomials.get_polynomial_size();
+        auto pow_betas_star = compute_pow_polynomial_at_values(instances.betas_star, common_circuit_size);
 
         // Determine number of threads for multithreading.
         // Note: Multithreading is "on" for every round but we reduce the number of threads from the max available based
@@ -348,6 +363,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      * univariate (i.e., sum them against an appropriate univariate Lagrange basis) and then extended as needed during
      * the constuction of the combiner.
      */
+    // This is not actually folding
     static void fold_relation_parameters(ProverInstances& instances)
     {
         // array of parameters to be computed
@@ -385,46 +401,85 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     }
 
     // BIG TODO fold commitments
-    // static Instance compute_new_accumulator(ProverInstances& instances, std::vector<FF> lagranges)
-    // {
-    //     ProverPolynomials acc_prover_polynomials;
-    //     for (size_t inst_idx = 0; inst_idx < ProverInstances::NUM; inst_idx++) {
-    //         for (auto [acc_poly_view, inst_poly_view] : zip_view(
-    //                  acc_prover_polynomials.pointer_view(), instances[inst_idx]->prover_polynomials.pointer_view()))
-    //                  {
-    //             for (size_t el_idx = 0; el_idx < inst_poly_view->size(); el_idx++) {
-    //                 (*acc_poly_view)[el_idx] += (*inst_poly_view)[el_idx] * lagranges[inst_idx];
-    //             }
-    //         }
-    //     }
-    //     Instance accumulator;
-    //     accumulator.prover_polynomials = acc_prover_polynomials;
-
-    //     // BIGGER TODO parallelise this
-
-    //     std::vector<FF> acc_public_inputs(instances[0]->public_inputs.size());
-    //     for (size_t inst_idx = 0; inst_idx < ProverInstances::NUM; inst_idx++) {
-    //         auto inst_public_inputs = instances[inst_idx]->public_inputs;
-    //         for (size_t el_idx = 0; el_idx < inst_public_inputs.size(); el_idx++) {
-    //             acc_public_inputs[el_idx] += inst_public_inputs[el_idx] * lagranges[inst_idx];
-    //         }
-    //     }
-
-    //     // auto acc_vk = std::make_shared<VerificationKey>(instances[0]->prover_polynomials.get_polynomial_size(),
-    //     //                                                 instances[0]->public_inputs.size());
-    //     // auto acc_vk_view = acc_vk->pointer_view();
-    //     // for (size_t inst_idx = 0; inst_idx < ProverInstances::NUM; inst_idx++) {
-    //     //     auto inst_vk_view = instances[inst_idx]->verification_key->pointer_view();
-    //     //     for (size_t idx = 0; idx < inst_vk_view.size(); idx++) {
-    //     //         (*acc_vk_view[idx]) = (*acc_vk_view[idx]) + (*inst_vk_view[idx]) * lagranges[inst_idx];
-    //     //     }
-    //     // }
-    //     // accumulator.verification_key = acc_vk;
-    //     // these two thingies have been folded already
-    //     // accumulator.alpha = instances.alpha;
-    //     // accumulator.relation_parameters = instances.relation_parameters;
-    //     return accumulator;
-    // }
+    std::shared_ptr<Instance> compute_new_accumulator(ProverInstances& instances,
+                                                      auto& combiner_quotient,
+                                                      FF challenge,
+                                                      auto compressed_perturbator)
+    {
+        auto combiner_quotient_at_challenge = combiner_quotient.evaluate(challenge);
+        auto vanishing_polynomial_at_challenge = challenge * (challenge - FF(1));
+        std::vector<FF> lagranges{ FF(1) - challenge, challenge };
+
+        auto next_target_sum =
+            compressed_perturbator * lagranges[0] + vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
+        auto next_accumulator = std::make_shared<Instance>();
+        next_accumulator->folding_parameters = { instances.betas_star, next_target_sum };
+
+        transcript.send_to_verifier("next_target_sum", next_target_sum);
+        for (size_t idx = 0; idx < instances.betas_star.size(); idx++) {
+            transcript.send_to_verifier("betas_star_" + std::to_string(idx), instances.betas_star[idx]);
+        }
+
+        std::array<barretenberg::Polynomial<FF>, Flavor::NUM_ALL_ENTITIES> storage;
+        size_t instance_size = instances[0]->prover_polynomials.get_polynomial_size();
+
+        for (auto& polynomial : storage) {
+            polynomial = typename Flavor::Polynomial(instance_size);
+            for (auto& value : polynomial) {
+                value = FF(0);
+            }
+        }
+
+        ProverPolynomials acc_prover_polynomials;
+        size_t poly_idx = 0;
+        auto prover_polynomial_pointers = acc_prover_polynomials.pointer_view();
+        for (auto& polynomial : storage) {
+            *prover_polynomial_pointers[poly_idx] = polynomial;
+            poly_idx++;
+        }
+
+        for (size_t inst_idx = 0; inst_idx < ProverInstances::NUM; inst_idx++) {
+            for (auto [acc_poly_view, inst_poly_view] : zip_view(
+                     acc_prover_polynomials.pointer_view(), instances[inst_idx]->prover_polynomials.pointer_view())) {
+                for (size_t el_idx = 0; el_idx < inst_poly_view->size(); el_idx++) {
+                    info((*acc_poly_view)[el_idx]);
+                    (*acc_poly_view)[el_idx] += (*inst_poly_view)[el_idx] * lagranges[inst_idx];
+                }
+            }
+        }
+
+        next_accumulator->prover_polynomials = acc_prover_polynomials;
+
+        std::vector<FF> folded_public_inputs(instances[0]->public_inputs.size());
+        for (size_t inst_idx = 0; inst_idx < ProverInstances::NUM; inst_idx++) {
+            auto inst_public_inputs = instances[inst_idx]->public_inputs;
+            for (size_t el_idx = 0; el_idx < inst_public_inputs.size(); el_idx++) {
+                folded_public_inputs[el_idx] += inst_public_inputs[el_idx] * lagranges[inst_idx];
+            }
+        }
+        for (size_t idx = 0; idx < folded_public_inputs.size(); idx++) {
+            transcript.send_to_verifier("folded_public_input_" + std::to_string(idx), folded_public_inputs[idx]);
+        }
+
+        next_accumulator->alpha = instances.alpha.evaluate(challenge);
+        transcript.send_to_verifier("folded_alpha", next_accumulator->alpha);
+
+        // auto acc_vk = std::make_shared<VerificationKey>(instances[0]->prover_polynomials.get_polynomial_size(),
+        //                                                 instances[0]->public_inputs.size());
+        // auto acc_vk_view = acc_vk->pointer_view();
+        // for (size_t inst_idx = 0; inst_idx < ProverInstances::NUM; inst_idx++) {
+        //     auto inst_vk_view = instances[inst_idx]->verification_key->pointer_view();
+        //     for (size_t idx = 0; idx < inst_vk_view.size(); idx++) {
+        //         (*acc_vk_view[idx]) = (*acc_vk_view[idx]) + (*inst_vk_view[idx]) * lagranges[inst_idx];
+        //     }
+        // }
+        // next_accumulator->verification_key = acc_vk;
+        //  TODO send vk to verifier too
+        // these two thingies have been folded already
+        // next_accumulator->alpha = instances.alpha;
+        // next_accumulator->relation_parameters = instances.relation_parameters;
+        return next_accumulator;
+    }
 };
 
 extern template class ProtoGalaxyProver_<ProverInstances_<honk::flavor::Ultra, 2>>;
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index 1e13cdd42f3..c78a8457831 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -7,7 +7,7 @@ void ProtoGalaxyVerifier_<VerifierInstances>::prepare_for_folding(std::vector<ui
 {
     transcript = BaseTranscript<FF>{ fold_data };
     auto index = 0;
-    for (auto it = verifier_instances.begin(); it != verifier_instances.end(); it++, index++) {
+    for (auto it = instances.begin(); it != instances.end(); it++, index++) {
         auto inst = *it;
         auto domain_separator = std::to_string(index);
         inst->instance_size = transcript.template receive_from_prover<uint32_t>(domain_separator + "_circuit_size");
@@ -29,15 +29,14 @@ void ProtoGalaxyVerifier_<VerifierInstances>::prepare_for_folding(std::vector<ui
         inst->relation_parameters =
             RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
         inst->alpha = transcript.get_challenge(domain_separator + "_alpha");
+
+        // WORKTODO does verifier need to also fold relation parameters and alpha
     }
 }
 
 template <class VerifierInstances>
-VerifierFoldingResult<typename VerifierInstances::Flavor> ProtoGalaxyVerifier_<
-    VerifierInstances>::fold_public_parameters(std::vector<uint8_t> fold_data)
+bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<uint8_t> fold_data)
 {
-    using Flavor = typename VerifierInstances::Flavor;
-
     prepare_for_folding(fold_data);
     auto delta = transcript.get_challenge("delta");
     auto accumulator = get_accumulator();
@@ -59,36 +58,53 @@ VerifierFoldingResult<typename VerifierInstances::Flavor> ProtoGalaxyVerifier_<
     }
     Univariate<FF, VerifierInstances::BATCHED_EXTENDED_LENGTH, VerifierInstances::NUM> combiner_quotient(
         combiner_quotient_evals);
+
     auto combiner_challenge = transcript.get_challenge("combiner_quotient_challenge");
     auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge);
+
     auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
-    auto lagrange_0_at_challenge = FF(1) - combiner_challenge;
-    auto lagrange_1_at_challenge = combiner_challenge;
+    auto lagranges = std::vector<FF>{ FF(1) - combiner_challenge, combiner_challenge };
+
+    auto expected_next_target_sum =
+        perturbator_at_challenge * lagranges[0] + vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
+    auto next_target_sum = transcript.template receive_from_prover<FF>("next_target_sum");
+    bool verified = (expected_next_target_sum == next_target_sum);
 
-    auto new_target_sum = perturbator_at_challenge * lagrange_0_at_challenge +
-                          vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
+    auto expected_betas_star = update_gate_separation_challenges(
+        perturbator_challenge, accumulator->folding_parameters.gate_separation_challenges, deltas);
+    for (size_t idx = 0; idx < log_instance_size; idx++) {
+        auto expected_beta_star = transcript.template receive_from_prover<FF>("betas_star_" + std::to_string(idx));
+        verified = verified & (expected_betas_star[idx] == expected_beta_star);
+    }
 
-    auto accumulator_public_inputs = accumulator->public_inputs;
-    auto other_public_inputs = verifier_instances[1]->public_inputs;
-    std::vector<FF> folded_public_inputs(accumulator->public_input_size);
+    std::vector<FF> folded_public_inputs(instances[0]->public_inputs.size());
+    auto folded_alpha = FF(0);
+    for (size_t inst_idx = 0; inst_idx < VerifierInstances::NUM; inst_idx++) {
+        auto instance = instances[inst_idx];
+        auto inst_public_inputs = instance->public_inputs;
+        for (size_t el_idx = 0; el_idx < inst_public_inputs.size(); el_idx++) {
+            folded_public_inputs[el_idx] += inst_public_inputs[el_idx] * lagranges[inst_idx];
+        }
+        folded_alpha += instance->alpha + lagranges[inst_idx];
+    }
 
     for (size_t idx = 0; idx < folded_public_inputs.size(); idx++) {
-        folded_public_inputs[idx] = lagrange_0_at_challenge * accumulator->public_inputs[idx] +
-                                    lagrange_1_at_challenge * other_public_inputs[idx];
+        auto public_input = transcript.template receive_from_prover<FF>("folded_public_input" + std::to_string(idx));
+        verified = verified & (public_input == folded_public_inputs[idx]);
     }
+
+    auto alpha = transcript.template receive_from_prover<FF>("folded_alpha");
+    verified = verified & (alpha == folded_alpha);
     // all verification keys have the same size
 
-    auto acc_vk_view = accumulator->verification_key->pointer_view();
-    auto inst_vk_view = verifier_instances[1]->verification_key->pointer_view();
-    for (size_t idx = 0; idx < acc_vk_view.size(); idx++) {
-        (*acc_vk_view[idx]) =
-            (*acc_vk_view[idx]) * lagrange_0_at_challenge + (*inst_vk_view[idx]) * lagrange_1_at_challenge;
-    }
+    // auto acc_vk_view = accumulator->verification_key->pointer_view();
+    // auto inst_vk_view = verifier_instances[1]->verification_key->pointer_view();
+    // for (size_t idx = 0; idx < acc_vk_view.size(); idx++) {
+    //     (*acc_vk_view[idx]) =
+    //         (*acc_vk_view[idx]) * lagrange_0_at_challenge + (*inst_vk_view[idx]) * lagrange_1_at_challenge;
+    // }
 
-    VerifierFoldingResult<Flavor> res;
-    res.parameters.target_sum = new_target_sum;
-    res.folded_public_inputs = folded_public_inputs;
-    return res;
+    return verified;
 }
 
 template class ProtoGalaxyVerifier_<VerifierInstances_<honk::flavor::Ultra, 2>>;
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
index e8f7032cb30..4101b482df8 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
@@ -13,11 +13,11 @@ template <class VerifierInstances> class ProtoGalaxyVerifier_ {
     using FF = typename Flavor::FF;
     using Instance = typename VerifierInstances::Instance;
     using VerificationKey = typename Flavor::VerificationKey;
-    VerifierInstances verifier_instances;
+    VerifierInstances instances;
     BaseTranscript<FF> transcript;
 
     ProtoGalaxyVerifier_(VerifierInstances insts)
-        : verifier_instances(insts){};
+        : instances(insts){};
     ~ProtoGalaxyVerifier_() = default;
     /**
      * @brief For a new round challenge δ at each iteration of the ProtoGalaxy protocol, compute the vector
@@ -33,7 +33,21 @@ template <class VerifierInstances> class ProtoGalaxyVerifier_ {
         return pows;
     }
 
-    std::shared_ptr<Instance> get_accumulator() { return verifier_instances[0]; }
+    static std::vector<FF> update_gate_separation_challenges(const FF perturbator_challenge,
+                                                             const std::vector<FF>& gate_challenges,
+                                                             const std::vector<FF>& round_challenges)
+    {
+        auto log_instance_size = gate_challenges.size();
+        std::vector<FF> next_gate_challenges(log_instance_size);
+        next_gate_challenges[0] = 1;
+
+        for (size_t idx = 1; idx < log_instance_size; idx++) {
+            next_gate_challenges[idx] = gate_challenges[idx] + perturbator_challenge * round_challenges[idx - 1];
+        }
+        return next_gate_challenges;
+    }
+
+    std::shared_ptr<Instance> get_accumulator() { return instances[0]; }
 
     /**
      * @brief Instatiate the VerifierInstances and the VerifierTranscript.
@@ -47,7 +61,7 @@ template <class VerifierInstances> class ProtoGalaxyVerifier_ {
      *
      * TODO(https://github.com/AztecProtocol/barretenberg/issues/690): finalise the implementation of this function
      */
-    VerifierFoldingResult<Flavor> fold_public_parameters(std::vector<uint8_t> fold_data);
+    bool verify_folding_proof(std::vector<uint8_t> fold_data);
 };
 
 extern template class ProtoGalaxyVerifier_<VerifierInstances_<honk::flavor::Ultra, 2>>;
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
index 9a29763e42a..e56ae49ee28 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
@@ -8,6 +8,7 @@ template <typename Flavor_, size_t NUM_> struct ProverInstances_ {
   public:
     static_assert(NUM_ > 0, "Must have at least one prover instance");
     using Flavor = Flavor_;
+    using FoldingParameters = Flavor::FoldingParameters;
     using FF = typename Flavor::FF;
     static constexpr size_t NUM = NUM_;
     using Instance = ProverInstance_<Flavor>;
@@ -22,6 +23,7 @@ template <typename Flavor_, size_t NUM_> struct ProverInstances_ {
     ArrayType _data;
     RelationParameters relation_parameters;
     AlphaType alpha;
+    std::vector<FF> betas_star;
 
     std::shared_ptr<Instance> const& operator[](size_t idx) const { return _data[idx]; }
     typename ArrayType::iterator begin() { return _data.begin(); };
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
index 3d0c700f140..21232f5f116 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
@@ -3,7 +3,6 @@
 #include "barretenberg/flavor/goblin_ultra.hpp"
 #include "barretenberg/flavor/ultra.hpp"
 #include "barretenberg/proof_system/composer/composer_lib.hpp"
-#include "barretenberg/protogalaxy/folding_result.hpp"
 #include "barretenberg/relations/relation_parameters.hpp"
 #include "barretenberg/srs/factories/file_crs_factory.hpp"
 
@@ -48,6 +47,7 @@ template <class Flavor> class ProverInstance_ {
     std::vector<uint32_t> recursive_proof_public_input_indices;
     // non-empty for the accumulated instances
     FoldingParameters folding_parameters;
+    bool is_accumulator = false;
 
     ProverInstance_(Circuit& circuit)
     {
@@ -56,11 +56,11 @@ template <class Flavor> class ProverInstance_ {
         compute_witness(circuit);
     }
 
-    ProverInstance_(FoldingResult<Flavor> result)
-        : verification_key(std::move(result.verification_key))
-        , prover_polynomials(result.folded_prover_polynomials)
-        , public_inputs(result.folded_public_inputs)
-        , folding_parameters(result.folding_parameters){};
+    // ProverInstance_(FoldingResult<Flavor> result)
+    //     : verification_key(std::move(result.verification_key))
+    //     , prover_polynomials(result.folded_prover_polynomials)
+    //     , public_inputs(result.folded_public_inputs)
+    //     , folding_parameters(result.folding_parameters){};
 
     ProverInstance_() = default;
     ~ProverInstance_() = default;
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
index 290e9afcd01..9edbfd4bfd1 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
@@ -159,11 +159,9 @@ TEST_F(ProtoGalaxyTests, PerturbatorPolynomial)
         target_sum += full_honk_evals[i] * pow_beta[i];
     }
 
-    auto accumulator = std::make_shared<Instance>(
-        FoldingResult<Flavor>{ .folded_prover_polynomials = full_polynomials,
-                               .folded_public_inputs = std::vector<FF>{},
-                               .verification_key = std::make_shared<Flavor::VerificationKey>(),
-                               .folding_parameters = { betas, target_sum } });
+    auto accumulator = std::make_shared<Instance>();
+    accumulator->prover_polynomials = full_polynomials;
+    accumulator->folding_parameters = { betas, target_sum };
     accumulator->relation_parameters = relation_parameters;
     accumulator->alpha = alpha;
 
@@ -252,22 +250,55 @@ TEST_F(ProtoGalaxyTests, FoldAlpha)
     EXPECT_EQ(instances.alpha, expected_alpha);
 }
 
-// TEST_F(ProtoGalaxyTests, ComputeNewAccumulator)
-// {
-//     using Instances = ProverInstances_<Flavor, 2>;
-//     using Instance = typename Instances::Instance;
+TEST_F(ProtoGalaxyTests, ComputeNewAccumulator)
+{
+    const size_t log_instance_size(4);
+    const size_t instance_size(1 << log_instance_size);
+
+    std::array<barretenberg::Polynomial<FF>, NUM_POLYNOMIALS> random_polynomials;
+    for (auto& poly : random_polynomials) {
+        poly = get_random_polynomial(instance_size);
+    }
+    auto full_polynomials = construct_ultra_full_polynomials(random_polynomials);
+    auto relation_parameters = proof_system::RelationParameters<FF>::get_random();
+    auto alpha = FF::random_element();
+
+    auto full_honk_evals =
+        ProtoGalaxyProver::compute_full_honk_evaluations(full_polynomials, alpha, relation_parameters);
+    std::vector<FF> betas(log_instance_size);
+    for (size_t idx = 0; idx < log_instance_size; idx++) {
+        betas[idx] = FF::random_element();
+    }
+
+    // Construct pow(\vec{betas}) as in the paper
+    auto pow_beta = ProtoGalaxyProver::compute_pow_polynomial_at_values(betas, instance_size);
+
+    // Compute the corresponding target sum and create a dummy accumulator
+    auto target_sum = FF(0);
+    for (size_t i = 0; i < instance_size; i++) {
+        target_sum += full_honk_evals[i] * pow_beta[i];
+    }
+
+    auto accumulator = std::make_shared<Instance>();
+    accumulator->prover_polynomials = full_polynomials;
+    accumulator->folding_parameters = { betas, target_sum };
+    accumulator->relation_parameters = relation_parameters;
+    accumulator->alpha = alpha;
+    accumulator->is_accumulator = true;
+    accumulator->public_inputs = std::vector<FF>{ FF::random_element() };
 
-//     Builder builder1;
-//     builder1.add_variable(3);
-//     auto instance1 = std::make_shared<Instance>(builder1);
+    auto builder = typename Flavor::CircuitBuilder();
+    auto composer = UltraComposer();
+    builder.add_public_variable(FF(1));
 
-//     Builder builder2;
-//     builder2.add_variable(3);
-//     auto instance2 = std::make_shared<Instance>(builder2);
+    auto instance = composer.create_instance(builder);
+    auto instances = std::vector<std::shared_ptr<Instance>>{ accumulator, instance };
+    auto folding_prover = composer.create_folding_prover(instances);
+    // auto folding_verifier = composer.create_folding_verifier(instances);
 
-//     Instances instances{ { instance1, instance2 } };
+    auto result = folding_prover.fold_instances();
 
-//     auto acc = ProtoGalaxyProver::compute_new_accumulator(instances, std::vector<FF>{ FF(1), FF(2) });
-// }
+    // folding_verifier.verify_folding_proof(result.folding_data);
+}
 
 } // namespace protogalaxy_tests
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.cpp
index ff8cf11b869..8726dd30d16 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.cpp
@@ -13,6 +13,7 @@ std::shared_ptr<ProverInstance_<Flavor>> UltraComposer_<Flavor>::create_instance
     circuit.finalize_circuit();
     auto instance = std::make_shared<Instance>(circuit);
     instance->commitment_key = compute_commitment_key(instance->proving_key->circuit_size);
+    instance->compute_verification_key();
     return instance;
 }
 
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp
index 3100efad8e9..2e8178f985e 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp
@@ -108,7 +108,7 @@ template <UltraFlavor Flavor> class UltraComposer_ {
     {
         std::vector<std::shared_ptr<VerificationKey>> vks;
         for (const auto& inst : instances) {
-            vks.emplace_back(inst->compute_verification_key());
+            vks.emplace_back(inst->verification_key);
         }
         VerifierInstances insts(vks);
         ProtoGalaxyVerifier_<VerifierInstances> output_state(insts);
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_transcript.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_transcript.test.cpp
index cd6cdf1b40d..72153b723c2 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_transcript.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_transcript.test.cpp
@@ -96,7 +96,7 @@ class UltraTranscriptTests : public ::testing::Test {
     void generate_test_circuit(auto& builder)
     {
         FF a = 1;
-        builder.add_variable(a);
+        // builder.add_variable(a);
         builder.add_public_variable(a);
     }
 
@@ -124,6 +124,7 @@ TEST_F(UltraTranscriptTests, ProverManifestConsistency)
     auto composer = UltraComposer();
     auto instance = composer.create_instance(builder);
     auto prover = composer.create_prover(instance);
+
     auto proof = prover.construct_proof();
 
     // Check that the prover generated manifest agrees with the manifest hard coded in this suite

From 985c006728f135e126d251b7fe72fa2cc6cdae47 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Tue, 28 Nov 2023 09:58:46 +0000
Subject: [PATCH 31/86] hmm

---
 .../src/barretenberg/flavor/goblin_ultra.hpp  |   4 +-
 .../cpp/src/barretenberg/flavor/ultra.hpp     |  69 ++++----
 .../protogalaxy/protogalaxy_prover.cpp        |  79 +++++++---
 .../protogalaxy/protogalaxy_prover.hpp        |  30 ++--
 .../protogalaxy/protogalaxy_verifier.cpp      | 149 ++++++++++++++----
 .../protogalaxy/protogalaxy_verifier.hpp      |   4 +-
 .../sumcheck/instance/instances.hpp           |  15 +-
 .../sumcheck/instance/prover_instance.hpp     |   4 +-
 .../sumcheck/instance/verifier_instance.hpp   |   3 +-
 .../ultra_honk/protogalaxy.test.cpp           |  47 +++++-
 .../ultra_honk/ultra_composer.hpp             |   9 +-
 11 files changed, 302 insertions(+), 111 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
index ad3b536ed47..01c38b294c5 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
@@ -85,12 +85,12 @@ class GoblinUltra {
      */
     class PrecomputedEntities : public PrecomputedEntities_<DataType, HandleType, NUM_PRECOMPUTED_ENTITIES> {
       public:
-        DataType q_m;             // column 0
         DataType q_c;             // column 1
         DataType q_l;             // column 2
         DataType q_r;             // column 3
         DataType q_o;             // column 4
         DataType q_4;             // column 5
+        DataType q_m;             // column 0
         DataType q_arith;         // column 6
         DataType q_sort;          // column 7
         DataType q_elliptic;      // column 8
@@ -115,12 +115,12 @@ class GoblinUltra {
         DataType databus_id;      // column 27 // id polynomial, i.e. id_i = i
 
         DEFINE_POINTER_VIEW(NUM_PRECOMPUTED_ENTITIES,
-                            &q_m,
                             &q_c,
                             &q_l,
                             &q_r,
                             &q_o,
                             &q_4,
+                            &q_m,
                             &q_arith,
                             &q_sort,
                             &q_elliptic,
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
index 06a08bd4ae5..05ae168760d 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
@@ -82,12 +82,12 @@ class Ultra {
      */
     class PrecomputedEntities : public PrecomputedEntities_<DataType, HandleType, NUM_PRECOMPUTED_ENTITIES> {
       public:
-        DataType q_m;            // column 0
         DataType q_c;            // column 1
         DataType q_l;            // column 2
         DataType q_r;            // column 3
         DataType q_o;            // column 4
         DataType q_4;            // column 5
+        DataType q_m;            // column 0
         DataType q_arith;        // column 6
         DataType q_sort;         // column 7
         DataType q_elliptic;     // column 8
@@ -109,12 +109,12 @@ class Ultra {
         DataType lagrange_last;  // column 24
 
         DEFINE_POINTER_VIEW(NUM_PRECOMPUTED_ENTITIES,
-                            &q_m,
                             &q_c,
                             &q_l,
                             &q_r,
                             &q_o,
                             &q_4,
+                            &q_m,
                             &q_arith,
                             &q_sort,
                             &q_elliptic,
@@ -196,12 +196,12 @@ class Ultra {
     template <typename DataType, typename HandleType>
     class AllEntities : public AllEntities_<DataType, HandleType, NUM_ALL_ENTITIES> {
       public:
+        DataType q_m;                // column 5
         DataType q_c;                // column 0
         DataType q_l;                // column 1
         DataType q_r;                // column 2
         DataType q_o;                // column 3
         DataType q_4;                // column 4
-        DataType q_m;                // column 5
         DataType q_arith;            // column 6
         DataType q_sort;             // column 7
         DataType q_elliptic;         // column 8
@@ -296,6 +296,16 @@ class Ultra {
 
             };
         };
+
+        std::vector<HandleType> get_precomputed()
+        {
+            return { q_c,          q_l,   q_r,      q_o,     q_4,     q_m,     q_arith, q_sort,
+                     q_elliptic,   q_aux, q_lookup, sigma_1, sigma_2, sigma_3, sigma_4, id_1,
+                     id_2,         id_3,  id_4,     table_1, table_2, table_3, table_4, lagrange_first,
+                     lagrange_last
+
+            };
+        }
         std::vector<HandleType> get_to_be_shifted() override
         {
             return { table_1, table_2, table_3, table_4, w_l, w_r, w_o, w_4, sorted_accum, z_perm, z_lookup };
@@ -411,32 +421,31 @@ class Ultra {
             z_lookup = "Z_LOOKUP";
             sorted_accum = "SORTED_ACCUM";
 
-            // The ones beginning with "__" are only used for debugging
-            q_c = "__Q_C";
-            q_l = "__Q_L";
-            q_r = "__Q_R";
-            q_o = "__Q_O";
-            q_4 = "__Q_4";
-            q_m = "__Q_M";
-            q_arith = "__Q_ARITH";
-            q_sort = "__Q_SORT";
-            q_elliptic = "__Q_ELLIPTIC";
-            q_aux = "__Q_AUX";
-            q_lookup = "__Q_LOOKUP";
-            sigma_1 = "__SIGMA_1";
-            sigma_2 = "__SIGMA_2";
-            sigma_3 = "__SIGMA_3";
-            sigma_4 = "__SIGMA_4";
-            id_1 = "__ID_1";
-            id_2 = "__ID_2";
-            id_3 = "__ID_3";
-            id_4 = "__ID_4";
-            table_1 = "__TABLE_1";
-            table_2 = "__TABLE_2";
-            table_3 = "__TABLE_3";
-            table_4 = "__TABLE_4";
-            lagrange_first = "__LAGRANGE_FIRST";
-            lagrange_last = "__LAGRANGE_LAST";
+            q_c = "Q_C";
+            q_l = "Q_L";
+            q_r = "Q_R";
+            q_o = "Q_O";
+            q_4 = "Q_4";
+            q_m = "Q_M";
+            q_arith = "Q_ARITH";
+            q_sort = "Q_SORT";
+            q_elliptic = "Q_ELLIPTIC";
+            q_aux = "Q_AUX";
+            q_lookup = "Q_LOOKUP";
+            sigma_1 = "SIGMA_1";
+            sigma_2 = "SIGMA_2";
+            sigma_3 = "SIGMA_3";
+            sigma_4 = "SIGMA_4";
+            id_1 = "ID_1";
+            id_2 = "ID_2";
+            id_3 = "ID_3";
+            id_4 = "ID_4";
+            table_1 = "TABLE_1";
+            table_2 = "TABLE_2";
+            table_3 = "TABLE_3";
+            table_4 = "TABLE_4";
+            lagrange_first = "LAGRANGE_FIRST";
+            lagrange_last = "LAGRANGE_LAST";
         };
     };
 
@@ -447,11 +456,11 @@ class Ultra {
         {
             static_cast<void>(transcript);
             q_m = verification_key->q_m;
+            q_c = verification_key->q_c;
             q_l = verification_key->q_l;
             q_r = verification_key->q_r;
             q_o = verification_key->q_o;
             q_4 = verification_key->q_4;
-            q_c = verification_key->q_c;
             q_arith = verification_key->q_arith;
             q_sort = verification_key->q_sort;
             q_elliptic = verification_key->q_elliptic;
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index 76a5a2b4a91..4a0eaa2c4f3 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -7,30 +7,62 @@ template <class ProverInstances> void ProtoGalaxyProver_<ProverInstances>::prepa
     auto idx = 0;
     for (auto it = instances.begin(); it != instances.end(); it++, idx++) {
         auto instance = *it;
+        auto domain_separator = std::to_string(idx);
+        transcript.send_to_verifier(domain_separator + "is_accumulator", instance->is_accumulator);
         if (static_cast<bool>(instance->is_accumulator)) {
-            continue;
-        }
-        instance->initialize_prover_polynomials();
+            const auto instance_size = static_cast<uint32_t>(instance->instance_size);
+            const auto num_public_inputs = static_cast<uint32_t>(instance->public_inputs.size());
 
-        auto domain_separator = std::to_string(idx);
-        const auto circuit_size = static_cast<uint32_t>(instance->proving_key->circuit_size);
-        const auto num_public_inputs = static_cast<uint32_t>(instance->proving_key->num_public_inputs);
+            transcript.send_to_verifier(domain_separator + "_instance_size", instance_size);
+            transcript.send_to_verifier(domain_separator + "_public_input_size", num_public_inputs);
 
-        transcript.send_to_verifier(domain_separator + "_circuit_size", circuit_size);
-        transcript.send_to_verifier(domain_separator + "_public_input_size", num_public_inputs);
-        transcript.send_to_verifier(domain_separator + "_pub_inputs_offset",
-                                    static_cast<uint32_t>(instance->pub_inputs_offset));
+            for (size_t i = 0; i < instance->public_inputs.size(); ++i) {
+                auto public_input_i = instance->public_inputs[i];
+                transcript.send_to_verifier(domain_separator + "_public_input_" + std::to_string(i), public_input_i);
+            }
+            transcript.send_to_verifier(domain_separator + "_eta", instance->relation_parameters.eta);
+            transcript.send_to_verifier(domain_separator + "_beta", instance->relation_parameters.beta);
+            transcript.send_to_verifier(domain_separator + "_gamma", instance->relation_parameters.gamma);
+            transcript.send_to_verifier(domain_separator + "_public_input_delta",
+                                        instance->relation_parameters.public_input_delta);
+            transcript.send_to_verifier(domain_separator + "_lookup_grand_product_delta",
+                                        instance->relation_parameters.lookup_grand_product_delta);
 
-        for (size_t i = 0; i < instance->proving_key->num_public_inputs; ++i) {
-            auto public_input_i = instance->public_inputs[i];
-            transcript.send_to_verifier(domain_separator + "_public_input_" + std::to_string(i), public_input_i);
-        }
+            transcript.send_to_verifier(domain_separator + "_alpha", instance->alpha);
+
+            auto folding_parameters = instance->folding_parameters;
+            transcript.send_to_verifier(domain_separator + "_target_sum", folding_parameters.target_sum);
+            for (size_t idx = 0; idx < folding_parameters.gate_separation_challenges.size(); idx++) {
+                transcript.send_to_verifier(domain_separator + "_gate_challenge_" + std::to_string(idx),
+                                            folding_parameters.gate_separation_challenges[idx]);
+            }
 
-        auto [eta, beta, gamma] = transcript.get_challenges(
-            domain_separator + "_eta", domain_separator + "_beta", domain_separator + "_gamma");
-        instance->compute_sorted_accumulator_polynomials(eta);
-        instance->compute_grand_product_polynomials(beta, gamma);
-        instance->alpha = transcript.get_challenge(domain_separator + "_alpha");
+        } else {
+            instance->initialize_prover_polynomials();
+            const auto instance_size = static_cast<uint32_t>(instance->instance_size);
+            const auto num_public_inputs = static_cast<uint32_t>(instance->public_inputs.size());
+
+            transcript.send_to_verifier(domain_separator + "_instance_size", instance_size);
+            transcript.send_to_verifier(domain_separator + "_public_input_size", num_public_inputs);
+            transcript.send_to_verifier(domain_separator + "_pub_inputs_offset",
+                                        static_cast<uint32_t>(instance->pub_inputs_offset));
+
+            for (size_t i = 0; i < num_public_inputs; ++i) {
+                auto public_input_i = instance->public_inputs[i];
+                transcript.send_to_verifier(domain_separator + "_public_input_" + std::to_string(i), public_input_i);
+            }
+
+            auto [eta, beta, gamma] = transcript.get_challenges(
+                domain_separator + "_eta", domain_separator + "_beta", domain_separator + "_gamma");
+            instance->compute_sorted_accumulator_polynomials(eta);
+            instance->compute_grand_product_polynomials(beta, gamma);
+            instance->alpha = transcript.get_challenge(domain_separator + "_alpha");
+        }
+        auto vk_view = instance->verification_key->pointer_view();
+        auto labels = typename Flavor::CommitmentLabels().get_precomputed();
+        for (size_t idx = 0; idx < labels.size(); idx++) {
+            transcript.send_to_verifier(domain_separator + labels[idx], (*vk_view[idx]));
+        }
     }
 
     fold_relation_parameters(instances);
@@ -46,6 +78,7 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
     // first time and accumulator is 0
     // TODO(#https://github.com/AztecProtocol/barretenberg/issues/763): Fold alpha
     auto delta = transcript.get_challenge("delta");
+    info(delta);
     auto accumulator = get_accumulator();
     auto instance_size = accumulator->prover_polynomials.get_polynomial_size();
     const auto log_instance_size = static_cast<size_t>(numeric::get_msb(instance_size));
@@ -55,8 +88,10 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
     for (size_t idx = 0; idx <= log_instance_size; idx++) {
         transcript.send_to_verifier("perturbator_" + std::to_string(idx), perturbator[idx]);
     }
+    assert(perturbator[0] == accumulator->folding_parameters.target_sum);
 
     auto perturbator_challenge = transcript.get_challenge("perturbator_challenge");
+    info("pc", perturbator_challenge);
     instances.betas_star = update_gate_separation_challenges(
         perturbator_challenge, accumulator->folding_parameters.gate_separation_challenges, deltas);
 
@@ -65,10 +100,12 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
     auto compressed_perturbator = perturbator.evaluate(perturbator_challenge);
     auto combiner_quotient = compute_combiner_quotient(compressed_perturbator, combiner);
 
-    for (size_t idx = ProverInstances::NUM; idx < combiner_quotient.size(); idx++) {
+    for (size_t idx = ProverInstances::NUM; idx < ProverInstances::BATCHED_EXTENDED_LENGTH; idx++) {
+        info(combiner_quotient.value_at(idx));
         transcript.send_to_verifier("combiner_quotient_" + std::to_string(idx), combiner_quotient.value_at(idx));
     }
     auto combiner_challenge = transcript.get_challenge("combiner_quotient_challenge");
+    info("cc", combiner_challenge);
     // TODO(https://github.com/AztecProtocol/barretenberg/issues/764): Generalize these formulas as well as computation
     // of Lagrange basis
 
@@ -80,5 +117,5 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
     return res;
 }
 template class ProtoGalaxyProver_<ProverInstances_<honk::flavor::Ultra, 2>>;
-template class ProtoGalaxyProver_<ProverInstances_<honk::flavor::GoblinUltra, 2>>;
+// template class ProtoGalaxyProver_<ProverInstances_<honk::flavor::GoblinUltra, 2>>;
 } // namespace proof_system::honk
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index cb5a3de1d68..38cb45eedff 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -442,7 +442,6 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
             for (auto [acc_poly_view, inst_poly_view] : zip_view(
                      acc_prover_polynomials.pointer_view(), instances[inst_idx]->prover_polynomials.pointer_view())) {
                 for (size_t el_idx = 0; el_idx < inst_poly_view->size(); el_idx++) {
-                    info((*acc_poly_view)[el_idx]);
                     (*acc_poly_view)[el_idx] += (*inst_poly_view)[el_idx] * lagranges[inst_idx];
                 }
             }
@@ -464,16 +463,23 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         next_accumulator->alpha = instances.alpha.evaluate(challenge);
         transcript.send_to_verifier("folded_alpha", next_accumulator->alpha);
 
-        // auto acc_vk = std::make_shared<VerificationKey>(instances[0]->prover_polynomials.get_polynomial_size(),
-        //                                                 instances[0]->public_inputs.size());
-        // auto acc_vk_view = acc_vk->pointer_view();
-        // for (size_t inst_idx = 0; inst_idx < ProverInstances::NUM; inst_idx++) {
-        //     auto inst_vk_view = instances[inst_idx]->verification_key->pointer_view();
-        //     for (size_t idx = 0; idx < inst_vk_view.size(); idx++) {
-        //         (*acc_vk_view[idx]) = (*acc_vk_view[idx]) + (*inst_vk_view[idx]) * lagranges[inst_idx];
-        //     }
-        // }
-        // next_accumulator->verification_key = acc_vk;
+        auto acc_vk = std::make_shared<VerificationKey>(instances[0]->prover_polynomials.get_polynomial_size(),
+                                                        instances[0]->public_inputs.size());
+        auto acc_vk_view = acc_vk->pointer_view();
+        for (size_t inst_idx = 0; inst_idx < ProverInstances::NUM; inst_idx++) {
+            auto inst_vk_view = instances[inst_idx]->verification_key->pointer_view();
+            for (size_t idx = 0; idx < inst_vk_view.size(); idx++) {
+                (*acc_vk_view[idx]) = (*acc_vk_view[idx]) + (*inst_vk_view[idx]) * lagranges[inst_idx];
+            }
+        }
+        next_accumulator->verification_key = acc_vk;
+        auto labels = typename Flavor::CommitmentLabels().get_precomputed();
+        for (size_t idx = 0; idx < labels.size(); idx++) {
+            info(labels[idx]);
+            info((*acc_vk_view[idx]));
+            transcript.send_to_verifier("new" + labels[idx], (*acc_vk_view[idx]));
+        }
+        // auto a = labels.get_precomputed();
         //  TODO send vk to verifier too
         // these two thingies have been folded already
         // next_accumulator->alpha = instances.alpha;
@@ -483,5 +489,5 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
 };
 
 extern template class ProtoGalaxyProver_<ProverInstances_<honk::flavor::Ultra, 2>>;
-extern template class ProtoGalaxyProver_<ProverInstances_<honk::flavor::GoblinUltra, 2>>;
+// extern template class ProtoGalaxyProver_<ProverInstances_<honk::flavor::GoblinUltra, 2>>;
 } // namespace proof_system::honk
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index c78a8457831..aed797067e1 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -10,27 +10,96 @@ void ProtoGalaxyVerifier_<VerifierInstances>::prepare_for_folding(std::vector<ui
     for (auto it = instances.begin(); it != instances.end(); it++, index++) {
         auto inst = *it;
         auto domain_separator = std::to_string(index);
-        inst->instance_size = transcript.template receive_from_prover<uint32_t>(domain_separator + "_circuit_size");
-        inst->public_input_size =
-            transcript.template receive_from_prover<uint32_t>(domain_separator + "_public_input_size");
-        inst->pub_inputs_offset =
-            transcript.template receive_from_prover<uint32_t>(domain_separator + "_pub_inputs_offset");
-
-        for (size_t i = 0; i < inst->public_input_size; ++i) {
-            auto public_input_i =
-                transcript.template receive_from_prover<FF>(domain_separator + "_public_input_" + std::to_string(i));
-            inst->public_inputs.emplace_back(public_input_i);
+        inst->is_accumulator = transcript.template receive_from_prover<uint32_t>(domain_separator + "is_accumulator");
+        if (static_cast<bool>(inst->is_accumulator)) {
+            inst->instance_size =
+                transcript.template receive_from_prover<uint32_t>(domain_separator + "_instance_size");
+            inst->public_input_size =
+                transcript.template receive_from_prover<uint32_t>(domain_separator + "_public_input_size");
+
+            for (size_t i = 0; i < inst->public_input_size; ++i) {
+                auto public_input_i = transcript.template receive_from_prover<FF>(domain_separator + "_public_input_" +
+                                                                                  std::to_string(i));
+                inst->public_inputs.emplace_back(public_input_i);
+            }
+
+            auto eta = transcript.template receive_from_prover<FF>(domain_separator + "_eta");
+            auto beta = transcript.template receive_from_prover<FF>(domain_separator + "_beta");
+            auto gamma = transcript.template receive_from_prover<FF>(domain_separator + "_gamma");
+            auto public_input_delta =
+                transcript.template receive_from_prover<FF>(domain_separator + "_public_input_delta");
+            auto lookup_grand_product_delta =
+                transcript.template receive_from_prover<FF>(domain_separator + "_lookup_grand_product_delta");
+            inst->relation_parameters =
+                RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
+            inst->alpha = transcript.template receive_from_prover<FF>(domain_separator + "_alpha");
+
+            inst->folding_parameters.target_sum =
+                transcript.template receive_from_prover<FF>(domain_separator + "_target_sum");
+
+            auto log_inst_size = static_cast<size_t>(numeric::get_msb(inst->instance_size));
+            inst->folding_parameters.gate_separation_challenges = std::vector<FF>(log_inst_size);
+            for (size_t idx = 0; idx < log_inst_size; idx++) {
+                inst->folding_parameters.gate_separation_challenges[idx] = transcript.template receive_from_prover<FF>(
+                    domain_separator + "_gate_challenge_" + std::to_string(idx));
+            }
+        } else {
+
+            inst->instance_size =
+                transcript.template receive_from_prover<uint32_t>(domain_separator + "_instance_size");
+            inst->public_input_size =
+                transcript.template receive_from_prover<uint32_t>(domain_separator + "_public_input_size");
+            inst->pub_inputs_offset =
+                transcript.template receive_from_prover<uint32_t>(domain_separator + "_pub_inputs_offset");
+
+            for (size_t i = 0; i < inst->public_input_size; ++i) {
+                auto public_input_i = transcript.template receive_from_prover<FF>(domain_separator + "_public_input_" +
+                                                                                  std::to_string(i));
+                inst->public_inputs.emplace_back(public_input_i);
+            }
+            auto [eta, beta, gamma] = transcript.get_challenges(
+                domain_separator + "_eta", domain_separator + "_beta", domain_separator + "_gamma");
+            const FF public_input_delta = compute_public_input_delta<Flavor>(
+                inst->public_inputs, beta, gamma, inst->instance_size, inst->pub_inputs_offset);
+            const FF lookup_grand_product_delta =
+                compute_lookup_grand_product_delta<FF>(beta, gamma, inst->instance_size);
+            inst->relation_parameters =
+                RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
+            inst->alpha = transcript.get_challenge(domain_separator + "_alpha");
         }
-        auto [eta, beta, gamma] = transcript.get_challenges(
-            domain_separator + "_eta", domain_separator + "_beta", domain_separator + "_gamma");
-        const FF public_input_delta = compute_public_input_delta<Flavor>(
-            inst->public_inputs, beta, gamma, inst->instance_size, inst->pub_inputs_offset);
-        const FF lookup_grand_product_delta = compute_lookup_grand_product_delta<FF>(beta, gamma, inst->instance_size);
-        inst->relation_parameters =
-            RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
-        inst->alpha = transcript.get_challenge(domain_separator + "_alpha");
-
-        // WORKTODO does verifier need to also fold relation parameters and alpha
+
+        auto labels = typename Flavor::CommitmentLabels();
+        auto vk = std::make_shared<VerificationKey>(inst->instance_size, inst->public_input_size);
+
+        vk->q_m = transcript.template receive_from_prover<Commitment>(domain_separator + labels.q_m);
+        vk->q_l = transcript.template receive_from_prover<Commitment>(domain_separator + labels.q_l);
+        vk->q_r = transcript.template receive_from_prover<Commitment>(domain_separator + labels.q_r);
+        vk->q_o = transcript.template receive_from_prover<Commitment>(domain_separator + labels.q_o);
+        vk->q_c = transcript.template receive_from_prover<Commitment>(domain_separator + labels.q_c);
+        vk->sigma_1 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.sigma_1);
+        vk->sigma_2 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.sigma_2);
+        vk->sigma_3 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.sigma_3);
+        vk->id_1 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.id_1);
+        vk->id_2 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.id_2);
+        vk->id_3 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.id_3);
+        vk->lagrange_first =
+            transcript.template receive_from_prover<Commitment>(domain_separator + labels.lagrange_first);
+        vk->lagrange_last =
+            transcript.template receive_from_prover<Commitment>(domain_separator + labels.lagrange_last);
+
+        vk->q_4 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.q_4);
+        vk->q_arith = transcript.template receive_from_prover<Commitment>(domain_separator + labels.q_arith);
+        vk->q_sort = transcript.template receive_from_prover<Commitment>(domain_separator + labels.q_sort);
+        vk->q_elliptic = transcript.template receive_from_prover<Commitment>(domain_separator + labels.q_elliptic);
+        vk->q_aux = transcript.template receive_from_prover<Commitment>(domain_separator + labels.q_aux);
+        vk->q_lookup = transcript.template receive_from_prover<Commitment>(domain_separator + labels.q_lookup);
+        vk->sigma_4 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.sigma_4);
+        vk->id_4 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.id_4);
+        vk->table_1 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.table_1);
+        vk->table_2 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.table_2);
+        vk->table_3 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.table_3);
+        vk->table_4 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.table_4);
+        inst->verification_key = std::move(vk);
     }
 }
 
@@ -39,6 +108,7 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
 {
     prepare_for_folding(fold_data);
     auto delta = transcript.get_challenge("delta");
+    info(delta);
     auto accumulator = get_accumulator();
     auto log_instance_size = static_cast<size_t>(numeric::get_msb(accumulator->instance_size));
     auto deltas = compute_round_challenge_pows(log_instance_size, delta);
@@ -46,20 +116,26 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
     for (size_t idx = 0; idx <= log_instance_size; idx++) {
         perturbator_coeffs[idx] = transcript.template receive_from_prover<FF>("perturbator_" + std::to_string(idx));
     }
+    assert(perturbator_coeffs[0] == accumulator->folding_parameters.target_sum);
     auto perturbator = Polynomial<FF>(perturbator_coeffs);
     auto perturbator_challenge = transcript.get_challenge("perturbator_challenge");
+    info("pc", perturbator_challenge);
     auto perturbator_at_challenge = perturbator.evaluate(perturbator_challenge);
 
     // Thed degree of K(X) is dk - k - 1 = k(d - 1) - 1. Hence we need  k(d - 1) evaluations to represent it.
-    std::array<FF, VerifierInstances::BATCHED_EXTENDED_LENGTH - VerifierInstances::NUM> combiner_quotient_evals = {};
+    std::array<FF, VerifierInstances::BATCHED_EXTENDED_LENGTH - VerifierInstances::NUM> combiner_quotient_evals;
+    info(VerifierInstances::BATCHED_EXTENDED_LENGTH);
+    info(VerifierInstances::NUM);
     for (size_t idx = 0; idx < VerifierInstances::BATCHED_EXTENDED_LENGTH - VerifierInstances::NUM; idx++) {
         combiner_quotient_evals[idx] = transcript.template receive_from_prover<FF>(
             "combiner_quotient_" + std::to_string(idx + VerifierInstances::NUM));
+        info(combiner_quotient_evals[idx]);
     }
     Univariate<FF, VerifierInstances::BATCHED_EXTENDED_LENGTH, VerifierInstances::NUM> combiner_quotient(
         combiner_quotient_evals);
 
     auto combiner_challenge = transcript.get_challenge("combiner_quotient_challenge");
+    info("cc", combiner_challenge);
     auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge);
 
     auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
@@ -69,12 +145,14 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
         perturbator_at_challenge * lagranges[0] + vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
     auto next_target_sum = transcript.template receive_from_prover<FF>("next_target_sum");
     bool verified = (expected_next_target_sum == next_target_sum);
+    assert(expected_next_target_sum == next_target_sum);
 
     auto expected_betas_star = update_gate_separation_challenges(
         perturbator_challenge, accumulator->folding_parameters.gate_separation_challenges, deltas);
     for (size_t idx = 0; idx < log_instance_size; idx++) {
         auto expected_beta_star = transcript.template receive_from_prover<FF>("betas_star_" + std::to_string(idx));
         verified = verified & (expected_betas_star[idx] == expected_beta_star);
+        assert(expected_betas_star[idx] == expected_beta_star);
     }
 
     std::vector<FF> folded_public_inputs(instances[0]->public_inputs.size());
@@ -85,28 +163,41 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
         for (size_t el_idx = 0; el_idx < inst_public_inputs.size(); el_idx++) {
             folded_public_inputs[el_idx] += inst_public_inputs[el_idx] * lagranges[inst_idx];
         }
-        folded_alpha += instance->alpha + lagranges[inst_idx];
+        folded_alpha += instance->alpha * lagranges[inst_idx];
     }
 
     for (size_t idx = 0; idx < folded_public_inputs.size(); idx++) {
         auto public_input = transcript.template receive_from_prover<FF>("folded_public_input" + std::to_string(idx));
         verified = verified & (public_input == folded_public_inputs[idx]);
+        assert(public_input == folded_public_inputs[idx]);
     }
 
     auto alpha = transcript.template receive_from_prover<FF>("folded_alpha");
     verified = verified & (alpha == folded_alpha);
+    assert(alpha == folded_alpha);
     // all verification keys have the same size
 
-    // auto acc_vk_view = accumulator->verification_key->pointer_view();
-    // auto inst_vk_view = verifier_instances[1]->verification_key->pointer_view();
-    // for (size_t idx = 0; idx < acc_vk_view.size(); idx++) {
-    //     (*acc_vk_view[idx]) =
-    //         (*acc_vk_view[idx]) * lagrange_0_at_challenge + (*inst_vk_view[idx]) * lagrange_1_at_challenge;
-    // }
+    auto acc_vk = std::make_shared<VerificationKey>(instances[0]->instance_size, instances[0]->public_input_size);
+    auto acc_vk_view = acc_vk->pointer_view();
+    for (size_t inst_idx = 0; inst_idx < VerifierInstances::NUM; inst_idx++) {
+        auto inst_vk_view = instances[inst_idx]->verification_key->pointer_view();
+        for (size_t idx = 0; idx < inst_vk_view.size(); idx++) {
+            (*acc_vk_view[idx]) = (*acc_vk_view[idx]) + (*inst_vk_view[idx]) * lagranges[inst_idx];
+        }
+    }
+
+    auto labels = typename Flavor::CommitmentLabels().get_precomputed();
+    for (size_t idx = 0; idx < labels.size(); idx++) {
+        auto vk = transcript.template receive_from_prover<Commitment>("new" + labels[idx]);
+        info(labels[idx]);
+        info((*acc_vk_view[idx]));
+        assert(vk == (*acc_vk_view[idx]));
+        verified = verified & (vk == (*acc_vk_view[idx]));
+    }
 
     return verified;
 }
 
 template class ProtoGalaxyVerifier_<VerifierInstances_<honk::flavor::Ultra, 2>>;
-template class ProtoGalaxyVerifier_<VerifierInstances_<honk::flavor::GoblinUltra, 2>>;
+// template class ProtoGalaxyVerifier_<VerifierInstances_<honk::flavor::GoblinUltra, 2>>;
 } // namespace proof_system::honk
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
index 4101b482df8..7e13939b238 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
@@ -11,8 +11,10 @@ template <class VerifierInstances> class ProtoGalaxyVerifier_ {
   public:
     using Flavor = typename VerifierInstances::Flavor;
     using FF = typename Flavor::FF;
+    using Commitment = typename Flavor::Commitment;
     using Instance = typename VerifierInstances::Instance;
     using VerificationKey = typename Flavor::VerificationKey;
+
     VerifierInstances instances;
     BaseTranscript<FF> transcript;
 
@@ -65,5 +67,5 @@ template <class VerifierInstances> class ProtoGalaxyVerifier_ {
 };
 
 extern template class ProtoGalaxyVerifier_<VerifierInstances_<honk::flavor::Ultra, 2>>;
-extern template class ProtoGalaxyVerifier_<VerifierInstances_<honk::flavor::GoblinUltra, 2>>;
+// extern template class ProtoGalaxyVerifier_<VerifierInstances_<honk::flavor::GoblinUltra, 2>>;
 } // namespace proof_system::honk
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
index e56ae49ee28..4e4e9e0334e 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
@@ -100,12 +100,19 @@ template <typename Flavor_, size_t NUM_> struct VerifierInstances_ {
     std::shared_ptr<Instance> const& operator[](size_t idx) const { return _data[idx]; }
     typename ArrayType::iterator begin() { return _data.begin(); };
     typename ArrayType::iterator end() { return _data.end(); };
-    VerifierInstances_(std::vector<std::shared_ptr<VerificationKey>> vks)
+    // VerifierInstances_(std::vector<std::shared_ptr<VerificationKey>> vks)
+    // {
+    //     ASSERT(vks.size() == NUM);
+    //     for (size_t idx = 0; idx < vks.size(); idx++) {
+    //         Instance inst;
+    //         inst.verification_key = std::move(vks[idx]);
+    //         _data[idx] = std::make_unique<Instance>(inst);
+    //     }
+    // };
+    VerifierInstances_()
     {
-        ASSERT(vks.size() == NUM);
-        for (size_t idx = 0; idx < vks.size(); idx++) {
+        for (size_t idx = 0; idx < NUM; idx++) {
             Instance inst;
-            inst.verification_key = std::move(vks[idx]);
             _data[idx] = std::make_unique<Instance>(inst);
         }
     };
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
index 21232f5f116..9100bba4226 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
@@ -47,13 +47,15 @@ template <class Flavor> class ProverInstance_ {
     std::vector<uint32_t> recursive_proof_public_input_indices;
     // non-empty for the accumulated instances
     FoldingParameters folding_parameters;
-    bool is_accumulator = false;
+    uint32_t is_accumulator = 0;
+    size_t instance_size;
 
     ProverInstance_(Circuit& circuit)
     {
         compute_circuit_size_parameters(circuit);
         compute_proving_key(circuit);
         compute_witness(circuit);
+        instance_size = proving_key->circuit_size;
     }
 
     // ProverInstance_(FoldingResult<Flavor> result)
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
index fb14bd32b8b..7cbc1f0d347 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
@@ -11,11 +11,12 @@ template <class Flavor> class VerifierInstance_ {
 
     std::shared_ptr<VerificationKey> verification_key;
     std::vector<FF> public_inputs;
-    size_t pub_inputs_offset;
+    size_t pub_inputs_offset = 0;
     size_t public_input_size;
     size_t instance_size;
     RelationParameters<FF> relation_parameters;
     FF alpha;
+    uint32_t is_accumulator = 0;
     FoldingParameters folding_parameters;
 };
 } // namespace proof_system::honk
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
index 9edbfd4bfd1..50166d2048e 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
@@ -6,10 +6,13 @@ using namespace barretenberg;
 using namespace proof_system::honk;
 
 using Flavor = flavor::Ultra;
+using VerificationKey = Flavor::VerificationKey;
 using Instance = ProverInstance_<Flavor>;
 using Instances = ProverInstances_<Flavor, 2>;
 using ProtoGalaxyProver = ProtoGalaxyProver_<Instances>;
 using FF = Flavor::FF;
+using Affine = Flavor::Commitment;
+using Projective = Flavor::GroupElement;
 using Builder = Flavor::CircuitBuilder;
 using Polynomial = typename Flavor::Polynomial;
 using ProverPolynomials = Flavor::ProverPolynomials;
@@ -81,6 +84,40 @@ ProverPolynomials construct_ultra_full_polynomials(auto& input_polynomials)
     return full_polynomials;
 }
 
+std::shared_ptr<VerificationKey> construct_ultra_verification_key(size_t instance_size, size_t num_public_inputs)
+{
+    auto verification_key = std::make_shared<typename Flavor::VerificationKey>(instance_size, num_public_inputs);
+    // Compute and store commitments to all precomputed polynomials
+    verification_key->q_m = Affine(Projective::random_element());
+    verification_key->q_l = Affine(Projective::random_element());
+    verification_key->q_r = Affine(Projective::random_element());
+    verification_key->q_o = Affine(Projective::random_element());
+    verification_key->q_c = Affine(Projective::random_element());
+    verification_key->sigma_1 = Affine(Projective::random_element());
+    verification_key->sigma_2 = Affine(Projective::random_element());
+    verification_key->sigma_3 = Affine(Projective::random_element());
+    verification_key->id_1 = Affine(Projective::random_element());
+    verification_key->id_2 = Affine(Projective::random_element());
+    verification_key->id_3 = Affine(Projective::random_element());
+    verification_key->lagrange_first = Affine(Projective::random_element());
+    verification_key->lagrange_last = Affine(Projective::random_element());
+
+    verification_key->q_4 = Affine(Projective::random_element());
+    verification_key->q_arith = Affine(Projective::random_element());
+    verification_key->q_sort = Affine(Projective::random_element());
+    verification_key->q_elliptic = Affine(Projective::random_element());
+    verification_key->q_aux = Affine(Projective::random_element());
+    verification_key->q_lookup = Affine(Projective::random_element());
+    verification_key->sigma_4 = Affine(Projective::random_element());
+    verification_key->id_4 = Affine(Projective::random_element());
+    verification_key->table_1 = Affine(Projective::random_element());
+    verification_key->table_2 = Affine(Projective::random_element());
+    verification_key->table_3 = Affine(Projective::random_element());
+    verification_key->table_4 = Affine(Projective::random_element());
+
+    return verification_key;
+}
+
 class ProtoGalaxyTests : public ::testing::Test {
   public:
     static void SetUpTestSuite() { barretenberg::srs::init_crs_factory("../srs_db/ignition"); }
@@ -280,12 +317,14 @@ TEST_F(ProtoGalaxyTests, ComputeNewAccumulator)
     }
 
     auto accumulator = std::make_shared<Instance>();
+    accumulator->instance_size = instance_size;
     accumulator->prover_polynomials = full_polynomials;
     accumulator->folding_parameters = { betas, target_sum };
     accumulator->relation_parameters = relation_parameters;
     accumulator->alpha = alpha;
-    accumulator->is_accumulator = true;
+    accumulator->is_accumulator = 1;
     accumulator->public_inputs = std::vector<FF>{ FF::random_element() };
+    accumulator->verification_key = construct_ultra_verification_key(instance_size, 1);
 
     auto builder = typename Flavor::CircuitBuilder();
     auto composer = UltraComposer();
@@ -294,11 +333,11 @@ TEST_F(ProtoGalaxyTests, ComputeNewAccumulator)
     auto instance = composer.create_instance(builder);
     auto instances = std::vector<std::shared_ptr<Instance>>{ accumulator, instance };
     auto folding_prover = composer.create_folding_prover(instances);
-    // auto folding_verifier = composer.create_folding_verifier(instances);
+    auto folding_verifier = composer.create_folding_verifier();
 
     auto result = folding_prover.fold_instances();
-
-    // folding_verifier.verify_folding_proof(result.folding_data);
+    auto a = folding_verifier.verify_folding_proof(result.folding_data);
+    EXPECT_EQ(a, true);
 }
 
 } // namespace protogalaxy_tests
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp
index 2e8178f985e..ae08c3ae9df 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp
@@ -104,13 +104,10 @@ template <UltraFlavor Flavor> class UltraComposer_ {
 
         return output_state;
     };
-    ProtoGalaxyVerifier_<VerifierInstances> create_folding_verifier(std::vector<std::shared_ptr<Instance>> instances)
+    ProtoGalaxyVerifier_<VerifierInstances> create_folding_verifier()
     {
-        std::vector<std::shared_ptr<VerificationKey>> vks;
-        for (const auto& inst : instances) {
-            vks.emplace_back(inst->verification_key);
-        }
-        VerifierInstances insts(vks);
+
+        auto insts = VerifierInstances();
         ProtoGalaxyVerifier_<VerifierInstances> output_state(insts);
 
         return output_state;

From a19ae07363828731cd939e39652f8ff790856d87 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Tue, 28 Nov 2023 15:19:41 +0000
Subject: [PATCH 32/86] works

---
 .../src/barretenberg/flavor/goblin_ultra.hpp  |   2 +-
 .../cpp/src/barretenberg/flavor/ultra.hpp     |  16 +-
 .../protogalaxy/combiner.test.cpp             | 490 +++++++-----------
 .../protogalaxy/protogalaxy_prover.cpp        |  79 ++-
 .../protogalaxy/protogalaxy_prover.hpp        |  37 +-
 .../protogalaxy/protogalaxy_verifier.cpp      | 121 ++---
 .../protogalaxy/protogalaxy_verifier.hpp      |   6 +-
 .../sumcheck/instance/prover_instance.hpp     |   2 +
 .../sumcheck/instance/verifier_instance.hpp   |   1 +
 .../ultra_honk/protogalaxy.test.cpp           |   1 +
 10 files changed, 300 insertions(+), 455 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
index 01c38b294c5..b6e21471041 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
@@ -577,7 +577,7 @@ class GoblinUltra {
 
     class FoldingParameters {
       public:
-        std::vector<FF> gate_separation_challenges;
+        std::vector<FF> gate_challenges;
         FF target_sum;
     };
 
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
index 05ae168760d..64ec8db57e8 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
@@ -82,12 +82,12 @@ class Ultra {
      */
     class PrecomputedEntities : public PrecomputedEntities_<DataType, HandleType, NUM_PRECOMPUTED_ENTITIES> {
       public:
+        DataType q_m;            // column 0
         DataType q_c;            // column 1
         DataType q_l;            // column 2
         DataType q_r;            // column 3
         DataType q_o;            // column 4
         DataType q_4;            // column 5
-        DataType q_m;            // column 0
         DataType q_arith;        // column 6
         DataType q_sort;         // column 7
         DataType q_elliptic;     // column 8
@@ -109,12 +109,12 @@ class Ultra {
         DataType lagrange_last;  // column 24
 
         DEFINE_POINTER_VIEW(NUM_PRECOMPUTED_ENTITIES,
+                            &q_m,
                             &q_c,
                             &q_l,
                             &q_r,
                             &q_o,
                             &q_4,
-                            &q_m,
                             &q_arith,
                             &q_sort,
                             &q_elliptic,
@@ -196,12 +196,12 @@ class Ultra {
     template <typename DataType, typename HandleType>
     class AllEntities : public AllEntities_<DataType, HandleType, NUM_ALL_ENTITIES> {
       public:
-        DataType q_m;                // column 5
         DataType q_c;                // column 0
         DataType q_l;                // column 1
         DataType q_r;                // column 2
         DataType q_o;                // column 3
         DataType q_4;                // column 4
+        DataType q_m;                // column 5
         DataType q_arith;            // column 6
         DataType q_sort;             // column 7
         DataType q_elliptic;         // column 8
@@ -299,13 +299,15 @@ class Ultra {
 
         std::vector<HandleType> get_precomputed()
         {
-            return { q_c,          q_l,   q_r,      q_o,     q_4,     q_m,     q_arith, q_sort,
+            return { q_m,          q_c,   q_l,      q_r,     q_o,     q_4,     q_arith, q_sort,
                      q_elliptic,   q_aux, q_lookup, sigma_1, sigma_2, sigma_3, sigma_4, id_1,
                      id_2,         id_3,  id_4,     table_1, table_2, table_3, table_4, lagrange_first,
                      lagrange_last
 
             };
         }
+
+        std::vector<HandleType> get_witness() { return { w_l, w_r, w_o, w_4, sorted_accum, z_perm, z_lookup }; };
         std::vector<HandleType> get_to_be_shifted() override
         {
             return { table_1, table_2, table_3, table_4, w_l, w_r, w_o, w_4, sorted_accum, z_perm, z_lookup };
@@ -349,8 +351,8 @@ class Ultra {
     using VerificationKey = VerificationKey_<PrecomputedEntities<Commitment, CommitmentHandle>>;
 
     /**
-     * @brief A field element for each entity of the flavor. These entities represent the prover polynomials evaluated
-     * at one point.
+     * @brief A field element for each entity of the flavor. These entities represent the prover polynomials
+     * evaluated at one point.
      */
     class AllValues : public AllEntities<FF, FF> {
       public:
@@ -485,7 +487,7 @@ class Ultra {
 
     class FoldingParameters {
       public:
-        std::vector<FF> gate_separation_challenges;
+        std::vector<FF> gate_challenges;
         FF target_sum;
     };
 
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp
index e1e9809778a..b467afb2503 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp
@@ -1,289 +1,201 @@
-// #include "barretenberg/flavor/ultra.hpp"
-// #include "barretenberg/honk/utils/testing.hpp"
-// #include "barretenberg/protogalaxy/protogalaxy_prover.hpp"
-// #include "barretenberg/relations/relation_parameters.hpp"
-// #include "barretenberg/sumcheck/instance/instances.hpp"
-// #include <gtest/gtest.h>
-
-// using namespace proof_system::honk;
-// namespace barretenberg::test_protogalaxy_prover {
-// using Flavor = proof_system::honk::flavor::Ultra;
-// using Polynomial = typename Flavor::Polynomial;
-// using FF = typename Flavor::FF;
-// using RelationParameters = proof_system::RelationParameters<FF>;
-
-// // TODO(https://github.com/AztecProtocol/barretenberg/issues/780): Improve combiner tests to check more than the
-// // arithmetic relation so we more than unit test folding relation parameters and alpha as well.
-// TEST(Protogalaxy, CombinerOn2Instances)
-// {
-//     constexpr size_t NUM_INSTANCES = 2;
-//     using ProverInstance = ProverInstance_<Flavor>;
-//     using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
-//     using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
-
-//     const auto restrict_to_standard_arithmetic_relation = [](auto& polys) {
-//         std::fill(polys.q_arith.begin(), polys.q_arith.end(), 1);
-//         std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
-//         std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
-//         std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
-//         std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
-//         std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
-//         std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
-//         std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
-//     };
-
-//     auto run_test = [&](bool is_random_input) {
-//         // Combiner test on prover polynomisls containing random values, restricted to only the standard arithmetic
-//         // relation.
-//         if (is_random_input) {
-//             std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-//             std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
-//             ProtoGalaxyProver prover;
-//             std::vector<FF> pow_betas = { FF(1), FF(2) };
-
-//             for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-//                 auto instance = std::make_shared<ProverInstance>();
-//                 auto [storage, prover_polynomials] = proof_system::honk::get_sequential_prover_polynomials<Flavor>(
-//                     /*log_circuit_size=*/1, idx * 128);
-//                 restrict_to_standard_arithmetic_relation(prover_polynomials);
-//                 storage_arrays[idx] = std::move(storage);
-//                 instance->prover_polynomials = prover_polynomials;
-//                 instance_data[idx] = instance;
-//             }
-
-//             ProverInstances instances{ instance_data };
-//             instances.alpha = Univariate<FF, 13>(FF(0)); // focus on the arithmetic relation only
-
-//             auto result = prover.compute_combiner(instances, pow_betas);
-//             auto expected_result =
-//                 barretenberg::Univariate<FF, 13>(std::array<FF, 13>{ 87706,
-//                                                                      13644570,
-//                                                                      76451738,
-//                                                                      226257946,
-//                                                                      static_cast<uint64_t>(500811930),
-//                                                                      static_cast<uint64_t>(937862426),
-//                                                                      static_cast<uint64_t>(1575158170),
-//                                                                      static_cast<uint64_t>(2450447898),
-//                                                                      static_cast<uint64_t>(3601480346),
-//                                                                      static_cast<uint64_t>(5066004250),
-//                                                                      static_cast<uint64_t>(6881768346),
-//                                                                      static_cast<uint64_t>(9086521370),
-//                                                                      static_cast<uint64_t>(11718012058) });
-//             EXPECT_EQ(result, expected_result);
-//         } else {
-
-//             std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-//             std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
-//             ProtoGalaxyProver prover;
-//             std::vector<FF> pow_betas = { FF(1), FF(2) };
-
-//             for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-//                 auto instance = std::make_shared<ProverInstance>();
-//                 auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
-//                     /*log_circuit_size=*/1);
-//                 restrict_to_standard_arithmetic_relation(prover_polynomials);
-//                 storage_arrays[idx] = std::move(storage);
-//                 instance->prover_polynomials = prover_polynomials;
-//                 instance_data[idx] = instance;
-//             }
-
-//             ProverInstances instances{ instance_data };
-//             instances.alpha = Univariate<FF, 13>(FF(0)); // focus on the arithmetic relation only
-
-//             const auto create_add_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
-//                 polys.w_l[idx] = w_l;
-//                 polys.w_r[idx] = w_r;
-//                 polys.w_o[idx] = w_l + w_r;
-//                 polys.q_l[idx] = 1;
-//                 polys.q_r[idx] = 1;
-//                 polys.q_o[idx] = -1;
-//             };
-
-//             const auto create_mul_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
-//                 polys.w_l[idx] = w_l;
-//                 polys.w_r[idx] = w_r;
-//                 polys.w_o[idx] = w_l * w_r;
-//                 polys.q_m[idx] = 1;
-//                 polys.q_o[idx] = -1;
-//             };
-
-//             create_add_gate(instances[0]->prover_polynomials, 0, 1, 2);
-//             create_add_gate(instances[0]->prover_polynomials, 1, 0, 4);
-//             create_add_gate(instances[1]->prover_polynomials, 0, 3, 4);
-//             create_mul_gate(instances[1]->prover_polynomials, 1, 1, 4);
-
-//             restrict_to_standard_arithmetic_relation(instances[0]->prover_polynomials);
-//             restrict_to_standard_arithmetic_relation(instances[1]->prover_polynomials);
-
-//             /* Instance 0                                    Instance 1
-//                 w_l w_r w_o q_m q_l q_r q_o q_c               w_l w_r w_o q_m q_l q_r q_o q_c
-//                 1   2   3   0   1   1   -1  0                 3   4   7   0   1   1   -1  0
-//                 0   4   4   0   1   1   -1  0                 1   4   4   1   0   0   -1  0             */
-
-//             /* Lagrange-combined values, row index 0         Lagrange-combined values, row index 1
-//                 in    0    1    2    3    4    5    6        in    0    1    2    3    4    5    6
-//                 w_l   1    3    5    7    9   11   13        w_l   0    1    2    3    4    5    6
-//                 w_r   2    4    6    8   10   12   14        w_r   4    4    4    4    4    4    4
-//                 w_o   3    7   11   15   19   23   27        w_o   4    4    4    4    4    4    0
-//                 q_m   0    0    0    0    0    0    0        q_m   0    1    2    3    4    5    6
-//                 q_l   1    1    1    1    1    1    1        q_l   1    0   -1   -2   -3   -4   -5
-//                 q_r   1    1    1    1    1    1    1        q_r   1    0   -1   -2   -3   -4   -5
-//                 q_o  -1   -1   -1   -1   -1   -1   -1        q_o  -1   -1   -1   -1   -1   -1   -1
-//                 q_c   0    0    0    0    0    0    0        q_c   0    0    0    0    0    0    0
-
-//             relation value:
-//                       0    0    0    0    0    0    0              0    0    6   18   36   60   90      */
-
-//             auto result = prover.compute_combiner(instances, pow_betas);
-//             auto expected_result = barretenberg::Univariate<FF, 13>(
-//                 std::array<FF, 13>{ 0, 0, 12, 36, 72, 120, 180, 252, 336, 432, 540, 660, 792 });
-
-//             EXPECT_EQ(result, expected_result);
-//         }
-//     };
-//     run_test(true);
-//     run_test(false);
-// };
-
-// TEST(Protogalaxy, CombinerOn4Instances)
-// {
-//     constexpr size_t NUM_INSTANCES = 4;
-//     using ProverInstance = ProverInstance_<Flavor>;
-//     using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
-//     using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
-
-//     const auto zero_all_selectors = [](auto& polys) {
-//         std::fill(polys.q_arith.begin(), polys.q_arith.end(), 0);
-//         std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
-//         std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
-//         std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
-//         std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
-//         std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
-//         std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
-//         std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
-//     };
-
-//     auto run_test = [&]() {
-//         std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-//         std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
-//         ProtoGalaxyProver prover;
-//         std::vector<FF> pow_betas = { FF(1), FF(2) };
-
-//         for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-//             auto instance = std::make_shared<ProverInstance>();
-//             auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
-//                 /*log_circuit_size=*/1);
-//             storage_arrays[idx] = std::move(storage);
-//             instance->prover_polynomials = prover_polynomials;
-//             instance_data[idx] = instance;
-//         }
-
-//         ProverInstances instances{ instance_data };
-//         instances.alpha = Univariate<FF, 43>(FF(0)); // focus on the arithmetic relation only
-
-//         zero_all_selectors(instances[0]->prover_polynomials);
-//         zero_all_selectors(instances[1]->prover_polynomials);
-//         zero_all_selectors(instances[2]->prover_polynomials);
-//         zero_all_selectors(instances[3]->prover_polynomials);
-
-//         auto result = prover.compute_combiner(instances, pow_betas);
-//         std::array<FF, 43> zeroes;
-//         std::fill(zeroes.begin(), zeroes.end(), 0);
-//         auto expected_result = barretenberg::Univariate<FF, 43>(zeroes);
-//         EXPECT_EQ(result, expected_result);
-//     };
-//     run_test();
-// };
-
-// TEST(Protogalaxy, FullTestMaybe)
-// {
-//     constexpr size_t NUM_INSTANCES = 2;
-//     using ProverInstance = ProverInstance_<Flavor>;
-//     using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
-//     using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
-
-//     const auto restrict_to_standard_arithmetic_relation = [](auto& polys) {
-//         std::fill(polys.q_arith.begin(), polys.q_arith.end(), 1);
-//         std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
-//         std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
-//         std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
-//         std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
-//         std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
-//         std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
-//         std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
-//     };
-
-//     std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-//     std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
-//     ProtoGalaxyProver prover;
-
-//     for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-//         auto instance = std::make_shared<ProverInstance>();
-//         auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
-//             /*log_circuit_size=*/1);
-//         restrict_to_standard_arithmetic_relation(prover_polynomials);
-//         storage_arrays[idx] = std::move(storage);
-//         instance->prover_polynomials = prover_polynomials;
-//         instance_data[idx] = instance;
-//     }
-
-//     ProverInstances instances{ instance_data };
-//     instances.alpha = Univariate<FF, 13>(FF(0)); // focus on the arithmetic relation only
-
-//     const auto create_add_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
-//         polys.w_l[idx] = w_l;
-//         polys.w_r[idx] = w_r;
-//         polys.w_o[idx] = w_l + w_r;
-//         polys.q_l[idx] = 1;
-//         polys.q_r[idx] = 1;
-//         polys.q_o[idx] = -1;
-//     };
-
-//     const auto create_mul_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
-//         polys.w_l[idx] = w_l;
-//         polys.w_r[idx] = w_r;
-//         polys.w_o[idx] = w_l * w_r;
-//         polys.q_m[idx] = 1;
-//         polys.q_o[idx] = -1;
-//     };
-
-//     create_add_gate(instances[0]->prover_polynomials, 0, 1, 2);
-//     create_add_gate(instances[0]->prover_polynomials, 1, 0, 4);
-//     create_add_gate(instances[1]->prover_polynomials, 0, 3, 4);
-//     create_mul_gate(instances[1]->prover_polynomials, 1, 1, 4);
-
-//     restrict_to_standard_arithmetic_relation(instances[0]->prover_polynomials);
-//     restrict_to_standard_arithmetic_relation(instances[1]->prover_polynomials);
-
-//     proof_system::RelationParameters<FF> params;
-//     auto full_honk_evals = prover.compute_full_honk_evaluations(instances[0]->prover_polynomials, FF(0), params);
-//     auto betas = std::vector<FF>{ FF(2) };
-//     auto pow_betas = prover.compute_pow_polynomial_at_values(betas);
-//     // Compute the corresponding target sum and create a dummy accumulator
-//     auto target_sum = FF(0);
-//     for (size_t i = 0; i < 2; i++) {
-//         target_sum += full_honk_evals[i] * pow_betas[i];
-//     }
-//     instances[0]->folding_parameters = { betas, target_sum };
-
-//     auto deltas = std::vector<FF>{ FF(2), FF(4) };
-//     auto perturbator = prover.compute_perturbator(instances[0], deltas);
-
-//     EXPECT_EQ(perturbator[0], target_sum);
-
-//     instances.alpha = Univariate<FF, 13>(FF(0));
-
-//     auto challenge = FF(2);
-
-//     auto betas_star = prover.update_gate_separation_challenges(challenge, betas, deltas);
-//     instances.folding_parameters.gate_separation_challenges = betas;
-//     auto combiner = prover.compute_combiner(instances);
-
-//     auto compressed_perturbator = perturbator.evaluate(challenge);
-//     auto combiner_quotient = prover.compute_combiner_quotient(compressed_perturbator, combiner);
-
-//     auto combiner_challenge = FF(2);
-//     auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge);
-// }
-
-// } // namespace barretenberg::test_protogalaxy_prover
\ No newline at end of file
+#include "barretenberg/flavor/ultra.hpp"
+#include "barretenberg/honk/utils/testing.hpp"
+#include "barretenberg/protogalaxy/protogalaxy_prover.hpp"
+#include "barretenberg/relations/relation_parameters.hpp"
+#include "barretenberg/sumcheck/instance/instances.hpp"
+#include <gtest/gtest.h>
+
+using namespace proof_system::honk;
+namespace barretenberg::test_protogalaxy_prover {
+using Flavor = proof_system::honk::flavor::Ultra;
+using Polynomial = typename Flavor::Polynomial;
+using FF = typename Flavor::FF;
+using RelationParameters = proof_system::RelationParameters<FF>;
+
+// TODO(https://github.com/AztecProtocol/barretenberg/issues/780): Improve combiner tests to check more than the
+// arithmetic relation so we more than unit test folding relation parameters and alpha as well.
+TEST(Protogalaxy, CombinerOn2Instances)
+{
+    constexpr size_t NUM_INSTANCES = 2;
+    using ProverInstance = ProverInstance_<Flavor>;
+    using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
+    using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
+
+    const auto restrict_to_standard_arithmetic_relation = [](auto& polys) {
+        std::fill(polys.q_arith.begin(), polys.q_arith.end(), 1);
+        std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
+        std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
+        std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
+        std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
+        std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
+        std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
+        std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
+    };
+
+    auto run_test = [&](bool is_random_input) {
+        // Combiner test on prover polynomisls containing random values, restricted to only the standard arithmetic
+        // relation.
+        if (is_random_input) {
+            std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+            std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
+            ProtoGalaxyProver prover;
+            std::vector<FF> pow_betas = { FF(1), FF(2) };
+
+            for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+                auto instance = std::make_shared<ProverInstance>();
+                auto [storage, prover_polynomials] = proof_system::honk::get_sequential_prover_polynomials<Flavor>(
+                    /*log_circuit_size=*/1, idx * 128);
+                restrict_to_standard_arithmetic_relation(prover_polynomials);
+                storage_arrays[idx] = std::move(storage);
+                instance->prover_polynomials = prover_polynomials;
+                instance->instance_size = 2;
+                instance_data[idx] = instance;
+            }
+
+            ProverInstances instances{ instance_data };
+            instances.alpha = Univariate<FF, 13>(FF(0)); // focus on the arithmetic relation only
+
+            auto result = prover.compute_combiner(instances, pow_betas);
+            auto expected_result =
+                barretenberg::Univariate<FF, 13>(std::array<FF, 13>{ 87706,
+                                                                     13644570,
+                                                                     76451738,
+                                                                     226257946,
+                                                                     static_cast<uint64_t>(500811930),
+                                                                     static_cast<uint64_t>(937862426),
+                                                                     static_cast<uint64_t>(1575158170),
+                                                                     static_cast<uint64_t>(2450447898),
+                                                                     static_cast<uint64_t>(3601480346),
+                                                                     static_cast<uint64_t>(5066004250),
+                                                                     static_cast<uint64_t>(6881768346),
+                                                                     static_cast<uint64_t>(9086521370),
+                                                                     static_cast<uint64_t>(11718012058) });
+            EXPECT_EQ(result, expected_result);
+        } else {
+
+            std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+            std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
+            ProtoGalaxyProver prover;
+            std::vector<FF> pow_betas = { FF(1), FF(2) };
+
+            for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+                auto instance = std::make_shared<ProverInstance>();
+                auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
+                    /*log_circuit_size=*/1);
+                restrict_to_standard_arithmetic_relation(prover_polynomials);
+                storage_arrays[idx] = std::move(storage);
+                instance->prover_polynomials = prover_polynomials;
+                instance->instance_size = 2;
+                instance_data[idx] = instance;
+            }
+
+            ProverInstances instances{ instance_data };
+            instances.alpha = Univariate<FF, 13>(FF(0)); // focus on the arithmetic relation only
+
+            const auto create_add_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
+                polys.w_l[idx] = w_l;
+                polys.w_r[idx] = w_r;
+                polys.w_o[idx] = w_l + w_r;
+                polys.q_l[idx] = 1;
+                polys.q_r[idx] = 1;
+                polys.q_o[idx] = -1;
+            };
+
+            const auto create_mul_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
+                polys.w_l[idx] = w_l;
+                polys.w_r[idx] = w_r;
+                polys.w_o[idx] = w_l * w_r;
+                polys.q_m[idx] = 1;
+                polys.q_o[idx] = -1;
+            };
+
+            create_add_gate(instances[0]->prover_polynomials, 0, 1, 2);
+            create_add_gate(instances[0]->prover_polynomials, 1, 0, 4);
+            create_add_gate(instances[1]->prover_polynomials, 0, 3, 4);
+            create_mul_gate(instances[1]->prover_polynomials, 1, 1, 4);
+
+            restrict_to_standard_arithmetic_relation(instances[0]->prover_polynomials);
+            restrict_to_standard_arithmetic_relation(instances[1]->prover_polynomials);
+
+            /* Instance 0                                    Instance 1
+                w_l w_r w_o q_m q_l q_r q_o q_c               w_l w_r w_o q_m q_l q_r q_o q_c
+                1   2   3   0   1   1   -1  0                 3   4   7   0   1   1   -1  0
+                0   4   4   0   1   1   -1  0                 1   4   4   1   0   0   -1  0             */
+
+            /* Lagrange-combined values, row index 0         Lagrange-combined values, row index 1
+                in    0    1    2    3    4    5    6        in    0    1    2    3    4    5    6
+                w_l   1    3    5    7    9   11   13        w_l   0    1    2    3    4    5    6
+                w_r   2    4    6    8   10   12   14        w_r   4    4    4    4    4    4    4
+                w_o   3    7   11   15   19   23   27        w_o   4    4    4    4    4    4    0
+                q_m   0    0    0    0    0    0    0        q_m   0    1    2    3    4    5    6
+                q_l   1    1    1    1    1    1    1        q_l   1    0   -1   -2   -3   -4   -5
+                q_r   1    1    1    1    1    1    1        q_r   1    0   -1   -2   -3   -4   -5
+                q_o  -1   -1   -1   -1   -1   -1   -1        q_o  -1   -1   -1   -1   -1   -1   -1
+                q_c   0    0    0    0    0    0    0        q_c   0    0    0    0    0    0    0
+
+            relation value:
+                      0    0    0    0    0    0    0              0    0    6   18   36   60   90      */
+
+            auto result = prover.compute_combiner(instances, pow_betas);
+            auto expected_result = barretenberg::Univariate<FF, 13>(
+                std::array<FF, 13>{ 0, 0, 12, 36, 72, 120, 180, 252, 336, 432, 540, 660, 792 });
+
+            EXPECT_EQ(result, expected_result);
+        }
+    };
+    run_test(true);
+    run_test(false);
+};
+
+TEST(Protogalaxy, CombinerOn4Instances)
+{
+    constexpr size_t NUM_INSTANCES = 4;
+    using ProverInstance = ProverInstance_<Flavor>;
+    using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
+    using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
+
+    const auto zero_all_selectors = [](auto& polys) {
+        std::fill(polys.q_arith.begin(), polys.q_arith.end(), 0);
+        std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
+        std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
+        std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
+        std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
+        std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
+        std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
+        std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
+    };
+
+    auto run_test = [&]() {
+        std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+        std::array<std::array<Polynomial, Flavor::NUM_ALL_ENTITIES>, NUM_INSTANCES> storage_arrays;
+        ProtoGalaxyProver prover;
+        std::vector<FF> pow_betas = { FF(1), FF(2) };
+
+        for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+            auto instance = std::make_shared<ProverInstance>();
+            auto [storage, prover_polynomials] = proof_system::honk::get_zero_prover_polynomials<Flavor>(
+                /*log_circuit_size=*/1);
+            storage_arrays[idx] = std::move(storage);
+            instance->prover_polynomials = prover_polynomials;
+            instance->instance_size = 2;
+            instance_data[idx] = instance;
+        }
+
+        ProverInstances instances{ instance_data };
+        instances.alpha = Univariate<FF, 43>(FF(0)); // focus on the arithmetic relation only
+
+        zero_all_selectors(instances[0]->prover_polynomials);
+        zero_all_selectors(instances[1]->prover_polynomials);
+        zero_all_selectors(instances[2]->prover_polynomials);
+        zero_all_selectors(instances[3]->prover_polynomials);
+
+        auto result = prover.compute_combiner(instances, pow_betas);
+        std::array<FF, 43> zeroes;
+        std::fill(zeroes.begin(), zeroes.end(), 0);
+        auto expected_result = barretenberg::Univariate<FF, 43>(zeroes);
+        EXPECT_EQ(result, expected_result);
+    };
+    run_test();
+};
+
+} // namespace barretenberg::test_protogalaxy_prover
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index 4a0eaa2c4f3..93b8a66e027 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -9,17 +9,29 @@ template <class ProverInstances> void ProtoGalaxyProver_<ProverInstances>::prepa
         auto instance = *it;
         auto domain_separator = std::to_string(idx);
         transcript.send_to_verifier(domain_separator + "is_accumulator", instance->is_accumulator);
-        if (static_cast<bool>(instance->is_accumulator)) {
-            const auto instance_size = static_cast<uint32_t>(instance->instance_size);
-            const auto num_public_inputs = static_cast<uint32_t>(instance->public_inputs.size());
 
-            transcript.send_to_verifier(domain_separator + "_instance_size", instance_size);
-            transcript.send_to_verifier(domain_separator + "_public_input_size", num_public_inputs);
+        if (!static_cast<bool>(instance->is_accumulator)) {
+            instance->initialize_prover_polynomials();
+        }
 
-            for (size_t i = 0; i < instance->public_inputs.size(); ++i) {
-                auto public_input_i = instance->public_inputs[i];
-                transcript.send_to_verifier(domain_separator + "_public_input_" + std::to_string(i), public_input_i);
-            }
+        const auto instance_size = static_cast<uint32_t>(instance->instance_size);
+        const auto num_public_inputs = static_cast<uint32_t>(instance->public_inputs.size());
+
+        transcript.send_to_verifier(domain_separator + "_instance_size", instance_size);
+        transcript.send_to_verifier(domain_separator + "_public_input_size", num_public_inputs);
+
+        for (size_t i = 0; i < instance->public_inputs.size(); ++i) {
+            auto public_input_i = instance->public_inputs[i];
+            transcript.send_to_verifier(domain_separator + "_public_input_" + std::to_string(i), public_input_i);
+        }
+
+        if (!static_cast<bool>(instance->is_accumulator)) {
+            auto [eta, beta, gamma] = transcript.get_challenges(
+                domain_separator + "_eta", domain_separator + "_beta", domain_separator + "_gamma");
+            instance->compute_sorted_accumulator_polynomials(eta);
+            instance->compute_grand_product_polynomials(beta, gamma);
+            instance->alpha = transcript.get_challenge(domain_separator + "_alpha");
+        } else {
             transcript.send_to_verifier(domain_separator + "_eta", instance->relation_parameters.eta);
             transcript.send_to_verifier(domain_separator + "_beta", instance->relation_parameters.beta);
             transcript.send_to_verifier(domain_separator + "_gamma", instance->relation_parameters.gamma);
@@ -32,36 +44,16 @@ template <class ProverInstances> void ProtoGalaxyProver_<ProverInstances>::prepa
 
             auto folding_parameters = instance->folding_parameters;
             transcript.send_to_verifier(domain_separator + "_target_sum", folding_parameters.target_sum);
-            for (size_t idx = 0; idx < folding_parameters.gate_separation_challenges.size(); idx++) {
+            for (size_t idx = 0; idx < folding_parameters.gate_challenges.size(); idx++) {
                 transcript.send_to_verifier(domain_separator + "_gate_challenge_" + std::to_string(idx),
-                                            folding_parameters.gate_separation_challenges[idx]);
+                                            folding_parameters.gate_challenges[idx]);
             }
-
-        } else {
-            instance->initialize_prover_polynomials();
-            const auto instance_size = static_cast<uint32_t>(instance->instance_size);
-            const auto num_public_inputs = static_cast<uint32_t>(instance->public_inputs.size());
-
-            transcript.send_to_verifier(domain_separator + "_instance_size", instance_size);
-            transcript.send_to_verifier(domain_separator + "_public_input_size", num_public_inputs);
-            transcript.send_to_verifier(domain_separator + "_pub_inputs_offset",
-                                        static_cast<uint32_t>(instance->pub_inputs_offset));
-
-            for (size_t i = 0; i < num_public_inputs; ++i) {
-                auto public_input_i = instance->public_inputs[i];
-                transcript.send_to_verifier(domain_separator + "_public_input_" + std::to_string(i), public_input_i);
-            }
-
-            auto [eta, beta, gamma] = transcript.get_challenges(
-                domain_separator + "_eta", domain_separator + "_beta", domain_separator + "_gamma");
-            instance->compute_sorted_accumulator_polynomials(eta);
-            instance->compute_grand_product_polynomials(beta, gamma);
-            instance->alpha = transcript.get_challenge(domain_separator + "_alpha");
         }
+
         auto vk_view = instance->verification_key->pointer_view();
         auto labels = typename Flavor::CommitmentLabels().get_precomputed();
         for (size_t idx = 0; idx < labels.size(); idx++) {
-            transcript.send_to_verifier(domain_separator + labels[idx], (*vk_view[idx]));
+            transcript.send_to_verifier(domain_separator + "_" + labels[idx], (*vk_view[idx]));
         }
     }
 
@@ -74,38 +66,31 @@ template <class ProverInstances>
 ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverInstances>::fold_instances()
 {
     prepare_for_folding();
+
     // TODO(#https://github.com/AztecProtocol/barretenberg/issues/740): Handle the case where we are folding for the
     // first time and accumulator is 0
-    // TODO(#https://github.com/AztecProtocol/barretenberg/issues/763): Fold alpha
     auto delta = transcript.get_challenge("delta");
-    info(delta);
     auto accumulator = get_accumulator();
-    auto instance_size = accumulator->prover_polynomials.get_polynomial_size();
-    const auto log_instance_size = static_cast<size_t>(numeric::get_msb(instance_size));
-    auto deltas = compute_round_challenge_pows(log_instance_size, delta);
+    auto deltas = compute_round_challenge_pows(accumulator->log_instance_size, delta);
 
     auto perturbator = compute_perturbator(accumulator, deltas);
-    for (size_t idx = 0; idx <= log_instance_size; idx++) {
+    for (size_t idx = 0; idx <= accumulator->log_instance_size; idx++) {
         transcript.send_to_verifier("perturbator_" + std::to_string(idx), perturbator[idx]);
     }
     assert(perturbator[0] == accumulator->folding_parameters.target_sum);
-
     auto perturbator_challenge = transcript.get_challenge("perturbator_challenge");
-    info("pc", perturbator_challenge);
-    instances.betas_star = update_gate_separation_challenges(
-        perturbator_challenge, accumulator->folding_parameters.gate_separation_challenges, deltas);
-
-    auto combiner = compute_combiner(instances);
+    instances.betas_star =
+        update_gate_challenges(perturbator_challenge, accumulator->folding_parameters.gate_challenges, deltas);
+    const auto pow_betas_star = compute_pow_polynomial_at_values(instances.betas_star, accumulator->instance_size);
+    auto combiner = compute_combiner(instances, pow_betas_star);
 
     auto compressed_perturbator = perturbator.evaluate(perturbator_challenge);
     auto combiner_quotient = compute_combiner_quotient(compressed_perturbator, combiner);
 
     for (size_t idx = ProverInstances::NUM; idx < ProverInstances::BATCHED_EXTENDED_LENGTH; idx++) {
-        info(combiner_quotient.value_at(idx));
         transcript.send_to_verifier("combiner_quotient_" + std::to_string(idx), combiner_quotient.value_at(idx));
     }
     auto combiner_challenge = transcript.get_challenge("combiner_quotient_challenge");
-    info("cc", combiner_challenge);
     // TODO(https://github.com/AztecProtocol/barretenberg/issues/764): Generalize these formulas as well as computation
     // of Lagrange basis
 
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index 38cb45eedff..cac12cce9df 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -89,9 +89,9 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         return pows;
     }
 
-    static std::vector<FF> update_gate_separation_challenges(const FF perturbator_challenge,
-                                                             const std::vector<FF>& gate_challenges,
-                                                             const std::vector<FF>& round_challenges)
+    static std::vector<FF> update_gate_challenges(const FF perturbator_challenge,
+                                                  const std::vector<FF>& gate_challenges,
+                                                  const std::vector<FF>& round_challenges)
     {
         auto log_instance_size = gate_challenges.size();
         std::vector<FF> next_gate_challenges(log_instance_size);
@@ -206,7 +206,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     {
         auto full_honk_evaluations = compute_full_honk_evaluations(
             accumulator->prover_polynomials, accumulator->alpha, accumulator->relation_parameters);
-        const auto betas = accumulator->folding_parameters.gate_separation_challenges;
+        const auto betas = accumulator->folding_parameters.gate_challenges;
         assert(betas.size() == deltas.size());
         auto coeffs = construct_perturbator_coefficients(betas, deltas, full_honk_evaluations);
         return Polynomial<FF>(coeffs);
@@ -256,10 +256,10 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      * @brief Compute the combiner polynomial $G$ in the Protogalaxy paper.
      *
      */
-    ExtendedUnivariateWithRandomization compute_combiner(const ProverInstances& instances)
+    ExtendedUnivariateWithRandomization compute_combiner(const ProverInstances& instances,
+                                                         const std::vector<FF>& pow_betas_star)
     {
-        size_t common_circuit_size = instances[0]->prover_polynomials.get_polynomial_size();
-        auto pow_betas_star = compute_pow_polynomial_at_values(instances.betas_star, common_circuit_size);
+        size_t common_instance_size = instances[0]->instance_size;
 
         // Determine number of threads for multithreading.
         // Note: Multithreading is "on" for every round but we reduce the number of threads from the max available based
@@ -267,10 +267,10 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         // For now we use a power of 2 number of threads simply to ensure the round size is evenly divided.
         size_t max_num_threads = get_num_cpus_pow2(); // number of available threads (power of 2)
         size_t min_iterations_per_thread = 1 << 6; // min number of iterations for which we'll spin up a unique thread
-        size_t desired_num_threads = common_circuit_size / min_iterations_per_thread;
+        size_t desired_num_threads = common_instance_size / min_iterations_per_thread;
         size_t num_threads = std::min(desired_num_threads, max_num_threads); // fewer than max if justified
         num_threads = num_threads > 0 ? num_threads : 1;                     // ensure num threads is >= 1
-        size_t iterations_per_thread = common_circuit_size / num_threads;    // actual iterations per thread
+        size_t iterations_per_thread = common_instance_size / num_threads;   // actual iterations per thread
 
         // Constuct univariate accumulator containers; one per thread
         std::vector<TupleOfTuplesOfUnivariates> thread_univariate_accumulators(num_threads);
@@ -417,14 +417,12 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
 
         transcript.send_to_verifier("next_target_sum", next_target_sum);
         for (size_t idx = 0; idx < instances.betas_star.size(); idx++) {
-            transcript.send_to_verifier("betas_star_" + std::to_string(idx), instances.betas_star[idx]);
+            transcript.send_to_verifier("next_gate_challenge_" + std::to_string(idx), instances.betas_star[idx]);
         }
 
         std::array<barretenberg::Polynomial<FF>, Flavor::NUM_ALL_ENTITIES> storage;
-        size_t instance_size = instances[0]->prover_polynomials.get_polynomial_size();
-
         for (auto& polynomial : storage) {
-            polynomial = typename Flavor::Polynomial(instance_size);
+            polynomial = typename Flavor::Polynomial(instances[0]->instance_size);
             for (auto& value : polynomial) {
                 value = FF(0);
             }
@@ -457,11 +455,11 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
             }
         }
         for (size_t idx = 0; idx < folded_public_inputs.size(); idx++) {
-            transcript.send_to_verifier("folded_public_input_" + std::to_string(idx), folded_public_inputs[idx]);
+            transcript.send_to_verifier("next_public_input_" + std::to_string(idx), folded_public_inputs[idx]);
         }
 
         next_accumulator->alpha = instances.alpha.evaluate(challenge);
-        transcript.send_to_verifier("folded_alpha", next_accumulator->alpha);
+        transcript.send_to_verifier("next_alpha", next_accumulator->alpha);
 
         auto acc_vk = std::make_shared<VerificationKey>(instances[0]->prover_polynomials.get_polynomial_size(),
                                                         instances[0]->public_inputs.size());
@@ -475,14 +473,9 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         next_accumulator->verification_key = acc_vk;
         auto labels = typename Flavor::CommitmentLabels().get_precomputed();
         for (size_t idx = 0; idx < labels.size(); idx++) {
-            info(labels[idx]);
-            info((*acc_vk_view[idx]));
-            transcript.send_to_verifier("new" + labels[idx], (*acc_vk_view[idx]));
+            transcript.send_to_verifier("next_" + labels[idx], (*acc_vk_view[idx]));
         }
-        // auto a = labels.get_precomputed();
-        //  TODO send vk to verifier too
-        // these two thingies have been folded already
-        // next_accumulator->alpha = instances.alpha;
+
         // next_accumulator->relation_parameters = instances.relation_parameters;
         return next_accumulator;
     }
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index aed797067e1..23d2910f76d 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -11,18 +11,19 @@ void ProtoGalaxyVerifier_<VerifierInstances>::prepare_for_folding(std::vector<ui
         auto inst = *it;
         auto domain_separator = std::to_string(index);
         inst->is_accumulator = transcript.template receive_from_prover<uint32_t>(domain_separator + "is_accumulator");
-        if (static_cast<bool>(inst->is_accumulator)) {
-            inst->instance_size =
-                transcript.template receive_from_prover<uint32_t>(domain_separator + "_instance_size");
-            inst->public_input_size =
-                transcript.template receive_from_prover<uint32_t>(domain_separator + "_public_input_size");
-
-            for (size_t i = 0; i < inst->public_input_size; ++i) {
-                auto public_input_i = transcript.template receive_from_prover<FF>(domain_separator + "_public_input_" +
-                                                                                  std::to_string(i));
-                inst->public_inputs.emplace_back(public_input_i);
-            }
 
+        inst->instance_size = transcript.template receive_from_prover<uint32_t>(domain_separator + "_instance_size");
+        inst->log_instance_size = static_cast<size_t>(numeric::get_msb(inst->instance_size));
+        inst->public_input_size =
+            transcript.template receive_from_prover<uint32_t>(domain_separator + "_public_input_size");
+
+        for (size_t i = 0; i < inst->public_input_size; ++i) {
+            auto public_input_i =
+                transcript.template receive_from_prover<FF>(domain_separator + "_public_input_" + std::to_string(i));
+            inst->public_inputs.emplace_back(public_input_i);
+        }
+
+        if (static_cast<bool>(inst->is_accumulator)) {
             auto eta = transcript.template receive_from_prover<FF>(domain_separator + "_eta");
             auto beta = transcript.template receive_from_prover<FF>(domain_separator + "_beta");
             auto gamma = transcript.template receive_from_prover<FF>(domain_separator + "_gamma");
@@ -37,26 +38,13 @@ void ProtoGalaxyVerifier_<VerifierInstances>::prepare_for_folding(std::vector<ui
             inst->folding_parameters.target_sum =
                 transcript.template receive_from_prover<FF>(domain_separator + "_target_sum");
 
-            auto log_inst_size = static_cast<size_t>(numeric::get_msb(inst->instance_size));
-            inst->folding_parameters.gate_separation_challenges = std::vector<FF>(log_inst_size);
-            for (size_t idx = 0; idx < log_inst_size; idx++) {
-                inst->folding_parameters.gate_separation_challenges[idx] = transcript.template receive_from_prover<FF>(
+            inst->folding_parameters.gate_challenges = std::vector<FF>(inst->log_instance_size);
+            for (size_t idx = 0; idx < inst->log_instance_size; idx++) {
+                inst->folding_parameters.gate_challenges[idx] = transcript.template receive_from_prover<FF>(
                     domain_separator + "_gate_challenge_" + std::to_string(idx));
             }
         } else {
 
-            inst->instance_size =
-                transcript.template receive_from_prover<uint32_t>(domain_separator + "_instance_size");
-            inst->public_input_size =
-                transcript.template receive_from_prover<uint32_t>(domain_separator + "_public_input_size");
-            inst->pub_inputs_offset =
-                transcript.template receive_from_prover<uint32_t>(domain_separator + "_pub_inputs_offset");
-
-            for (size_t i = 0; i < inst->public_input_size; ++i) {
-                auto public_input_i = transcript.template receive_from_prover<FF>(domain_separator + "_public_input_" +
-                                                                                  std::to_string(i));
-                inst->public_inputs.emplace_back(public_input_i);
-            }
             auto [eta, beta, gamma] = transcript.get_challenges(
                 domain_separator + "_eta", domain_separator + "_beta", domain_separator + "_gamma");
             const FF public_input_delta = compute_public_input_delta<Flavor>(
@@ -68,38 +56,12 @@ void ProtoGalaxyVerifier_<VerifierInstances>::prepare_for_folding(std::vector<ui
             inst->alpha = transcript.get_challenge(domain_separator + "_alpha");
         }
 
-        auto labels = typename Flavor::CommitmentLabels();
-        auto vk = std::make_shared<VerificationKey>(inst->instance_size, inst->public_input_size);
-
-        vk->q_m = transcript.template receive_from_prover<Commitment>(domain_separator + labels.q_m);
-        vk->q_l = transcript.template receive_from_prover<Commitment>(domain_separator + labels.q_l);
-        vk->q_r = transcript.template receive_from_prover<Commitment>(domain_separator + labels.q_r);
-        vk->q_o = transcript.template receive_from_prover<Commitment>(domain_separator + labels.q_o);
-        vk->q_c = transcript.template receive_from_prover<Commitment>(domain_separator + labels.q_c);
-        vk->sigma_1 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.sigma_1);
-        vk->sigma_2 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.sigma_2);
-        vk->sigma_3 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.sigma_3);
-        vk->id_1 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.id_1);
-        vk->id_2 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.id_2);
-        vk->id_3 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.id_3);
-        vk->lagrange_first =
-            transcript.template receive_from_prover<Commitment>(domain_separator + labels.lagrange_first);
-        vk->lagrange_last =
-            transcript.template receive_from_prover<Commitment>(domain_separator + labels.lagrange_last);
-
-        vk->q_4 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.q_4);
-        vk->q_arith = transcript.template receive_from_prover<Commitment>(domain_separator + labels.q_arith);
-        vk->q_sort = transcript.template receive_from_prover<Commitment>(domain_separator + labels.q_sort);
-        vk->q_elliptic = transcript.template receive_from_prover<Commitment>(domain_separator + labels.q_elliptic);
-        vk->q_aux = transcript.template receive_from_prover<Commitment>(domain_separator + labels.q_aux);
-        vk->q_lookup = transcript.template receive_from_prover<Commitment>(domain_separator + labels.q_lookup);
-        vk->sigma_4 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.sigma_4);
-        vk->id_4 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.id_4);
-        vk->table_1 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.table_1);
-        vk->table_2 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.table_2);
-        vk->table_3 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.table_3);
-        vk->table_4 = transcript.template receive_from_prover<Commitment>(domain_separator + labels.table_4);
-        inst->verification_key = std::move(vk);
+        inst->verification_key = std::make_shared<VerificationKey>(inst->instance_size, inst->public_input_size);
+        auto vk_view = inst->verification_key->pointer_view();
+        auto labels = typename Flavor::CommitmentLabels().get_precomputed();
+        for (size_t idx = 0; idx < labels.size(); idx++) {
+            (*vk_view[idx]) = transcript.template receive_from_prover<Commitment>(domain_separator + "_" + labels[idx]);
+        }
     }
 }
 
@@ -107,35 +69,30 @@ template <class VerifierInstances>
 bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<uint8_t> fold_data)
 {
     prepare_for_folding(fold_data);
+
     auto delta = transcript.get_challenge("delta");
-    info(delta);
     auto accumulator = get_accumulator();
-    auto log_instance_size = static_cast<size_t>(numeric::get_msb(accumulator->instance_size));
-    auto deltas = compute_round_challenge_pows(log_instance_size, delta);
-    std::vector<FF> perturbator_coeffs(log_instance_size + 1);
-    for (size_t idx = 0; idx <= log_instance_size; idx++) {
+    auto deltas = compute_round_challenge_pows(accumulator->log_instance_size, delta);
+
+    std::vector<FF> perturbator_coeffs(accumulator->log_instance_size + 1);
+    for (size_t idx = 0; idx <= accumulator->log_instance_size; idx++) {
         perturbator_coeffs[idx] = transcript.template receive_from_prover<FF>("perturbator_" + std::to_string(idx));
     }
     assert(perturbator_coeffs[0] == accumulator->folding_parameters.target_sum);
     auto perturbator = Polynomial<FF>(perturbator_coeffs);
     auto perturbator_challenge = transcript.get_challenge("perturbator_challenge");
-    info("pc", perturbator_challenge);
     auto perturbator_at_challenge = perturbator.evaluate(perturbator_challenge);
 
     // Thed degree of K(X) is dk - k - 1 = k(d - 1) - 1. Hence we need  k(d - 1) evaluations to represent it.
     std::array<FF, VerifierInstances::BATCHED_EXTENDED_LENGTH - VerifierInstances::NUM> combiner_quotient_evals;
-    info(VerifierInstances::BATCHED_EXTENDED_LENGTH);
-    info(VerifierInstances::NUM);
     for (size_t idx = 0; idx < VerifierInstances::BATCHED_EXTENDED_LENGTH - VerifierInstances::NUM; idx++) {
         combiner_quotient_evals[idx] = transcript.template receive_from_prover<FF>(
             "combiner_quotient_" + std::to_string(idx + VerifierInstances::NUM));
-        info(combiner_quotient_evals[idx]);
     }
     Univariate<FF, VerifierInstances::BATCHED_EXTENDED_LENGTH, VerifierInstances::NUM> combiner_quotient(
         combiner_quotient_evals);
 
     auto combiner_challenge = transcript.get_challenge("combiner_quotient_challenge");
-    info("cc", combiner_challenge);
     auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge);
 
     auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
@@ -145,14 +102,12 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
         perturbator_at_challenge * lagranges[0] + vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
     auto next_target_sum = transcript.template receive_from_prover<FF>("next_target_sum");
     bool verified = (expected_next_target_sum == next_target_sum);
-    assert(expected_next_target_sum == next_target_sum);
-
-    auto expected_betas_star = update_gate_separation_challenges(
-        perturbator_challenge, accumulator->folding_parameters.gate_separation_challenges, deltas);
-    for (size_t idx = 0; idx < log_instance_size; idx++) {
-        auto expected_beta_star = transcript.template receive_from_prover<FF>("betas_star_" + std::to_string(idx));
-        verified = verified & (expected_betas_star[idx] == expected_beta_star);
-        assert(expected_betas_star[idx] == expected_beta_star);
+
+    auto expected_betas_star =
+        update_gate_challenges(perturbator_challenge, accumulator->folding_parameters.gate_challenges, deltas);
+    for (size_t idx = 0; idx < accumulator->log_instance_size; idx++) {
+        auto beta_star = transcript.template receive_from_prover<FF>("next_gate_challenge_" + std::to_string(idx));
+        verified = verified & (expected_betas_star[idx] == beta_star);
     }
 
     std::vector<FF> folded_public_inputs(instances[0]->public_inputs.size());
@@ -167,15 +122,12 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
     }
 
     for (size_t idx = 0; idx < folded_public_inputs.size(); idx++) {
-        auto public_input = transcript.template receive_from_prover<FF>("folded_public_input" + std::to_string(idx));
+        auto public_input = transcript.template receive_from_prover<FF>("next_public_input" + std::to_string(idx));
         verified = verified & (public_input == folded_public_inputs[idx]);
-        assert(public_input == folded_public_inputs[idx]);
     }
 
-    auto alpha = transcript.template receive_from_prover<FF>("folded_alpha");
+    auto alpha = transcript.template receive_from_prover<FF>("next_alpha");
     verified = verified & (alpha == folded_alpha);
-    assert(alpha == folded_alpha);
-    // all verification keys have the same size
 
     auto acc_vk = std::make_shared<VerificationKey>(instances[0]->instance_size, instances[0]->public_input_size);
     auto acc_vk_view = acc_vk->pointer_view();
@@ -188,10 +140,7 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
 
     auto labels = typename Flavor::CommitmentLabels().get_precomputed();
     for (size_t idx = 0; idx < labels.size(); idx++) {
-        auto vk = transcript.template receive_from_prover<Commitment>("new" + labels[idx]);
-        info(labels[idx]);
-        info((*acc_vk_view[idx]));
-        assert(vk == (*acc_vk_view[idx]));
+        auto vk = transcript.template receive_from_prover<Commitment>("next" + labels[idx]);
         verified = verified & (vk == (*acc_vk_view[idx]));
     }
 
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
index 7e13939b238..2ae89ceac40 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
@@ -35,9 +35,9 @@ template <class VerifierInstances> class ProtoGalaxyVerifier_ {
         return pows;
     }
 
-    static std::vector<FF> update_gate_separation_challenges(const FF perturbator_challenge,
-                                                             const std::vector<FF>& gate_challenges,
-                                                             const std::vector<FF>& round_challenges)
+    static std::vector<FF> update_gate_challenges(const FF perturbator_challenge,
+                                                  const std::vector<FF>& gate_challenges,
+                                                  const std::vector<FF>& round_challenges)
     {
         auto log_instance_size = gate_challenges.size();
         std::vector<FF> next_gate_challenges(log_instance_size);
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
index 9100bba4226..cd7e3a0092f 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
@@ -49,6 +49,7 @@ template <class Flavor> class ProverInstance_ {
     FoldingParameters folding_parameters;
     uint32_t is_accumulator = 0;
     size_t instance_size;
+    size_t log_instance_size;
 
     ProverInstance_(Circuit& circuit)
     {
@@ -56,6 +57,7 @@ template <class Flavor> class ProverInstance_ {
         compute_proving_key(circuit);
         compute_witness(circuit);
         instance_size = proving_key->circuit_size;
+        log_instance_size = static_cast<size_t>(numeric::get_msb(instance_size));
     }
 
     // ProverInstance_(FoldingResult<Flavor> result)
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
index 7cbc1f0d347..c231f44dd2a 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
@@ -14,6 +14,7 @@ template <class Flavor> class VerifierInstance_ {
     size_t pub_inputs_offset = 0;
     size_t public_input_size;
     size_t instance_size;
+    size_t log_instance_size;
     RelationParameters<FF> relation_parameters;
     FF alpha;
     uint32_t is_accumulator = 0;
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
index 50166d2048e..5e48177f505 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
@@ -318,6 +318,7 @@ TEST_F(ProtoGalaxyTests, ComputeNewAccumulator)
 
     auto accumulator = std::make_shared<Instance>();
     accumulator->instance_size = instance_size;
+    accumulator->log_instance_size = log_instance_size;
     accumulator->prover_polynomials = full_polynomials;
     accumulator->folding_parameters = { betas, target_sum };
     accumulator->relation_parameters = relation_parameters;

From 4581a0cb29a18f68f972e2eb11568b548c25a9c8 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Wed, 29 Nov 2023 14:16:19 +0000
Subject: [PATCH 33/86] confused

---
 .../protogalaxy/protogalaxy_prover.hpp        | 28 +++++++++++----
 .../protogalaxy/protogalaxy_verifier.cpp      | 35 +++++++++++++++++--
 .../ultra_honk/ultra_composer.hpp             |  3 +-
 3 files changed, 55 insertions(+), 11 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index cac12cce9df..667527f72a7 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -45,8 +45,8 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     BaseTranscript<FF> transcript;
 
     ProtoGalaxyProver_() = default;
-    ProtoGalaxyProver_(ProverInstances insts)
-        : instances(insts){};
+    ProtoGalaxyProver_(std::vector<std::shared_ptr<Instance>> insts)
+        : instances(ProverInstances(insts)){};
     ~ProtoGalaxyProver_() = default;
 
     /**
@@ -367,13 +367,12 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     static void fold_relation_parameters(ProverInstances& instances)
     {
         // array of parameters to be computed
-        auto& folded_parameters = instances.relation_parameters.to_fold;
         size_t param_idx = 0;
-        for (auto& folded_parameter : folded_parameters) {
+        for (auto& folded_parameter : instances.relation_parameters.to_fold) {
             Univariate<FF, ProverInstances::NUM> tmp(0);
             size_t instance_idx = 0;
             for (auto& instance : instances) {
-                tmp.value_at(instance_idx) = instance->relation_parameters.to_fold[param_idx];
+                tmp.value_at(instance_idx) = instance->relation_parameters.to_fold[param_idx].get();
                 instance_idx++;
             }
             folded_parameter.get() = tmp.template extend_to<ProverInstances::EXTENDED_LENGTH>();
@@ -461,6 +460,24 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         next_accumulator->alpha = instances.alpha.evaluate(challenge);
         transcript.send_to_verifier("next_alpha", next_accumulator->alpha);
 
+        auto combined_relation_parameters = instances.relation_parameters;
+        auto folded_relation_parameters = proof_system::RelationParameters<FF>{
+            combined_relation_parameters.eta.evaluate(challenge),
+            combined_relation_parameters.beta.evaluate(challenge),
+            combined_relation_parameters.gamma.evaluate(challenge),
+            combined_relation_parameters.public_input_delta.evaluate(challenge),
+            combined_relation_parameters.lookup_grand_product_delta.evaluate(challenge),
+        };
+        transcript.send_to_verifier("next_eta", folded_relation_parameters.eta);
+        info(folded_relation_parameters.eta);
+        transcript.send_to_verifier("next_beta", folded_relation_parameters.beta);
+        transcript.send_to_verifier("next_gamma", folded_relation_parameters.gamma);
+        transcript.send_to_verifier("next_public_input_delta", folded_relation_parameters.public_input_delta);
+        transcript.send_to_verifier("next_lookup_grand_product_delta",
+                                    folded_relation_parameters.lookup_grand_product_delta);
+
+        next_accumulator->relation_parameters = folded_relation_parameters;
+
         auto acc_vk = std::make_shared<VerificationKey>(instances[0]->prover_polynomials.get_polynomial_size(),
                                                         instances[0]->public_inputs.size());
         auto acc_vk_view = acc_vk->pointer_view();
@@ -476,7 +493,6 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
             transcript.send_to_verifier("next_" + labels[idx], (*acc_vk_view[idx]));
         }
 
-        // next_accumulator->relation_parameters = instances.relation_parameters;
         return next_accumulator;
     }
 };
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index 23d2910f76d..66e961dda46 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -112,6 +112,7 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
 
     std::vector<FF> folded_public_inputs(instances[0]->public_inputs.size());
     auto folded_alpha = FF(0);
+    auto folded_parameters = proof_system::RelationParameters<FF>{};
     for (size_t inst_idx = 0; inst_idx < VerifierInstances::NUM; inst_idx++) {
         auto instance = instances[inst_idx];
         auto inst_public_inputs = instance->public_inputs;
@@ -119,6 +120,12 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
             folded_public_inputs[el_idx] += inst_public_inputs[el_idx] * lagranges[inst_idx];
         }
         folded_alpha += instance->alpha * lagranges[inst_idx];
+        folded_parameters.eta += instance->relation_parameters.eta * lagranges[inst_idx];
+        folded_parameters.beta += instance->relation_parameters.beta * lagranges[inst_idx];
+        folded_parameters.gamma += instance->relation_parameters.gamma * lagranges[inst_idx];
+        folded_parameters.public_input_delta += instance->relation_parameters.public_input_delta * lagranges[inst_idx];
+        folded_parameters.lookup_grand_product_delta +=
+            instance->relation_parameters.lookup_grand_product_delta * lagranges[inst_idx];
     }
 
     for (size_t idx = 0; idx < folded_public_inputs.size(); idx++) {
@@ -126,9 +133,31 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
         verified = verified & (public_input == folded_public_inputs[idx]);
     }
 
-    auto alpha = transcript.template receive_from_prover<FF>("next_alpha");
-    verified = verified & (alpha == folded_alpha);
-
+    auto next_alpha = transcript.template receive_from_prover<FF>("next_alpha");
+    verified = verified & (next_alpha == folded_alpha);
+
+    auto next_eta = transcript.template receive_from_prover<FF>("next_eta");
+    info(next_eta);
+    info(folded_parameters.eta);
+    verified = verified & (next_eta == folded_parameters.eta);
+    auto next_beta = transcript.template receive_from_prover<FF>("next_beta");
+    info(next_beta);
+    info(folded_parameters.beta);
+    verified = verified & (next_beta == folded_parameters.beta);
+    auto next_gamma = transcript.template receive_from_prover<FF>("next_gamma");
+    verified = verified & (next_gamma == folded_parameters.gamma);
+    info(next_gamma);
+    info(folded_parameters.gamma);
+    auto next_public_input_delta = transcript.template receive_from_prover<FF>("next_public_input_delta");
+    info(next_public_input_delta);
+    info(folded_parameters.public_input_delta);
+    verified = verified & (next_public_input_delta == folded_parameters.public_input_delta);
+    auto next_lookup_grand_product_delta =
+        transcript.template receive_from_prover<FF>("next_lookup_grand_product_delta");
+    verified = verified & (next_lookup_grand_product_delta == folded_parameters.lookup_grand_product_delta);
+    info(next_lookup_grand_product_delta);
+    info(folded_parameters.lookup_grand_product_delta);
+    info(verified);
     auto acc_vk = std::make_shared<VerificationKey>(instances[0]->instance_size, instances[0]->public_input_size);
     auto acc_vk_view = acc_vk->pointer_view();
     for (size_t inst_idx = 0; inst_idx < VerifierInstances::NUM; inst_idx++) {
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp
index 35c5a092274..9cef5e9b7e5 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp
@@ -95,8 +95,7 @@ template <UltraFlavor Flavor> class UltraComposer_ {
 
     ProtoGalaxyProver_<ProverInstances> create_folding_prover(std::vector<std::shared_ptr<Instance>> instances)
     {
-        ProverInstances insts(instances);
-        ProtoGalaxyProver_<ProverInstances> output_state(insts);
+        ProtoGalaxyProver_<ProverInstances> output_state(instances);
 
         return output_state;
     };

From aa0acca898de3b55c85cdf5029eb23b31cb981ee Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Wed, 29 Nov 2023 14:27:29 +0000
Subject: [PATCH 34/86] works

---
 .../cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp    | 2 ++
 .../cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp  | 3 ++-
 2 files changed, 4 insertions(+), 1 deletion(-)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index 93b8a66e027..ecdf6853d65 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -26,6 +26,8 @@ template <class ProverInstances> void ProtoGalaxyProver_<ProverInstances>::prepa
         }
 
         if (!static_cast<bool>(instance->is_accumulator)) {
+            transcript.send_to_verifier(domain_separator + "_pub_inputs_offset",
+                                        static_cast<uint32_t>(instance->pub_inputs_offset));
             auto [eta, beta, gamma] = transcript.get_challenges(
                 domain_separator + "_eta", domain_separator + "_beta", domain_separator + "_gamma");
             instance->compute_sorted_accumulator_polynomials(eta);
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index 66e961dda46..44e3443d14f 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -44,7 +44,8 @@ void ProtoGalaxyVerifier_<VerifierInstances>::prepare_for_folding(std::vector<ui
                     domain_separator + "_gate_challenge_" + std::to_string(idx));
             }
         } else {
-
+            inst->pub_inputs_offset =
+                transcript.template receive_from_prover<uint32_t>(domain_separator + "_pub_inputs_offset");
             auto [eta, beta, gamma] = transcript.get_challenges(
                 domain_separator + "_eta", domain_separator + "_beta", domain_separator + "_gamma");
             const FF public_input_delta = compute_public_input_delta<Flavor>(

From a13ee932338069e1fd63b4f028cbc4383712d1f7 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Wed, 29 Nov 2023 20:37:01 +0000
Subject: [PATCH 35/86] create witness commitments structure

---
 .../src/barretenberg/flavor/goblin_ultra.hpp  | 14 +----
 .../flavor/goblin_ultra_recursive.hpp         | 12 +---
 .../cpp/src/barretenberg/flavor/ultra.hpp     | 23 ++------
 .../barretenberg/flavor/ultra_recursive.hpp   | 21 +------
 .../sumcheck/instance/prover_instance.cpp     | 56 +++++++++++++------
 .../sumcheck/instance/prover_instance.hpp     |  6 ++
 .../instance/prover_instance.test.cpp         |  2 +-
 .../barretenberg/ultra_honk/ultra_prover.cpp  | 35 +++++-------
 8 files changed, 68 insertions(+), 101 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
index 66127524328..eb10eb909fd 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
@@ -40,7 +40,7 @@ class GoblinUltra {
     // assignment of witnesses. We again choose a neutral name.
     static constexpr size_t NUM_PRECOMPUTED_ENTITIES = 28;
     // The total number of witness entities not including shifts.
-    static constexpr size_t NUM_WITNESS_ENTITIES = 18;
+    static constexpr size_t NUM_WITNESS_ENTITIES = 14;
 
     using GrandProductRelations =
         std::tuple<proof_system::UltraPermutationRelation<FF>, proof_system::LookupRelation<FF>>;
@@ -167,10 +167,6 @@ class GoblinUltra {
         DataType w_r;                  // column 1
         DataType w_o;                  // column 2
         DataType w_4;                  // column 3
-        DataType sorted_1;             // column 4
-        DataType sorted_2;             // column 5
-        DataType sorted_3;             // column 6
-        DataType sorted_4;             // column 7
         DataType sorted_accum;         // column 8
         DataType z_perm;               // column 9
         DataType z_lookup;             // column 10
@@ -187,10 +183,6 @@ class GoblinUltra {
                             &w_r,
                             &w_o,
                             &w_4,
-                            &sorted_1,
-                            &sorted_2,
-                            &sorted_3,
-                            &sorted_4,
                             &sorted_accum,
                             &z_perm,
                             &z_lookup,
@@ -207,8 +199,6 @@ class GoblinUltra {
         {
             return { ecc_op_wire_1, ecc_op_wire_2, ecc_op_wire_3, ecc_op_wire_4 };
         };
-        // The sorted concatenations of table and witness data needed for plookup.
-        std::vector<HandleType> get_sorted_polynomials() { return { sorted_1, sorted_2, sorted_3, sorted_4 }; };
     };
 
     /**
@@ -482,6 +472,8 @@ class GoblinUltra {
         }
     };
 
+    using WitnessCommitments = WitnessEntities<Commitment, CommitmentHandle>;
+
     /**
      * @brief A container for commitment labels.
      * @note It's debatable whether this should inherit from AllEntities. since most entries are not strictly needed. It
diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
index f37d26b3adc..9c1060ead79 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
@@ -66,7 +66,7 @@ template <typename BuilderType> class GoblinUltraRecursive_ {
     // assignment of witnesses. We again choose a neutral name.
     static constexpr size_t NUM_PRECOMPUTED_ENTITIES = 28;
     // The total number of witness entities not including shifts.
-    static constexpr size_t NUM_WITNESS_ENTITIES = 18;
+    static constexpr size_t NUM_WITNESS_ENTITIES = 14;
 
     // define the tuple of Relations that comprise the Sumcheck relation
     using Relations = std::tuple<proof_system::UltraArithmeticRelation<FF>,
@@ -180,10 +180,6 @@ template <typename BuilderType> class GoblinUltraRecursive_ {
         DataType w_r;                  // column 1
         DataType w_o;                  // column 2
         DataType w_4;                  // column 3
-        DataType sorted_1;             // column 4
-        DataType sorted_2;             // column 5
-        DataType sorted_3;             // column 6
-        DataType sorted_4;             // column 7
         DataType sorted_accum;         // column 8
         DataType z_perm;               // column 9
         DataType z_lookup;             // column 10
@@ -200,10 +196,6 @@ template <typename BuilderType> class GoblinUltraRecursive_ {
                             &w_r,
                             &w_o,
                             &w_4,
-                            &sorted_1,
-                            &sorted_2,
-                            &sorted_3,
-                            &sorted_4,
                             &sorted_accum,
                             &z_perm,
                             &z_lookup,
@@ -220,8 +212,6 @@ template <typename BuilderType> class GoblinUltraRecursive_ {
         {
             return { ecc_op_wire_1, ecc_op_wire_2, ecc_op_wire_3, ecc_op_wire_4 };
         };
-        // The sorted concatenations of table and witness data needed for plookup.
-        std::vector<HandleType> get_sorted_polynomials() { return { sorted_1, sorted_2, sorted_3, sorted_4 }; };
     };
 
     /**
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
index da12a3994ef..9603d6f58e6 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
@@ -40,7 +40,7 @@ class Ultra {
     // assignment of witnesses. We again choose a neutral name.
     static constexpr size_t NUM_PRECOMPUTED_ENTITIES = 25;
     // The total number of witness entities not including shifts.
-    static constexpr size_t NUM_WITNESS_ENTITIES = 11;
+    static constexpr size_t NUM_WITNESS_ENTITIES = 7;
 
     using GrandProductRelations =
         std::tuple<proof_system::UltraPermutationRelation<FF>, proof_system::LookupRelation<FF>>;
@@ -158,30 +158,13 @@ class Ultra {
         DataType w_r;          // column 1
         DataType w_o;          // column 2
         DataType w_4;          // column 3
-        DataType sorted_1;     // column 4
-        DataType sorted_2;     // column 5
-        DataType sorted_3;     // column 6
-        DataType sorted_4;     // column 7
         DataType sorted_accum; // column 8
         DataType z_perm;       // column 9
         DataType z_lookup;     // column 10
 
-        DEFINE_POINTER_VIEW(NUM_WITNESS_ENTITIES,
-                            &w_l,
-                            &w_r,
-                            &w_o,
-                            &w_4,
-                            &sorted_1,
-                            &sorted_2,
-                            &sorted_3,
-                            &sorted_4,
-                            &sorted_accum,
-                            &z_perm,
-                            &z_lookup)
+        DEFINE_POINTER_VIEW(NUM_WITNESS_ENTITIES, &w_l, &w_r, &w_o, &w_4, &sorted_accum, &z_perm, &z_lookup)
 
         std::vector<HandleType> get_wires() override { return { w_l, w_r, w_o, w_4 }; };
-        // The sorted concatenations of table and witness data needed for plookup.
-        std::vector<HandleType> get_sorted_polynomials() { return { sorted_1, sorted_2, sorted_3, sorted_4 }; };
     };
 
     /**
@@ -392,6 +375,8 @@ class Ultra {
      */
     using ExtendedEdges = ProverUnivariates<MAX_PARTIAL_RELATION_LENGTH>;
 
+    using WitnessCommitments = WitnessEntities<Commitment, CommitmentHandle>;
+
     /**
      * @brief A container for commitment labels.
      * @note It's debatable whether this should inherit from AllEntities. since most entries are not strictly needed. It
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
index e780007eb36..aa186f355c0 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
@@ -66,7 +66,7 @@ template <typename BuilderType> class UltraRecursive_ {
     // assignment of witnesses. We again choose a neutral name.
     static constexpr size_t NUM_PRECOMPUTED_ENTITIES = 25;
     // The total number of witness entities not including shifts.
-    static constexpr size_t NUM_WITNESS_ENTITIES = 11;
+    static constexpr size_t NUM_WITNESS_ENTITIES = 7;
 
     // define the tuple of Relations that comprise the Sumcheck relation
     using Relations = std::tuple<proof_system::UltraArithmeticRelation<FF>,
@@ -143,30 +143,13 @@ template <typename BuilderType> class UltraRecursive_ {
         DataType w_r;          // column 1
         DataType w_o;          // column 2
         DataType w_4;          // column 3
-        DataType sorted_1;     // column 4
-        DataType sorted_2;     // column 5
-        DataType sorted_3;     // column 6
-        DataType sorted_4;     // column 7
         DataType sorted_accum; // column 8
         DataType z_perm;       // column 9
         DataType z_lookup;     // column 10
 
-        DEFINE_POINTER_VIEW(NUM_WITNESS_ENTITIES,
-                            &w_l,
-                            &w_r,
-                            &w_o,
-                            &w_4,
-                            &sorted_1,
-                            &sorted_2,
-                            &sorted_3,
-                            &sorted_4,
-                            &sorted_accum,
-                            &z_perm,
-                            &z_lookup, )
+        DEFINE_POINTER_VIEW(NUM_WITNESS_ENTITIES, &w_l, &w_r, &w_o, &w_4, &sorted_accum, &z_perm, &z_lookup, )
 
         std::vector<HandleType> get_wires() override { return { w_l, w_r, w_o, w_4 }; };
-        // The sorted concatenations of table and witness data needed for plookup.
-        std::vector<HandleType> get_sorted_polynomials() { return { sorted_1, sorted_2, sorted_3, sorted_4 }; };
     };
 
     /**
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.cpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.cpp
index 34f2842eede..3290c14814c 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.cpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.cpp
@@ -66,11 +66,10 @@ template <class Flavor> void ProverInstance_<Flavor>::compute_witness(Circuit& c
         construct_databus_polynomials(circuit);
     }
 
-    // Construct the sorted concatenated list polynomials for the lookup argument
-    polynomial s_1(dyadic_circuit_size);
-    polynomial s_2(dyadic_circuit_size);
-    polynomial s_3(dyadic_circuit_size);
-    polynomial s_4(dyadic_circuit_size);
+    // Initialise the sorted concatenated list polynomials for the lookup argument
+    for (auto& s_i : sorted_polynomials) {
+        s_i = Polynomial(dyadic_circuit_size);
+    }
 
     // The sorted list polynomials have (tables_size + lookups_size) populated entries. We define the index below so
     // that these entries are written into the last indices of the polynomials. The values on the first
@@ -116,21 +115,14 @@ template <class Flavor> void ProverInstance_<Flavor>::compute_witness(Circuit& c
 
         for (const auto& entry : lookup_gates) {
             const auto components = entry.to_sorted_list_components(table.use_twin_keys);
-            s_1[s_index] = components[0];
-            s_2[s_index] = components[1];
-            s_3[s_index] = components[2];
-            s_4[s_index] = table_index;
+            sorted_polynomials[0][s_index] = components[0];
+            sorted_polynomials[1][s_index] = components[1];
+            sorted_polynomials[2][s_index] = components[2];
+            sorted_polynomials[3][s_index] = table_index;
             ++s_index;
         }
     }
 
-    // Polynomial memory is zeroed out when constructed with size hint, so we don't have to initialize trailing
-    // space
-    proving_key->sorted_1 = s_1;
-    proving_key->sorted_2 = s_2;
-    proving_key->sorted_3 = s_3;
-    proving_key->sorted_4 = s_4;
-
     // Copy memory read/write record data into proving key. Prover needs to know which gates contain a read/write
     // 'record' witness on the 4th wire. This wire value can only be fully computed once the first 3 wire
     // polynomials have been committed to. The 4th wire on these gates will be a random linear combination of the
@@ -361,6 +353,27 @@ template <class Flavor> void ProverInstance_<Flavor>::initialize_prover_polynomi
     }
 }
 
+/**
+ * @brief Commit to the wire polynomials which are part of the witness with the exception of the fourth wire, which is
+ * only commited to after adding memory records.
+ */
+template <class Flavor> void ProverInstance_<Flavor>::compute_wire_commitments()
+{
+
+    witness_commitments.w_l = commitment_key->commit(proving_key->w_l);
+    witness_commitments.w_r = commitment_key->commit(proving_key->w_r);
+    witness_commitments.w_o = commitment_key->commit(proving_key->w_o);
+
+    if constexpr (IsGoblinFlavor<Flavor>) {
+        witness_commitments.ecc_op_wire_1 = commitment_key->commit(proving_key->ecc_op_wire_1);
+        witness_commitments.ecc_op_wire_2 = commitment_key->commit(proving_key->ecc_op_wire_2);
+        witness_commitments.ecc_op_wire_3 = commitment_key->commit(proving_key->ecc_op_wire_3);
+        witness_commitments.ecc_op_wire_4 = commitment_key->commit(proving_key->ecc_op_wire_4);
+        witness_commitments.calldata = commitment_key->commit(proving_key->calldata);
+        witness_commitments.calldata_read_counts = commitment_key->commit(proving_key->calldata_read_counts);
+    }
+}
+
 template <class Flavor> void ProverInstance_<Flavor>::compute_sorted_accumulator_polynomials(FF eta)
 {
     relation_parameters.eta = eta;
@@ -369,10 +382,14 @@ template <class Flavor> void ProverInstance_<Flavor>::compute_sorted_accumulator
     prover_polynomials.sorted_accum = proving_key->sorted_accum;
     prover_polynomials.sorted_accum_shift = proving_key->sorted_accum.shifted();
 
+    witness_commitments.sorted_accum = commitment_key->commit(prover_polynomials.sorted_accum);
+
     // Finalize fourth wire polynomial by adding lookup memory records
     add_plookup_memory_records_to_wire_4(eta);
     prover_polynomials.w_4 = proving_key->w_4;
     prover_polynomials.w_4_shift = proving_key->w_4.shifted();
+
+    witness_commitments.w_4 = commitment_key->commit(prover_polynomials.w_4);
 }
 
 /**
@@ -392,8 +409,6 @@ template <class Flavor> void ProverInstance_<Flavor>::compute_sorted_list_accumu
 
     auto sorted_list_accumulator = Polynomial{ circuit_size };
 
-    auto sorted_polynomials = proving_key->get_sorted_polynomials();
-
     // Construct s via Horner, i.e. s = s_1 + η(s_2 + η(s_3 + η*s_4))
     for (size_t i = 0; i < circuit_size; ++i) {
         FF T0 = sorted_polynomials[3][i];
@@ -463,6 +478,8 @@ void ProverInstance_<Flavor>::compute_logderivative_inverse(FF beta, FF gamma)
     // Compute permutation and lookup grand product polynomials
     lookup_library::compute_logderivative_inverse<Flavor, typename Flavor::LogDerivLookupRelation>(
         prover_polynomials, relation_parameters, proving_key->circuit_size);
+
+    witness_commitments.lookup_inverses = commitment_key->commit(prover_polynomials.lookup_inverses);
 }
 
 template <class Flavor> void ProverInstance_<Flavor>::compute_grand_product_polynomials(FF beta, FF gamma)
@@ -479,6 +496,9 @@ template <class Flavor> void ProverInstance_<Flavor>::compute_grand_product_poly
 
     // Compute permutation and lookup grand product polynomials
     grand_product_library::compute_grand_products<Flavor>(proving_key, prover_polynomials, relation_parameters);
+
+    witness_commitments.z_perm = commitment_key->commit(prover_polynomials.z_perm);
+    witness_commitments.z_lookup = commitment_key->commit(prover_polynomials.z_lookup);
 }
 
 /**
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
index c72e940a206..758ddcc085a 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
@@ -27,6 +27,7 @@ template <class Flavor> class ProverInstance_ {
     using FoldingParameters = typename Flavor::FoldingParameters;
     using ProverPolynomials = typename Flavor::ProverPolynomials;
     using Polynomial = typename Flavor::Polynomial;
+    using WitnessCommitments = typename Flavor::WitnessCommitments;
 
   public:
     std::shared_ptr<ProvingKey> proving_key;
@@ -34,6 +35,9 @@ template <class Flavor> class ProverInstance_ {
     std::shared_ptr<CommitmentKey> commitment_key;
 
     ProverPolynomials prover_polynomials;
+    WitnessCommitments witness_commitments;
+
+    std::array<Polynomial, 4> sorted_polynomials;
 
     // The number of public inputs has to be the same for all instances because they are
     // folded element by element.
@@ -68,6 +72,8 @@ template <class Flavor> class ProverInstance_ {
 
     void initialize_prover_polynomials();
 
+    void compute_wire_commitments();
+
     void compute_sorted_accumulator_polynomials(FF);
 
     void compute_sorted_list_accumulator(FF);
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.test.cpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.test.cpp
index a4c2884904e..20c8fbd394e 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.test.cpp
@@ -62,7 +62,7 @@ template <class Flavor> class InstanceTests : public testing::Test {
 
         // Construct mock sorted list polynomials.
         std::vector<Polynomial> sorted_lists;
-        auto sorted_list_polynomials = instance.proving_key->get_sorted_polynomials();
+        auto sorted_list_polynomials = instance.sorted_polynomials;
         for (auto& sorted_list_poly : sorted_list_polynomials) {
             Polynomial random_polynomial = get_random_polynomial(instance.proving_key->circuit_size);
             sorted_lists.emplace_back(random_polynomial);
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp
index 0dcdb608e2b..95231330e2f 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp
@@ -45,26 +45,25 @@ template <UltraFlavor Flavor> void UltraProver_<Flavor>::execute_preamble_round(
  */
 template <UltraFlavor Flavor> void UltraProver_<Flavor>::execute_wire_commitments_round()
 {
-    // Commit to the first three wire polynomials
-    // We only commit to the fourth wire polynomial after adding memory records
-    auto wire_polys = instance->proving_key->get_wires();
+    instance->compute_wire_commitments();
+
+    auto wire_comms = instance->witness_commitments.get_wires();
     auto labels = commitment_labels.get_wires();
     for (size_t idx = 0; idx < 3; ++idx) {
-        transcript.send_to_verifier(labels[idx], commitment_key->commit(wire_polys[idx]));
+        transcript.send_to_verifier(labels[idx], wire_comms[idx]);
     }
 
     if constexpr (IsGoblinFlavor<Flavor>) {
         // Commit to Goblin ECC op wires
-        auto op_wire_polys = instance->proving_key->get_ecc_op_wires();
+        auto op_wire_comms = instance->witness_commitments.get_ecc_op_wires();
         auto labels = commitment_labels.get_ecc_op_wires();
         for (size_t idx = 0; idx < Flavor::NUM_WIRES; ++idx) {
-            transcript.send_to_verifier(labels[idx], commitment_key->commit(op_wire_polys[idx]));
+            transcript.send_to_verifier(labels[idx], op_wire_comms[idx]);
         }
         // Commit to DataBus columns
-        transcript.send_to_verifier(commitment_labels.calldata,
-                                    commitment_key->commit(instance->proving_key->calldata));
+        transcript.send_to_verifier(commitment_labels.calldata, instance->witness_commitments.calldata);
         transcript.send_to_verifier(commitment_labels.calldata_read_counts,
-                                    commitment_key->commit(instance->proving_key->calldata_read_counts));
+                                    instance->witness_commitments.calldata_read_counts);
     }
 }
 
@@ -78,12 +77,8 @@ template <UltraFlavor Flavor> void UltraProver_<Flavor>::execute_sorted_list_acc
 
     instance->compute_sorted_accumulator_polynomials(eta);
 
-    // Commit to the sorted withness-table accumulator and the finalized (i.e. with memory records) fourth wire
-    // polynomial
-    auto sorted_accum_commitment = commitment_key->commit(instance->proving_key->sorted_accum);
-    auto w_4_commitment = commitment_key->commit(instance->proving_key->w_4);
-    transcript.send_to_verifier(commitment_labels.sorted_accum, sorted_accum_commitment);
-    transcript.send_to_verifier(commitment_labels.w_4, w_4_commitment);
+    transcript.send_to_verifier(commitment_labels.sorted_accum, instance->witness_commitments.sorted_accum);
+    transcript.send_to_verifier(commitment_labels.w_4, instance->witness_commitments.w_4);
 }
 
 /**
@@ -99,9 +94,7 @@ template <UltraFlavor Flavor> void UltraProver_<Flavor>::execute_log_derivative_
 
     if constexpr (IsGoblinFlavor<Flavor>) {
         instance->compute_logderivative_inverse(beta, gamma);
-
-        auto lookup_inverses_commitment = commitment_key->commit(instance->proving_key->lookup_inverses);
-        transcript.send_to_verifier(commitment_labels.lookup_inverses, lookup_inverses_commitment);
+        transcript.send_to_verifier(commitment_labels.lookup_inverses, instance->witness_commitments.lookup_inverses);
     }
 }
 
@@ -114,10 +107,8 @@ template <UltraFlavor Flavor> void UltraProver_<Flavor>::execute_grand_product_c
 
     instance->compute_grand_product_polynomials(relation_parameters.beta, relation_parameters.gamma);
 
-    auto z_perm_commitment = commitment_key->commit(instance->proving_key->z_perm);
-    auto z_lookup_commitment = commitment_key->commit(instance->proving_key->z_lookup);
-    transcript.send_to_verifier(commitment_labels.z_perm, z_perm_commitment);
-    transcript.send_to_verifier(commitment_labels.z_lookup, z_lookup_commitment);
+    transcript.send_to_verifier(commitment_labels.z_perm, instance->witness_commitments.z_perm);
+    transcript.send_to_verifier(commitment_labels.z_lookup, instance->witness_commitments.z_lookup);
 }
 
 /**

From 5634ed1b358d996b1494ac4eba49f33f683855ea Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Wed, 29 Nov 2023 20:44:16 +0000
Subject: [PATCH 36/86] add missing comments

---
 .../src/barretenberg/flavor/goblin_ultra.hpp  | 24 +++++++++++--------
 .../flavor/goblin_ultra_recursive.hpp         | 20 ++++++++--------
 .../cpp/src/barretenberg/flavor/ultra.hpp     | 10 +++++---
 .../barretenberg/flavor/ultra_recursive.hpp   |  6 ++---
 4 files changed, 34 insertions(+), 26 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
index eb10eb909fd..b07ed616ebf 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
@@ -167,16 +167,16 @@ class GoblinUltra {
         DataType w_r;                  // column 1
         DataType w_o;                  // column 2
         DataType w_4;                  // column 3
-        DataType sorted_accum;         // column 8
-        DataType z_perm;               // column 9
-        DataType z_lookup;             // column 10
-        DataType ecc_op_wire_1;        // column 11
-        DataType ecc_op_wire_2;        // column 12
-        DataType ecc_op_wire_3;        // column 13
-        DataType ecc_op_wire_4;        // column 14
-        DataType calldata;             // column 15
-        DataType calldata_read_counts; // column 16
-        DataType lookup_inverses;      // column 17
+        DataType sorted_accum;         // column 4
+        DataType z_perm;               // column 5
+        DataType z_lookup;             // column 6
+        DataType ecc_op_wire_1;        // column 7
+        DataType ecc_op_wire_2;        // column 8
+        DataType ecc_op_wire_3;        // column 9
+        DataType ecc_op_wire_4;        // column 10
+        DataType calldata;             // column 11
+        DataType calldata_read_counts; // column 12
+        DataType lookup_inverses;      // column 13
 
         DEFINE_POINTER_VIEW(NUM_WITNESS_ENTITIES,
                             &w_l,
@@ -472,6 +472,10 @@ class GoblinUltra {
         }
     };
 
+    /**
+     * @brief A container for the witness commitments.
+     *
+     */
     using WitnessCommitments = WitnessEntities<Commitment, CommitmentHandle>;
 
     /**
diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
index 9c1060ead79..68c53a37754 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
@@ -180,16 +180,16 @@ template <typename BuilderType> class GoblinUltraRecursive_ {
         DataType w_r;                  // column 1
         DataType w_o;                  // column 2
         DataType w_4;                  // column 3
-        DataType sorted_accum;         // column 8
-        DataType z_perm;               // column 9
-        DataType z_lookup;             // column 10
-        DataType ecc_op_wire_1;        // column 11
-        DataType ecc_op_wire_2;        // column 12
-        DataType ecc_op_wire_3;        // column 13
-        DataType ecc_op_wire_4;        // column 14
-        DataType calldata;             // column 15
-        DataType calldata_read_counts; // column 16
-        DataType lookup_inverses;      // column 17
+        DataType sorted_accum;         // column 4
+        DataType z_perm;               // column 5
+        DataType z_lookup;             // column 6
+        DataType ecc_op_wire_1;        // column 7
+        DataType ecc_op_wire_2;        // column 8
+        DataType ecc_op_wire_3;        // column 9
+        DataType ecc_op_wire_4;        // column 10
+        DataType calldata;             // column 11
+        DataType calldata_read_counts; // column 12
+        DataType lookup_inverses;      // column 13
 
         DEFINE_POINTER_VIEW(NUM_WITNESS_ENTITIES,
                             &w_l,
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
index 9603d6f58e6..aa53e1c6b26 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
@@ -158,9 +158,9 @@ class Ultra {
         DataType w_r;          // column 1
         DataType w_o;          // column 2
         DataType w_4;          // column 3
-        DataType sorted_accum; // column 8
-        DataType z_perm;       // column 9
-        DataType z_lookup;     // column 10
+        DataType sorted_accum; // column 4
+        DataType z_perm;       // column 5
+        DataType z_lookup;     // column 6
 
         DEFINE_POINTER_VIEW(NUM_WITNESS_ENTITIES, &w_l, &w_r, &w_o, &w_4, &sorted_accum, &z_perm, &z_lookup)
 
@@ -375,6 +375,10 @@ class Ultra {
      */
     using ExtendedEdges = ProverUnivariates<MAX_PARTIAL_RELATION_LENGTH>;
 
+    /**
+     * @brief A container for the witness commitments.
+     *
+     */
     using WitnessCommitments = WitnessEntities<Commitment, CommitmentHandle>;
 
     /**
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
index aa186f355c0..297f2520556 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
@@ -143,9 +143,9 @@ template <typename BuilderType> class UltraRecursive_ {
         DataType w_r;          // column 1
         DataType w_o;          // column 2
         DataType w_4;          // column 3
-        DataType sorted_accum; // column 8
-        DataType z_perm;       // column 9
-        DataType z_lookup;     // column 10
+        DataType sorted_accum; // column 4
+        DataType z_perm;       // column 5
+        DataType z_lookup;     // column 6
 
         DEFINE_POINTER_VIEW(NUM_WITNESS_ENTITIES, &w_l, &w_r, &w_o, &w_4, &sorted_accum, &z_perm, &z_lookup, )
 

From ae559028f9592649092e9dd05b07e1c06d38643b Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Wed, 29 Nov 2023 20:52:11 +0000
Subject: [PATCH 37/86] more missing docs

---
 barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp  | 1 -
 barretenberg/cpp/src/barretenberg/flavor/ultra.hpp         | 1 -
 .../src/barretenberg/sumcheck/instance/prover_instance.cpp | 7 ++++---
 3 files changed, 4 insertions(+), 5 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
index b07ed616ebf..f3a8fb273fd 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
@@ -474,7 +474,6 @@ class GoblinUltra {
 
     /**
      * @brief A container for the witness commitments.
-     *
      */
     using WitnessCommitments = WitnessEntities<Commitment, CommitmentHandle>;
 
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
index aa53e1c6b26..dcdf3566c85 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
@@ -377,7 +377,6 @@ class Ultra {
 
     /**
      * @brief A container for the witness commitments.
-     *
      */
     using WitnessCommitments = WitnessEntities<Commitment, CommitmentHandle>;
 
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.cpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.cpp
index 3290c14814c..2dc18bb594e 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.cpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.cpp
@@ -354,12 +354,12 @@ template <class Flavor> void ProverInstance_<Flavor>::initialize_prover_polynomi
 }
 
 /**
- * @brief Commit to the wire polynomials which are part of the witness with the exception of the fourth wire, which is
- * only commited to after adding memory records.
+ * @brief Commit to the wire polynomials (part of the witness), with the exception of the fourth wire, which is
+ * only commited to after adding memory records. In the Goblin Flavor, we also commit to the ECC OP wires and the
+ * DataBus columns
  */
 template <class Flavor> void ProverInstance_<Flavor>::compute_wire_commitments()
 {
-
     witness_commitments.w_l = commitment_key->commit(proving_key->w_l);
     witness_commitments.w_r = commitment_key->commit(proving_key->w_r);
     witness_commitments.w_o = commitment_key->commit(proving_key->w_o);
@@ -369,6 +369,7 @@ template <class Flavor> void ProverInstance_<Flavor>::compute_wire_commitments()
         witness_commitments.ecc_op_wire_2 = commitment_key->commit(proving_key->ecc_op_wire_2);
         witness_commitments.ecc_op_wire_3 = commitment_key->commit(proving_key->ecc_op_wire_3);
         witness_commitments.ecc_op_wire_4 = commitment_key->commit(proving_key->ecc_op_wire_4);
+
         witness_commitments.calldata = commitment_key->commit(proving_key->calldata);
         witness_commitments.calldata_read_counts = commitment_key->commit(proving_key->calldata_read_counts);
     }

From 4fa6c3d960d41c383b94356bf7b4d540e695eebf Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Thu, 30 Nov 2023 12:48:25 +0000
Subject: [PATCH 38/86] refactor based on discussion

---
 .../honk/verifier/goblin_verifier.test.cpp    | 25 +++---
 .../recursion/honk/verifier/verifier.test.cpp | 23 +++--
 .../sumcheck/instance/prover_instance.cpp     | 85 -------------------
 .../sumcheck/instance/prover_instance.hpp     |  3 -
 .../instance/prover_instance.test.cpp         | 14 +--
 .../ultra_honk/ultra_composer.cpp             | 71 ++++++++++++++--
 .../ultra_honk/ultra_composer.hpp             | 17 ++--
 .../barretenberg/ultra_honk/ultra_prover.cpp  | 40 +++++++--
 .../barretenberg/ultra_honk/ultra_prover.hpp  |  2 +-
 9 files changed, 135 insertions(+), 145 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/goblin_verifier.test.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/goblin_verifier.test.cpp
index 4dbdf3bc178..d1957a6d4ff 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/goblin_verifier.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/goblin_verifier.test.cpp
@@ -135,21 +135,20 @@ template <typename BuilderType> class GoblinRecursiveVerifierTest : public testi
         InnerComposer inner_composer;
         auto instance = inner_composer.create_instance(inner_circuit);
         auto prover = inner_composer.create_prover(instance); // A prerequisite for computing VK
-        const auto native_verification_key = instance->compute_verification_key();
 
         // Instantiate the recursive verification key from the native verification key
         OuterBuilder outer_circuit;
-        auto verification_key = std::make_shared<VerificationKey>(&outer_circuit, native_verification_key);
+        auto verification_key = std::make_shared<VerificationKey>(&outer_circuit, instance->verification_key);
 
         // Spot check some values in the recursive VK to ensure it was constructed correctly
-        EXPECT_EQ(verification_key->circuit_size, native_verification_key->circuit_size);
-        EXPECT_EQ(verification_key->log_circuit_size, native_verification_key->log_circuit_size);
-        EXPECT_EQ(verification_key->num_public_inputs, native_verification_key->num_public_inputs);
-        EXPECT_EQ(verification_key->q_m.get_value(), native_verification_key->q_m);
-        EXPECT_EQ(verification_key->q_r.get_value(), native_verification_key->q_r);
-        EXPECT_EQ(verification_key->sigma_1.get_value(), native_verification_key->sigma_1);
-        EXPECT_EQ(verification_key->id_3.get_value(), native_verification_key->id_3);
-        EXPECT_EQ(verification_key->lagrange_ecc_op.get_value(), native_verification_key->lagrange_ecc_op);
+        EXPECT_EQ(verification_key->circuit_size, instance->verification_key->circuit_size);
+        EXPECT_EQ(verification_key->log_circuit_size, instance->verification_key->log_circuit_size);
+        EXPECT_EQ(verification_key->num_public_inputs, instance->verification_key->num_public_inputs);
+        EXPECT_EQ(verification_key->q_m.get_value(), instance->verification_key->q_m);
+        EXPECT_EQ(verification_key->q_r.get_value(), instance->verification_key->q_r);
+        EXPECT_EQ(verification_key->sigma_1.get_value(), instance->verification_key->sigma_1);
+        EXPECT_EQ(verification_key->id_3.get_value(), instance->verification_key->id_3);
+        EXPECT_EQ(verification_key->lagrange_ecc_op.get_value(), instance->verification_key->lagrange_ecc_op);
     }
 
     /**
@@ -166,11 +165,10 @@ template <typename BuilderType> class GoblinRecursiveVerifierTest : public testi
         auto instance = inner_composer.create_instance(inner_circuit);
         auto inner_prover = inner_composer.create_prover(instance);
         auto inner_proof = inner_prover.construct_proof();
-        const auto native_verification_key = instance->compute_verification_key();
 
         // Create a recursive verification circuit for the proof of the inner circuit
         OuterBuilder outer_circuit;
-        auto verification_key = std::make_shared<VerificationKey>(&outer_circuit, native_verification_key);
+        auto verification_key = std::make_shared<VerificationKey>(&outer_circuit, instance->verification_key);
         RecursiveVerifier verifier(&outer_circuit, verification_key);
         auto pairing_points = verifier.verify_proof(inner_proof);
 
@@ -213,7 +211,6 @@ template <typename BuilderType> class GoblinRecursiveVerifierTest : public testi
         auto instance = inner_composer.create_instance(inner_circuit);
         auto inner_prover = inner_composer.create_prover(instance);
         auto inner_proof = inner_prover.construct_proof();
-        const auto native_verification_key = instance->compute_verification_key();
 
         // Arbitrarily tamper with the proof to be verified
         inner_prover.transcript.deserialize_full_transcript();
@@ -223,7 +220,7 @@ template <typename BuilderType> class GoblinRecursiveVerifierTest : public testi
 
         // Create a recursive verification circuit for the proof of the inner circuit
         OuterBuilder outer_circuit;
-        auto verification_key = std::make_shared<VerificationKey>(&outer_circuit, native_verification_key);
+        auto verification_key = std::make_shared<VerificationKey>(&outer_circuit, instance->verification_key);
         RecursiveVerifier verifier(&outer_circuit, verification_key);
         verifier.verify_proof(inner_proof);
 
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/verifier.test.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/verifier.test.cpp
index 29c5b9bd639..0852c45cc70 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/verifier.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/verifier.test.cpp
@@ -120,19 +120,18 @@ template <typename BuilderType> class RecursiveVerifierTest : public testing::Te
         InnerComposer inner_composer;
         auto instance = inner_composer.create_instance(inner_circuit);
         auto prover = inner_composer.create_prover(instance); // A prerequisite for computing VK
-        const auto native_verification_key = instance->compute_verification_key();
 
         // Instantiate the recursive verification key from the native verification key
-        auto verification_key = std::make_shared<VerificationKey>(&outer_circuit, native_verification_key);
+        auto verification_key = std::make_shared<VerificationKey>(&outer_circuit, instance->verification_key);
 
         // Spot check some values in the recursive VK to ensure it was constructed correctly
-        EXPECT_EQ(verification_key->circuit_size, native_verification_key->circuit_size);
-        EXPECT_EQ(verification_key->log_circuit_size, native_verification_key->log_circuit_size);
-        EXPECT_EQ(verification_key->num_public_inputs, native_verification_key->num_public_inputs);
-        EXPECT_EQ(verification_key->q_m.get_value(), native_verification_key->q_m);
-        EXPECT_EQ(verification_key->q_r.get_value(), native_verification_key->q_r);
-        EXPECT_EQ(verification_key->sigma_1.get_value(), native_verification_key->sigma_1);
-        EXPECT_EQ(verification_key->id_3.get_value(), native_verification_key->id_3);
+        EXPECT_EQ(verification_key->circuit_size, instance->verification_key->circuit_size);
+        EXPECT_EQ(verification_key->log_circuit_size, instance->verification_key->log_circuit_size);
+        EXPECT_EQ(verification_key->num_public_inputs, instance->verification_key->num_public_inputs);
+        EXPECT_EQ(verification_key->q_m.get_value(), instance->verification_key->q_m);
+        EXPECT_EQ(verification_key->q_r.get_value(), instance->verification_key->q_r);
+        EXPECT_EQ(verification_key->sigma_1.get_value(), instance->verification_key->sigma_1);
+        EXPECT_EQ(verification_key->id_3.get_value(), instance->verification_key->id_3);
     }
 
     /**
@@ -150,11 +149,10 @@ template <typename BuilderType> class RecursiveVerifierTest : public testing::Te
         auto instance = inner_composer.create_instance(inner_circuit);
         auto inner_prover = inner_composer.create_prover(instance);
         auto inner_proof = inner_prover.construct_proof();
-        const auto native_verification_key = instance->compute_verification_key();
 
         // Create a recursive verification circuit for the proof of the inner circuit
         OuterBuilder outer_circuit;
-        auto verification_key = std::make_shared<VerificationKey>(&outer_circuit, native_verification_key);
+        auto verification_key = std::make_shared<VerificationKey>(&outer_circuit, instance->verification_key);
         RecursiveVerifier verifier(&outer_circuit, verification_key);
         auto pairing_points = verifier.verify_proof(inner_proof);
 
@@ -198,7 +196,6 @@ template <typename BuilderType> class RecursiveVerifierTest : public testing::Te
         auto instance = inner_composer.create_instance(inner_circuit);
         auto inner_prover = inner_composer.create_prover(instance);
         auto inner_proof = inner_prover.construct_proof();
-        const auto native_verification_key = instance->compute_verification_key();
 
         // Arbitrarily tamper with the proof to be verified
         inner_prover.transcript.deserialize_full_transcript();
@@ -208,7 +205,7 @@ template <typename BuilderType> class RecursiveVerifierTest : public testing::Te
 
         // Create a recursive verification circuit for the proof of the inner circuit
         OuterBuilder outer_circuit;
-        auto verification_key = std::make_shared<VerificationKey>(&outer_circuit, native_verification_key);
+        auto verification_key = std::make_shared<VerificationKey>(&outer_circuit, instance->verification_key);
         RecursiveVerifier verifier(&outer_circuit, verification_key);
         verifier.verify_proof(inner_proof);
 
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.cpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.cpp
index 2dc18bb594e..41924b71875 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.cpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.cpp
@@ -353,28 +353,6 @@ template <class Flavor> void ProverInstance_<Flavor>::initialize_prover_polynomi
     }
 }
 
-/**
- * @brief Commit to the wire polynomials (part of the witness), with the exception of the fourth wire, which is
- * only commited to after adding memory records. In the Goblin Flavor, we also commit to the ECC OP wires and the
- * DataBus columns
- */
-template <class Flavor> void ProverInstance_<Flavor>::compute_wire_commitments()
-{
-    witness_commitments.w_l = commitment_key->commit(proving_key->w_l);
-    witness_commitments.w_r = commitment_key->commit(proving_key->w_r);
-    witness_commitments.w_o = commitment_key->commit(proving_key->w_o);
-
-    if constexpr (IsGoblinFlavor<Flavor>) {
-        witness_commitments.ecc_op_wire_1 = commitment_key->commit(proving_key->ecc_op_wire_1);
-        witness_commitments.ecc_op_wire_2 = commitment_key->commit(proving_key->ecc_op_wire_2);
-        witness_commitments.ecc_op_wire_3 = commitment_key->commit(proving_key->ecc_op_wire_3);
-        witness_commitments.ecc_op_wire_4 = commitment_key->commit(proving_key->ecc_op_wire_4);
-
-        witness_commitments.calldata = commitment_key->commit(proving_key->calldata);
-        witness_commitments.calldata_read_counts = commitment_key->commit(proving_key->calldata_read_counts);
-    }
-}
-
 template <class Flavor> void ProverInstance_<Flavor>::compute_sorted_accumulator_polynomials(FF eta)
 {
     relation_parameters.eta = eta;
@@ -383,14 +361,10 @@ template <class Flavor> void ProverInstance_<Flavor>::compute_sorted_accumulator
     prover_polynomials.sorted_accum = proving_key->sorted_accum;
     prover_polynomials.sorted_accum_shift = proving_key->sorted_accum.shifted();
 
-    witness_commitments.sorted_accum = commitment_key->commit(prover_polynomials.sorted_accum);
-
     // Finalize fourth wire polynomial by adding lookup memory records
     add_plookup_memory_records_to_wire_4(eta);
     prover_polynomials.w_4 = proving_key->w_4;
     prover_polynomials.w_4_shift = proving_key->w_4.shifted();
-
-    witness_commitments.w_4 = commitment_key->commit(prover_polynomials.w_4);
 }
 
 /**
@@ -479,8 +453,6 @@ void ProverInstance_<Flavor>::compute_logderivative_inverse(FF beta, FF gamma)
     // Compute permutation and lookup grand product polynomials
     lookup_library::compute_logderivative_inverse<Flavor, typename Flavor::LogDerivLookupRelation>(
         prover_polynomials, relation_parameters, proving_key->circuit_size);
-
-    witness_commitments.lookup_inverses = commitment_key->commit(prover_polynomials.lookup_inverses);
 }
 
 template <class Flavor> void ProverInstance_<Flavor>::compute_grand_product_polynomials(FF beta, FF gamma)
@@ -497,63 +469,6 @@ template <class Flavor> void ProverInstance_<Flavor>::compute_grand_product_poly
 
     // Compute permutation and lookup grand product polynomials
     grand_product_library::compute_grand_products<Flavor>(proving_key, prover_polynomials, relation_parameters);
-
-    witness_commitments.z_perm = commitment_key->commit(prover_polynomials.z_perm);
-    witness_commitments.z_lookup = commitment_key->commit(prover_polynomials.z_lookup);
-}
-
-/**
- * Compute verification key consisting of selector precommitments.
- *
- * @return Pointer to the resulting verification key of the Instance.
- * */
-template <class Flavor>
-std::shared_ptr<typename Flavor::VerificationKey> ProverInstance_<Flavor>::compute_verification_key()
-{
-    if (verification_key) {
-        return verification_key;
-    }
-
-    verification_key =
-        std::make_shared<typename Flavor::VerificationKey>(proving_key->circuit_size, proving_key->num_public_inputs);
-
-    // Compute and store commitments to all precomputed polynomials
-    verification_key->q_m = commitment_key->commit(proving_key->q_m);
-    verification_key->q_l = commitment_key->commit(proving_key->q_l);
-    verification_key->q_r = commitment_key->commit(proving_key->q_r);
-    verification_key->q_o = commitment_key->commit(proving_key->q_o);
-    verification_key->q_c = commitment_key->commit(proving_key->q_c);
-    verification_key->sigma_1 = commitment_key->commit(proving_key->sigma_1);
-    verification_key->sigma_2 = commitment_key->commit(proving_key->sigma_2);
-    verification_key->sigma_3 = commitment_key->commit(proving_key->sigma_3);
-    verification_key->id_1 = commitment_key->commit(proving_key->id_1);
-    verification_key->id_2 = commitment_key->commit(proving_key->id_2);
-    verification_key->id_3 = commitment_key->commit(proving_key->id_3);
-    verification_key->lagrange_first = commitment_key->commit(proving_key->lagrange_first);
-    verification_key->lagrange_last = commitment_key->commit(proving_key->lagrange_last);
-
-    verification_key->q_4 = commitment_key->commit(proving_key->q_4);
-    verification_key->q_arith = commitment_key->commit(proving_key->q_arith);
-    verification_key->q_sort = commitment_key->commit(proving_key->q_sort);
-    verification_key->q_elliptic = commitment_key->commit(proving_key->q_elliptic);
-    verification_key->q_aux = commitment_key->commit(proving_key->q_aux);
-    verification_key->q_lookup = commitment_key->commit(proving_key->q_lookup);
-    verification_key->sigma_4 = commitment_key->commit(proving_key->sigma_4);
-    verification_key->id_4 = commitment_key->commit(proving_key->id_4);
-    verification_key->table_1 = commitment_key->commit(proving_key->table_1);
-    verification_key->table_2 = commitment_key->commit(proving_key->table_2);
-    verification_key->table_3 = commitment_key->commit(proving_key->table_3);
-    verification_key->table_4 = commitment_key->commit(proving_key->table_4);
-
-    // TODO(luke): Similar to the lagrange_first/last polynomials, we dont really need to commit to these polynomials
-    // due to their simple structure.
-    if constexpr (IsGoblinFlavor<Flavor>) {
-        verification_key->lagrange_ecc_op = commitment_key->commit(proving_key->lagrange_ecc_op);
-        verification_key->q_busread = commitment_key->commit(proving_key->q_busread);
-        verification_key->databus_id = commitment_key->commit(proving_key->databus_id);
-    }
-
-    return verification_key;
 }
 
 template class ProverInstance_<honk::flavor::Ultra>;
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
index 758ddcc085a..251448bd497 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
@@ -32,7 +32,6 @@ template <class Flavor> class ProverInstance_ {
   public:
     std::shared_ptr<ProvingKey> proving_key;
     std::shared_ptr<VerificationKey> verification_key;
-    std::shared_ptr<CommitmentKey> commitment_key;
 
     ProverPolynomials prover_polynomials;
     WitnessCommitments witness_commitments;
@@ -68,8 +67,6 @@ template <class Flavor> class ProverInstance_ {
     ProverInstance_() = default;
     ~ProverInstance_() = default;
 
-    std::shared_ptr<VerificationKey> compute_verification_key();
-
     void initialize_prover_polynomials();
 
     void compute_wire_commitments();
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.test.cpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.test.cpp
index 20c8fbd394e..6fb7eeab3ed 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.test.cpp
@@ -60,14 +60,7 @@ template <class Flavor> class InstanceTests : public testing::Test {
         // Get random challenge eta
         auto eta = FF::random_element();
 
-        // Construct mock sorted list polynomials.
-        std::vector<Polynomial> sorted_lists;
         auto sorted_list_polynomials = instance.sorted_polynomials;
-        for (auto& sorted_list_poly : sorted_list_polynomials) {
-            Polynomial random_polynomial = get_random_polynomial(instance.proving_key->circuit_size);
-            sorted_lists.emplace_back(random_polynomial);
-            populate_span(sorted_list_poly, random_polynomial);
-        }
 
         // Method 1: computed sorted list accumulator polynomial using prover library method
         instance.compute_sorted_list_accumulator(eta);
@@ -78,10 +71,11 @@ template <class Flavor> class InstanceTests : public testing::Test {
         const FF eta_cube = eta_sqr * eta;
 
         // Compute s = s_1 + η*s_2 + η²*s_3 + η³*s_4
-        Polynomial sorted_list_accumulator_expected{ sorted_lists[0] };
+        Polynomial sorted_list_accumulator_expected{ sorted_list_polynomials[0] };
         for (size_t i = 0; i < instance.proving_key->circuit_size; ++i) {
-            sorted_list_accumulator_expected[i] +=
-                sorted_lists[1][i] * eta + sorted_lists[2][i] * eta_sqr + sorted_lists[3][i] * eta_cube;
+            sorted_list_accumulator_expected[i] += sorted_list_polynomials[1][i] * eta +
+                                                   sorted_list_polynomials[2][i] * eta_sqr +
+                                                   sorted_list_polynomials[3][i] * eta_cube;
         }
 
         EXPECT_EQ(sorted_list_accumulator, sorted_list_accumulator_expected);
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.cpp
index ff8cf11b869..8bb986cce27 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.cpp
@@ -6,20 +6,81 @@
 
 namespace proof_system::honk {
 
+/**
+ * Compute verification key consisting of selector precommitments.
+ *
+ * @return Pointer to the resulting verification key of the Instance.
+ * */
+template <UltraFlavor Flavor>
+void UltraComposer_<Flavor>::compute_verification_key(std::shared_ptr<ProverInstance_<Flavor>> instance)
+{
+    if (instance->verification_key) {
+        return;
+    }
+
+    auto& proving_key = instance->proving_key;
+
+    auto verification_key =
+        std::make_shared<typename Flavor::VerificationKey>(proving_key->circuit_size, proving_key->num_public_inputs);
+
+    // Compute and store commitments to all precomputed polynomials
+    verification_key->q_m = commitment_key->commit(proving_key->q_m);
+    verification_key->q_l = commitment_key->commit(proving_key->q_l);
+    verification_key->q_r = commitment_key->commit(proving_key->q_r);
+    verification_key->q_o = commitment_key->commit(proving_key->q_o);
+    verification_key->q_c = commitment_key->commit(proving_key->q_c);
+    verification_key->sigma_1 = commitment_key->commit(proving_key->sigma_1);
+    verification_key->sigma_2 = commitment_key->commit(proving_key->sigma_2);
+    verification_key->sigma_3 = commitment_key->commit(proving_key->sigma_3);
+    verification_key->id_1 = commitment_key->commit(proving_key->id_1);
+    verification_key->id_2 = commitment_key->commit(proving_key->id_2);
+    verification_key->id_3 = commitment_key->commit(proving_key->id_3);
+    verification_key->lagrange_first = commitment_key->commit(proving_key->lagrange_first);
+    verification_key->lagrange_last = commitment_key->commit(proving_key->lagrange_last);
+
+    verification_key->q_4 = commitment_key->commit(proving_key->q_4);
+    verification_key->q_arith = commitment_key->commit(proving_key->q_arith);
+    verification_key->q_sort = commitment_key->commit(proving_key->q_sort);
+    verification_key->q_elliptic = commitment_key->commit(proving_key->q_elliptic);
+    verification_key->q_aux = commitment_key->commit(proving_key->q_aux);
+    verification_key->q_lookup = commitment_key->commit(proving_key->q_lookup);
+    verification_key->sigma_4 = commitment_key->commit(proving_key->sigma_4);
+    verification_key->id_4 = commitment_key->commit(proving_key->id_4);
+    verification_key->table_1 = commitment_key->commit(proving_key->table_1);
+    verification_key->table_2 = commitment_key->commit(proving_key->table_2);
+    verification_key->table_3 = commitment_key->commit(proving_key->table_3);
+    verification_key->table_4 = commitment_key->commit(proving_key->table_4);
+
+    // TODO(luke): Similar to the lagrange_first/last polynomials, we dont really need to commit to these polynomials
+    // due to their simple structure.
+    if constexpr (IsGoblinFlavor<Flavor>) {
+        verification_key->lagrange_ecc_op = commitment_key->commit(proving_key->lagrange_ecc_op);
+        verification_key->q_busread = commitment_key->commit(proving_key->q_busread);
+        verification_key->databus_id = commitment_key->commit(proving_key->databus_id);
+    }
+
+    instance->verification_key = std::move(verification_key);
+}
+
 template <UltraFlavor Flavor>
 std::shared_ptr<ProverInstance_<Flavor>> UltraComposer_<Flavor>::create_instance(CircuitBuilder& circuit)
 {
     circuit.add_gates_to_ensure_all_polys_are_non_zero();
     circuit.finalize_circuit();
     auto instance = std::make_shared<Instance>(circuit);
-    instance->commitment_key = compute_commitment_key(instance->proving_key->circuit_size);
+
+    if (!commitment_key) {
+        commitment_key = compute_commitment_key(instance->proving_key->circuit_size);
+    }
+
+    compute_verification_key(instance);
     return instance;
 }
 
 template <UltraFlavor Flavor>
 UltraProver_<Flavor> UltraComposer_<Flavor>::create_prover(std::shared_ptr<Instance> instance)
 {
-    UltraProver_<Flavor> output_state(instance);
+    UltraProver_<Flavor> output_state(instance, commitment_key);
 
     return output_state;
 }
@@ -27,9 +88,9 @@ UltraProver_<Flavor> UltraComposer_<Flavor>::create_prover(std::shared_ptr<Insta
 template <UltraFlavor Flavor>
 UltraVerifier_<Flavor> UltraComposer_<Flavor>::create_verifier(std::shared_ptr<Instance> instance)
 {
-    auto verification_key = instance->compute_verification_key();
-    UltraVerifier_<Flavor> output_state(verification_key);
-    auto pcs_verification_key = std::make_unique<VerifierCommitmentKey>(verification_key->circuit_size, crs_factory_);
+    UltraVerifier_<Flavor> output_state(instance->verification_key);
+    auto pcs_verification_key =
+        std::make_unique<VerifierCommitmentKey>(instance->verification_key->circuit_size, crs_factory_);
     output_state.pcs_verification_key = std::move(pcs_verification_key);
 
     return output_state;
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp
index 3cd04b03ff1..33840e32bc7 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp
@@ -29,8 +29,6 @@ template <UltraFlavor Flavor> class UltraComposer_ {
     static constexpr size_t num_zero_rows = Flavor::has_zero_row ? 1 : 0;
     static constexpr std::string_view NAME_STRING = "UltraHonk";
     static constexpr size_t NUM_WIRES = CircuitBuilder::NUM_WIRES;
-    std::shared_ptr<ProvingKey> proving_key;
-    std::shared_ptr<VerificationKey> verification_key;
 
     // The crs_factory holds the path to the srs and exposes methods to extract the srs elements
     std::shared_ptr<srs::factories::CrsFactory<typename Flavor::Curve>> crs_factory_;
@@ -43,11 +41,6 @@ template <UltraFlavor Flavor> class UltraComposer_ {
         : crs_factory_(std::move(crs_factory))
     {}
 
-    UltraComposer_(std::shared_ptr<ProvingKey> p_key, std::shared_ptr<VerificationKey> v_key)
-        : proving_key(std::move(p_key))
-        , verification_key(std::move(v_key))
-    {}
-
     UltraComposer_(UltraComposer_&& other) noexcept = default;
     UltraComposer_(UltraComposer_ const& other) noexcept = default;
     UltraComposer_& operator=(UltraComposer_&& other) noexcept = default;
@@ -104,13 +97,21 @@ template <UltraFlavor Flavor> class UltraComposer_ {
     {
         std::vector<std::shared_ptr<VerificationKey>> vks;
         for (const auto& inst : instances) {
-            vks.emplace_back(inst->compute_verification_key());
+            vks.emplace_back(inst->verification_key);
         }
         VerifierInstances insts(vks);
         ProtoGalaxyVerifier_<VerifierInstances> output_state(insts);
 
         return output_state;
     };
+
+  private:
+    /**
+     * @brief Compute the verification key of an Instance, produced from a finalised circuit.
+     *
+     * @param inst
+     */
+    void compute_verification_key(std::shared_ptr<Instance>);
 };
 extern template class UltraComposer_<honk::flavor::Ultra>;
 extern template class UltraComposer_<honk::flavor::GoblinUltra>;
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp
index 5aa5f351cec..70d3e464566 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp
@@ -12,9 +12,9 @@ namespace proof_system::honk {
  * @tparam a type of UltraFlavor
  * */
 template <UltraFlavor Flavor>
-UltraProver_<Flavor>::UltraProver_(std::shared_ptr<Instance> inst)
+UltraProver_<Flavor>::UltraProver_(std::shared_ptr<Instance> inst, std::shared_ptr<CommitmentKey> commitment_key)
     : instance(std::move(inst))
-    , commitment_key(instance->commitment_key)
+    , commitment_key(commitment_key)
 {
     instance->initialize_prover_polynomials();
 }
@@ -40,26 +40,43 @@ template <UltraFlavor Flavor> void UltraProver_<Flavor>::execute_preamble_round(
 }
 
 /**
- * @brief Compute commitments to the first three wire polynomials (and ECC op wires if using Goblin).
- *
+ * @brief Commit to the wire polynomials (part of the witness), with the exception of the fourth wire, which is
+ * only commited to after adding memory records. In the Goblin Flavor, we also commit to the ECC OP wires and the
+ * DataBus columns.
  */
 template <UltraFlavor Flavor> void UltraProver_<Flavor>::execute_wire_commitments_round()
 {
-    instance->compute_wire_commitments();
+    auto& witness_commitments = instance->witness_commitments;
+    auto& proving_key = instance->proving_key;
+
+    // Commit to the first three wire polynomials
+    // We only commit to the fourth wire polynomial after adding memory recordss
+    witness_commitments.w_l = commitment_key->commit(proving_key->w_l);
+    witness_commitments.w_r = commitment_key->commit(proving_key->w_r);
+    witness_commitments.w_o = commitment_key->commit(proving_key->w_o);
 
-    auto wire_comms = instance->witness_commitments.get_wires();
+    auto wire_comms = witness_commitments.get_wires();
     auto labels = commitment_labels.get_wires();
     for (size_t idx = 0; idx < 3; ++idx) {
         transcript.send_to_verifier(labels[idx], wire_comms[idx]);
     }
 
     if constexpr (IsGoblinFlavor<Flavor>) {
+        // Commit to Goblin ECC op wires
+        witness_commitments.ecc_op_wire_1 = commitment_key->commit(proving_key->ecc_op_wire_1);
+        witness_commitments.ecc_op_wire_2 = commitment_key->commit(proving_key->ecc_op_wire_2);
+        witness_commitments.ecc_op_wire_3 = commitment_key->commit(proving_key->ecc_op_wire_3);
+        witness_commitments.ecc_op_wire_4 = commitment_key->commit(proving_key->ecc_op_wire_4);
+
         auto op_wire_comms = instance->witness_commitments.get_ecc_op_wires();
         auto labels = commitment_labels.get_ecc_op_wires();
         for (size_t idx = 0; idx < Flavor::NUM_WIRES; ++idx) {
             transcript.send_to_verifier(labels[idx], op_wire_comms[idx]);
         }
 
+        // Commit to DataBus columns
+        witness_commitments.calldata = commitment_key->commit(proving_key->calldata);
+        witness_commitments.calldata_read_counts = commitment_key->commit(proving_key->calldata_read_counts);
         transcript.send_to_verifier(commitment_labels.calldata, instance->witness_commitments.calldata);
         transcript.send_to_verifier(commitment_labels.calldata_read_counts,
                                     instance->witness_commitments.calldata_read_counts);
@@ -76,6 +93,12 @@ template <UltraFlavor Flavor> void UltraProver_<Flavor>::execute_sorted_list_acc
 
     instance->compute_sorted_accumulator_polynomials(eta);
 
+    auto& witness_commitments = instance->witness_commitments;
+    // Commit to the sorted withness-table accumulator and the finalized (i.e. with memory records) fourth wire
+    // polynomial
+    witness_commitments.sorted_accum = commitment_key->commit(instance->prover_polynomials.sorted_accum);
+    witness_commitments.w_4 = commitment_key->commit(instance->prover_polynomials.w_4);
+
     transcript.send_to_verifier(commitment_labels.sorted_accum, instance->witness_commitments.sorted_accum);
     transcript.send_to_verifier(commitment_labels.w_4, instance->witness_commitments.w_4);
 }
@@ -93,6 +116,8 @@ template <UltraFlavor Flavor> void UltraProver_<Flavor>::execute_log_derivative_
 
     if constexpr (IsGoblinFlavor<Flavor>) {
         instance->compute_logderivative_inverse(beta, gamma);
+        instance->witness_commitments.lookup_inverses =
+            commitment_key->commit(instance->prover_polynomials.lookup_inverses);
         transcript.send_to_verifier(commitment_labels.lookup_inverses, instance->witness_commitments.lookup_inverses);
     }
 }
@@ -106,6 +131,9 @@ template <UltraFlavor Flavor> void UltraProver_<Flavor>::execute_grand_product_c
 
     instance->compute_grand_product_polynomials(relation_parameters.beta, relation_parameters.gamma);
 
+    auto& witness_commitments = instance->witness_commitments;
+    witness_commitments.z_perm = commitment_key->commit(instance->prover_polynomials.z_perm);
+    witness_commitments.z_lookup = commitment_key->commit(instance->prover_polynomials.z_lookup);
     transcript.send_to_verifier(commitment_labels.z_perm, instance->witness_commitments.z_perm);
     transcript.send_to_verifier(commitment_labels.z_lookup, instance->witness_commitments.z_lookup);
 }
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.hpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.hpp
index fe50c8e0c80..9d660067229 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.hpp
@@ -23,7 +23,7 @@ template <UltraFlavor Flavor> class UltraProver_ {
     using Transcript = typename Flavor::Transcript;
 
   public:
-    explicit UltraProver_(std::shared_ptr<Instance>);
+    explicit UltraProver_(std::shared_ptr<Instance>, std::shared_ptr<CommitmentKey>);
     BBERG_PROFILE void execute_preamble_round();
     BBERG_PROFILE void execute_wire_commitments_round();
     BBERG_PROFILE void execute_sorted_list_accumulator_round();

From c2e78b51fde27b3257aef74f4643126c6d947b13 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Fri, 1 Dec 2023 12:30:13 +0000
Subject: [PATCH 39/86] cleaning up

---
 .../src/barretenberg/flavor/goblin_ultra.hpp  |  53 ++++++++-
 .../protogalaxy/protogalaxy_prover.cpp        |  32 +++--
 .../protogalaxy/protogalaxy_prover.hpp        | 112 +++++++++++-------
 .../protogalaxy/protogalaxy_verifier.cpp      |   6 +-
 .../protogalaxy/protogalaxy_verifier.hpp      |   5 +-
 .../sumcheck/instance/instances.hpp           |  12 +-
 .../sumcheck/instance/prover_instance.cpp     |   3 +
 .../sumcheck/instance/prover_instance.hpp     |   8 --
 .../ultra_honk/protogalaxy.test.cpp           |   4 +-
 9 files changed, 145 insertions(+), 90 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
index 5b146f68322..9d6f45806d5 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
@@ -85,12 +85,12 @@ class GoblinUltra {
      */
     class PrecomputedEntities : public PrecomputedEntities_<DataType, HandleType, NUM_PRECOMPUTED_ENTITIES> {
       public:
+        DataType q_m;             // column 0
         DataType q_c;             // column 1
         DataType q_l;             // column 2
         DataType q_r;             // column 3
         DataType q_o;             // column 4
         DataType q_4;             // column 5
-        DataType q_m;             // column 0
         DataType q_arith;         // column 6
         DataType q_sort;          // column 7
         DataType q_elliptic;      // column 8
@@ -374,6 +374,24 @@ class GoblinUltra {
                      calldata_read_counts,
                      lookup_inverses };
         };
+
+        std::vector<HandleType> get_witness()
+        {
+            return { w_l,
+                     w_r,
+                     w_o,
+                     w_4,
+                     sorted_accum,
+                     z_perm,
+                     z_lookup,
+                     ecc_op_wire_1,
+                     ecc_op_wire_2,
+                     ecc_op_wire_3,
+                     ecc_op_wire_4,
+                     calldata,
+                     calldata_read_counts,
+                     lookup_inverses };
+        };
         std::vector<HandleType> get_to_be_shifted() override
         {
             return { table_1, table_2, table_3, table_4, w_l, w_r, w_o, w_4, sorted_accum, z_perm, z_lookup };
@@ -383,6 +401,37 @@ class GoblinUltra {
             return { table_1_shift, table_2_shift, table_3_shift,      table_4_shift, w_l_shift,     w_r_shift,
                      w_o_shift,     w_4_shift,     sorted_accum_shift, z_perm_shift,  z_lookup_shift };
         };
+        std::vector<HandleType> get_precomputed()
+        {
+            return { q_m,
+                     q_c,
+                     q_l,
+                     q_r,
+                     q_o,
+                     q_4,
+                     q_arith,
+                     q_sort,
+                     q_elliptic,
+                     q_aux,
+                     q_lookup,
+                     q_busread,
+                     sigma_1,
+                     sigma_2,
+                     sigma_3,
+                     sigma_4,
+                     id_1,
+                     id_2,
+                     id_3,
+                     id_4,
+                     table_1,
+                     table_2,
+                     table_3,
+                     table_4,
+                     lagrange_first,
+                     lagrange_last,
+                     lagrange_ecc_op,
+                     databus_id };
+        }
     };
 
   public:
@@ -572,7 +621,7 @@ class GoblinUltra {
 
     class FoldingParameters {
       public:
-        std::vector<FF> gate_separation_challenges;
+        std::vector<FF> gate_challenges;
         FF target_sum;
     };
 
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index 73729198fe8..d2ca5f4af45 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -16,9 +16,9 @@ template <class ProverInstances> void ProtoGalaxyProver_<ProverInstances>::prepa
 
         const auto instance_size = static_cast<uint32_t>(instance->instance_size);
         const auto num_public_inputs = static_cast<uint32_t>(instance->public_inputs.size());
-
         transcript.send_to_verifier(domain_separator + "_instance_size", instance_size);
         transcript.send_to_verifier(domain_separator + "_public_input_size", num_public_inputs);
+
         for (size_t i = 0; i < instance->public_inputs.size(); ++i) {
             auto public_input_i = instance->public_inputs[i];
             transcript.send_to_verifier(domain_separator + "_public_input_" + std::to_string(i), public_input_i);
@@ -30,14 +30,13 @@ template <class ProverInstances> void ProtoGalaxyProver_<ProverInstances>::prepa
 
             auto& witness_commitments = instance->witness_commitments;
 
-            // Commit to the first three wire polynomials
+            // Commit to the first three wire polynomials of the instance
             // We only commit to the fourth wire polynomial after adding memory recordss
             witness_commitments.w_l = commitment_key->commit(instance->proving_key->w_l);
             witness_commitments.w_r = commitment_key->commit(instance->proving_key->w_r);
             witness_commitments.w_o = commitment_key->commit(instance->proving_key->w_o);
 
             auto wire_comms = witness_commitments.get_wires();
-
             auto commitment_labels = instance->commitment_labels;
             auto wire_labels = commitment_labels.get_wires();
             for (size_t idx = 0; idx < 3; ++idx) {
@@ -46,6 +45,9 @@ template <class ProverInstances> void ProtoGalaxyProver_<ProverInstances>::prepa
 
             auto eta = transcript.get_challenge(domain_separator + "_eta");
             instance->compute_sorted_accumulator_polynomials(eta);
+
+            // Commit to the sorted withness-table accumulator and the finalized (i.e. with memory records) fourth wire
+            // polynomial
             witness_commitments.sorted_accum = commitment_key->commit(instance->prover_polynomials.sorted_accum);
             witness_commitments.w_4 = commitment_key->commit(instance->prover_polynomials.w_4);
 
@@ -97,12 +99,6 @@ template <class ProverInstances> void ProtoGalaxyProver_<ProverInstances>::prepa
             transcript.send_to_verifier(domain_separator + "_" + labels[idx], (*vk_view[idx]));
         }
     }
-
-    for (auto w : instances[0]->witness_commitments.pointer_view()) {
-        info((*w));
-    }
-    fold_relation_parameters(instances);
-    fold_alpha(instances);
 }
 
 // TODO(#https://github.com/AztecProtocol/barretenberg/issues/689): finalise implementation this function
@@ -123,9 +119,13 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
     }
     assert(perturbator[0] == accumulator->folding_parameters.target_sum);
     auto perturbator_challenge = transcript.get_challenge("perturbator_challenge");
-    instances.betas_star =
+    instances.next_gate_challenges =
         update_gate_challenges(perturbator_challenge, accumulator->folding_parameters.gate_challenges, deltas);
-    const auto pow_betas_star = compute_pow_polynomial_at_values(instances.betas_star, accumulator->instance_size);
+    const auto pow_betas_star =
+        compute_pow_polynomial_at_values(instances.next_gate_challenges, accumulator->instance_size);
+
+    combine_relation_parameters(instances);
+    combine_alpha(instances);
     auto combiner = compute_combiner(instances, pow_betas_star);
 
     auto compressed_perturbator = perturbator.evaluate(perturbator_challenge);
@@ -135,16 +135,14 @@ ProverFoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverI
         transcript.send_to_verifier("combiner_quotient_" + std::to_string(idx), combiner_quotient.value_at(idx));
     }
     auto combiner_challenge = transcript.get_challenge("combiner_quotient_challenge");
-    // TODO(https://github.com/AztecProtocol/barretenberg/issues/764): Generalize these formulas as well as computation
-    // of Lagrange basis
-
-    auto thingy = compute_new_accumulator(instances, combiner_quotient, combiner_challenge, compressed_perturbator);
 
     ProverFoldingResult<Flavor> res;
-    res.accumulator = thingy;
+    res.accumulator =
+        compute_next_accumulator(instances, combiner_quotient, combiner_challenge, compressed_perturbator);
     res.folding_data = transcript.proof_data;
+
     return res;
 }
 template class ProtoGalaxyProver_<ProverInstances_<honk::flavor::Ultra, 2>>;
-// template class ProtoGalaxyProver_<ProverInstances_<honk::flavor::GoblinUltra, 2>>;
+template class ProtoGalaxyProver_<ProverInstances_<honk::flavor::GoblinUltra, 2>>;
 } // namespace proof_system::honk
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index a932ab6a0f6..bdd29b03f83 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -364,13 +364,13 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     }
 
     /**
-     * @brief Create folded (univariate) relation parameters.
+     * @brief Combine each relation parameter, in part, from all the instances into univariates, used in the computation
+     * of combiner.
      * @details For a given relation parameter type, extract that parameter from each instance, place the values in a
      * univariate (i.e., sum them against an appropriate univariate Lagrange basis) and then extended as needed during
      * the constuction of the combiner.
      */
-    // This is not actually folding
-    static void fold_relation_parameters(ProverInstances& instances)
+    static void combine_relation_parameters(ProverInstances& instances)
     {
         // array of parameters to be computed
         size_t param_idx = 0;
@@ -387,14 +387,15 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     }
 
     /**
-     * @brief Create folded univariate for the relation batching parameter (alpha).
+     * @brief Combine the relation batching parameter (named alpha) from each instance into a univariate, used in the
+     * computation of combiner.
      *
      */
     // TODO(https://github.com/AztecProtocol/barretenberg/issues/772): At the moment we have a single α per Instance, we
     // fold them and then we use the unique folded_α for each folded subrelation that is batched in the combiner. This
     // is obviously insecure. We need to generate α_i for each subrelation_i, fold them and then use folded_α_i when
     // batching the i-th folded subrelation in the combiner.
-    static void fold_alpha(ProverInstances& instances)
+    static void combine_alpha(ProverInstances& instances)
     {
         Univariate<FF, ProverInstances::NUM> accumulated_alpha;
         size_t instance_idx = 0;
@@ -405,26 +406,42 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         instances.alpha = accumulated_alpha.template extend_to<ProverInstances::BATCHED_EXTENDED_LENGTH>();
     }
 
-    // BIG TODO fold commitments
-    std::shared_ptr<Instance> compute_new_accumulator(ProverInstances& instances,
-                                                      auto& combiner_quotient,
-                                                      FF challenge,
-                                                      auto compressed_perturbator)
+    /**
+     * @brief Compute the next accumulator (ϕ, ω), send the public data (ϕ) to the verifier and return the accumulator
+     * as part of the result of the folding prover
+     *
+     * @details At this stage, we assume that the instances have the same size and the same number of public parameter.s
+     * @param instances
+     * @param combiner_quotient polynomial K in the paper
+     * @param challenge
+     * @param compressed_perturbator
+     */
+    std::shared_ptr<Instance> compute_next_accumulator(ProverInstances& instances,
+                                                       auto& combiner_quotient,
+                                                       FF challenge,
+                                                       auto compressed_perturbator)
     {
         auto combiner_quotient_at_challenge = combiner_quotient.evaluate(challenge);
+
+        // Given the challenge \gamma, compute Z(\gamma) and {L_0(\gamma),L_1(\gamma)}
+        // TODO(https://github.com/AztecProtocol/barretenberg/issues/764): Generalize the vanishing polynomial formula
+        // and the computation of Lagrange basis for k instances
         auto vanishing_polynomial_at_challenge = challenge * (challenge - FF(1));
         std::vector<FF> lagranges{ FF(1) - challenge, challenge };
 
-        auto next_target_sum =
-            compressed_perturbator * lagranges[0] + vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
         auto next_accumulator = std::make_shared<Instance>();
-        next_accumulator->folding_parameters = { instances.betas_star, next_target_sum };
 
+        // Compute the next target sum and send the next folding parameters to the verifier
+        auto next_target_sum =
+            compressed_perturbator * lagranges[0] + vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
+        next_accumulator->folding_parameters = { instances.next_gate_challenges, next_target_sum };
         transcript.send_to_verifier("next_target_sum", next_target_sum);
-        for (size_t idx = 0; idx < instances.betas_star.size(); idx++) {
-            transcript.send_to_verifier("next_gate_challenge_" + std::to_string(idx), instances.betas_star[idx]);
+        for (size_t idx = 0; idx < instances.next_gate_challenges.size(); idx++) {
+            transcript.send_to_verifier("next_gate_challenge_" + std::to_string(idx),
+                                        instances.next_gate_challenges[idx]);
         }
 
+        // Allocate space, initialised to 0, for the prover polynomials of the next accumulator
         std::array<barretenberg::Polynomial<FF>, Flavor::NUM_ALL_ENTITIES> storage;
         for (auto& polynomial : storage) {
             polynomial = typename Flavor::Polynomial(instances[0]->instance_size);
@@ -432,7 +449,6 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
                 value = FF(0);
             }
         }
-
         ProverPolynomials acc_prover_polynomials;
         size_t poly_idx = 0;
         auto prover_polynomial_pointers = acc_prover_polynomials.pointer_view();
@@ -441,6 +457,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
             poly_idx++;
         }
 
+        // Fold the prover polynomials
         for (size_t inst_idx = 0; inst_idx < ProverInstances::NUM; inst_idx++) {
             for (auto [acc_poly_view, inst_poly_view] : zip_view(
                      acc_prover_polynomials.pointer_view(), instances[inst_idx]->prover_polynomials.pointer_view())) {
@@ -451,39 +468,46 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         }
         next_accumulator->prover_polynomials = acc_prover_polynomials;
 
+        // Allocate space, initialised to 0, for the folded witness commitments
         WitnessCommitments acc_witness_commitments;
         auto acc_comm_view = acc_witness_commitments.pointer_view();
         for (auto c : acc_comm_view) {
             (*c) = Commitment::infinity();
         }
-        for (size_t inst_idx = 0; inst_idx < ProverInstances::NUM; inst_idx++) {
-            auto inst_comm_view = instances[inst_idx]->witness_commitments.pointer_view();
-            for (size_t comm_idx = 0; comm_idx < inst_comm_view.size(); comm_idx++) {
+
+        // Fold the witness commtimenets and send them to the verifier
+        auto witness_labels = next_accumulator->commitment_labels.get_witness();
+        for (size_t comm_idx = 0; comm_idx < acc_comm_view.size(); comm_idx++) {
+            size_t inst_idx = 0;
+            for (auto& instance : instances) {
                 (*acc_comm_view[comm_idx]) =
-                    (*acc_comm_view[comm_idx]) + (*inst_comm_view[comm_idx]) * lagranges[inst_idx];
+                    (*acc_comm_view[comm_idx]) +
+                    (*instance->witness_commitments.pointer_view()[comm_idx]) * lagranges[inst_idx];
+                inst_idx++;
             }
-        }
-        next_accumulator->witness_commitments = acc_witness_commitments;
-        auto witness_labels = next_accumulator->commitment_labels.get_witness();
-        for (size_t idx = 0; idx < witness_labels.size(); idx++) {
-            transcript.send_to_verifier("next_" + witness_labels[idx], (*acc_comm_view[idx]));
-            info((*acc_comm_view[idx]));
+            transcript.send_to_verifier("next_" + witness_labels[comm_idx], (*acc_comm_view[comm_idx]));
         }
 
-        std::vector<FF> folded_public_inputs(instances[0]->public_inputs.size());
-        for (size_t inst_idx = 0; inst_idx < ProverInstances::NUM; inst_idx++) {
-            auto inst_public_inputs = instances[inst_idx]->public_inputs;
-            for (size_t el_idx = 0; el_idx < inst_public_inputs.size(); el_idx++) {
-                folded_public_inputs[el_idx] += inst_public_inputs[el_idx] * lagranges[inst_idx];
+        // Fold the public inputs and send to the verifier
+        next_accumulator->public_inputs = std::vector<FF>(instances[0]->public_inputs.size(), 0);
+        size_t el_idx = 0;
+        for (auto& el : next_accumulator->public_inputs) {
+            size_t inst = 0;
+            for (auto& instance : instances) {
+                el += instance->public_inputs[el_idx] * lagranges[inst];
+                inst++;
             }
-        }
-        for (size_t idx = 0; idx < folded_public_inputs.size(); idx++) {
-            transcript.send_to_verifier("next_public_input_" + std::to_string(idx), folded_public_inputs[idx]);
+            transcript.send_to_verifier("next_public_input_" + std::to_string(el_idx), el);
+            el_idx++;
         }
 
+        // Evaluate the combined batching challenge α univariate at challenge to obtain next α and send it to the
+        // verifier
         next_accumulator->alpha = instances.alpha.evaluate(challenge);
         transcript.send_to_verifier("next_alpha", next_accumulator->alpha);
 
+        // Evaluate each relation parameter univariate at challenge to obtain the folded relation parameters and send to
+        // the verifier
         auto combined_relation_parameters = instances.relation_parameters;
         auto folded_relation_parameters = proof_system::RelationParameters<FF>{
             combined_relation_parameters.eta.evaluate(challenge),
@@ -498,28 +522,28 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         transcript.send_to_verifier("next_public_input_delta", folded_relation_parameters.public_input_delta);
         transcript.send_to_verifier("next_lookup_grand_product_delta",
                                     folded_relation_parameters.lookup_grand_product_delta);
-
         next_accumulator->relation_parameters = folded_relation_parameters;
 
+        // Fold the verification key and send it to the verifier
         auto acc_vk = std::make_shared<VerificationKey>(instances[0]->prover_polynomials.get_polynomial_size(),
                                                         instances[0]->public_inputs.size());
         auto acc_vk_view = acc_vk->pointer_view();
-        for (size_t inst_idx = 0; inst_idx < ProverInstances::NUM; inst_idx++) {
-            auto inst_vk_view = instances[inst_idx]->verification_key->pointer_view();
-            for (size_t idx = 0; idx < inst_vk_view.size(); idx++) {
-                (*acc_vk_view[idx]) = (*acc_vk_view[idx]) + (*inst_vk_view[idx]) * lagranges[inst_idx];
+        auto labels = next_accumulator->commitment_labels.get_precomputed();
+        for (size_t vk_idx = 0; vk_idx < acc_vk_view.size(); vk_idx++) {
+            size_t inst = 0;
+            for (auto& instance : instances) {
+                (*acc_vk_view[vk_idx]) =
+                    (*acc_vk_view[vk_idx]) + (*instance->verification_key->pointer_view()[vk_idx]) * lagranges[inst];
+                inst++;
             }
+            transcript.send_to_verifier("next_" + labels[vk_idx], (*acc_vk_view[vk_idx]));
         }
         next_accumulator->verification_key = acc_vk;
-        auto labels = typename Flavor::CommitmentLabels().get_precomputed();
-        for (size_t idx = 0; idx < labels.size(); idx++) {
-            transcript.send_to_verifier("next_" + labels[idx], (*acc_vk_view[idx]));
-        }
 
         return next_accumulator;
     }
 };
 
 extern template class ProtoGalaxyProver_<ProverInstances_<honk::flavor::Ultra, 2>>;
-// extern template class ProtoGalaxyProver_<ProverInstances_<honk::flavor::GoblinUltra, 2>>;
+extern template class ProtoGalaxyProver_<ProverInstances_<honk::flavor::GoblinUltra, 2>>;
 } // namespace proof_system::honk
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index 012a9d24b29..2bc6c3d28fd 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -138,7 +138,6 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
         verified = verified & (expected_betas_star[idx] == beta_star);
     }
 
-    info("verified: ", verified);
     WitnessCommitments acc_witness_commitments;
     auto acc_comm_view = acc_witness_commitments.pointer_view();
     for (auto c : acc_comm_view) {
@@ -154,9 +153,6 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
     auto witness_labels = typename Flavor::CommitmentLabels().get_witness();
     for (size_t idx = 0; idx < witness_labels.size(); idx++) {
         auto c = transcript.template receive_from_prover<Commitment>("next_" + witness_labels[idx]);
-        info(c);
-        info((*acc_comm_view[idx]));
-        assert(c == (*acc_comm_view[idx]));
         verified = verified & (c == (*acc_comm_view[idx]));
     }
 
@@ -221,5 +217,5 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
 }
 
 template class ProtoGalaxyVerifier_<VerifierInstances_<honk::flavor::Ultra, 2>>;
-// template class ProtoGalaxyVerifier_<VerifierInstances_<honk::flavor::GoblinUltra, 2>>;
+template class ProtoGalaxyVerifier_<VerifierInstances_<honk::flavor::GoblinUltra, 2>>;
 } // namespace proof_system::honk
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
index 0833256e8bc..8dbb372962a 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
@@ -60,7 +60,8 @@ template <class VerifierInstances> class ProtoGalaxyVerifier_ {
     void prepare_for_folding(std::vector<uint8_t> fold_data);
 
     /**
-     * @brief Run the folding protocol on the verifier side.
+     * @brief Run the folding protocol on the verifier side to verify the public data ϕ of the new accumulator, received
+     * from the prover, is correct.
      *
      * TODO(https://github.com/AztecProtocol/barretenberg/issues/690): finalise the implementation of this function
      */
@@ -68,5 +69,5 @@ template <class VerifierInstances> class ProtoGalaxyVerifier_ {
 };
 
 extern template class ProtoGalaxyVerifier_<VerifierInstances_<honk::flavor::Ultra, 2>>;
-// extern template class ProtoGalaxyVerifier_<VerifierInstances_<honk::flavor::GoblinUltra, 2>>;
+extern template class ProtoGalaxyVerifier_<VerifierInstances_<honk::flavor::GoblinUltra, 2>>;
 } // namespace proof_system::honk
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
index 4e4e9e0334e..f9d1a217b71 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
@@ -23,7 +23,7 @@ template <typename Flavor_, size_t NUM_> struct ProverInstances_ {
     ArrayType _data;
     RelationParameters relation_parameters;
     AlphaType alpha;
-    std::vector<FF> betas_star;
+    std::vector<FF> next_gate_challenges;
 
     std::shared_ptr<Instance> const& operator[](size_t idx) const { return _data[idx]; }
     typename ArrayType::iterator begin() { return _data.begin(); };
@@ -100,15 +100,7 @@ template <typename Flavor_, size_t NUM_> struct VerifierInstances_ {
     std::shared_ptr<Instance> const& operator[](size_t idx) const { return _data[idx]; }
     typename ArrayType::iterator begin() { return _data.begin(); };
     typename ArrayType::iterator end() { return _data.end(); };
-    // VerifierInstances_(std::vector<std::shared_ptr<VerificationKey>> vks)
-    // {
-    //     ASSERT(vks.size() == NUM);
-    //     for (size_t idx = 0; idx < vks.size(); idx++) {
-    //         Instance inst;
-    //         inst.verification_key = std::move(vks[idx]);
-    //         _data[idx] = std::make_unique<Instance>(inst);
-    //     }
-    // };
+
     VerifierInstances_()
     {
         for (size_t idx = 0; idx < NUM; idx++) {
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.cpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.cpp
index 41924b71875..9dfaa5baff3 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.cpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.cpp
@@ -351,6 +351,9 @@ template <class Flavor> void ProverInstance_<Flavor>::initialize_prover_polynomi
         size_t idx = i + pub_inputs_offset;
         public_inputs.emplace_back(public_wires_source[idx]);
     }
+
+    instance_size = proving_key->circuit_size;
+    log_instance_size = static_cast<size_t>(numeric::get_msb(instance_size));
 }
 
 template <class Flavor> void ProverInstance_<Flavor>::compute_sorted_accumulator_polynomials(FF eta)
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
index 477a9da49b9..b347055885d 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
@@ -60,16 +60,8 @@ template <class Flavor> class ProverInstance_ {
         compute_circuit_size_parameters(circuit);
         compute_proving_key(circuit);
         compute_witness(circuit);
-        instance_size = proving_key->circuit_size;
-        log_instance_size = static_cast<size_t>(numeric::get_msb(instance_size));
     }
 
-    // ProverInstance_(FoldingResult<Flavor> result)
-    //     : verification_key(std::move(result.verification_key))
-    //     , prover_polynomials(result.folded_prover_polynomials)
-    //     , public_inputs(result.folded_public_inputs)
-    //     , folding_parameters(result.folding_parameters){};
-
     ProverInstance_() = default;
     ~ProverInstance_() = default;
 
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
index 72d5570493e..213d1b648e5 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
@@ -247,7 +247,7 @@ TEST_F(ProtoGalaxyTests, FoldChallenges)
     instance2->relation_parameters.eta = 3;
 
     Instances instances{ { instance1, instance2 } };
-    ProtoGalaxyProver::fold_relation_parameters(instances);
+    ProtoGalaxyProver::combine_relation_parameters(instances);
 
     Univariate<FF, 12> expected_eta{ { 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 } };
     EXPECT_EQ(instances.relation_parameters.eta, expected_eta);
@@ -268,7 +268,7 @@ TEST_F(ProtoGalaxyTests, FoldAlpha)
     instance2->alpha = 4;
 
     Instances instances{ { instance1, instance2 } };
-    ProtoGalaxyProver::fold_alpha(instances);
+    ProtoGalaxyProver::combine_alpha(instances);
 
     Univariate<FF, 13> expected_alpha{ { 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26 } };
     EXPECT_EQ(instances.alpha, expected_alpha);

From 5c29867e8b6b6ef0a788780035238215aec18eea Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Fri, 1 Dec 2023 13:05:10 +0000
Subject: [PATCH 40/86] more refactoring

---
 .../protogalaxy/protogalaxy_verifier.cpp      | 98 ++++++++++---------
 .../protogalaxy/protogalaxy_verifier.hpp      |  5 +-
 2 files changed, 55 insertions(+), 48 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index 2bc6c3d28fd..16928326905 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -126,6 +126,7 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
     auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
     auto lagranges = std::vector<FF>{ FF(1) - combiner_challenge, combiner_challenge };
 
+    // Compute next folding parameters and verify against the ones received from the prover
     auto expected_next_target_sum =
         perturbator_at_challenge * lagranges[0] + vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
     auto next_target_sum = transcript.template receive_from_prover<FF>("next_target_sum");
@@ -138,79 +139,82 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
         verified = verified & (expected_betas_star[idx] == beta_star);
     }
 
+    // Compute ϕ and verify against the data received from the prover
+
     WitnessCommitments acc_witness_commitments;
-    auto acc_comm_view = acc_witness_commitments.pointer_view();
-    for (auto c : acc_comm_view) {
-        (*c) = Commitment::infinity();
-    }
-    for (size_t inst_idx = 0; inst_idx < VerifierInstances::NUM; inst_idx++) {
-        auto inst_comm_view = instances[inst_idx]->witness_commitments.pointer_view();
-        for (size_t comm_idx = 0; comm_idx < inst_comm_view.size(); comm_idx++) {
-            (*acc_comm_view[comm_idx]) = (*acc_comm_view[comm_idx]) + (*inst_comm_view[comm_idx]) * lagranges[inst_idx];
+    auto witness_labels = commitment_labels.get_witness();
+    size_t comm_idx = 0;
+    for (auto expected_comm : acc_witness_commitments.pointer_view()) {
+        (*expected_comm) = Commitment::infinity();
+        size_t inst = 0;
+        for (auto& instance : instances) {
+            (*expected_comm) =
+                (*expected_comm) + (*instance->witness_commitments.pointer_view()[comm_idx]) * lagranges[inst];
+            inst++;
         }
+        auto comm = transcript.template receive_from_prover<Commitment>("next_" + witness_labels[comm_idx]);
+        verified = verified & (comm == (*expected_comm));
+        comm_idx++;
     }
 
-    auto witness_labels = typename Flavor::CommitmentLabels().get_witness();
-    for (size_t idx = 0; idx < witness_labels.size(); idx++) {
-        auto c = transcript.template receive_from_prover<Commitment>("next_" + witness_labels[idx]);
-        verified = verified & (c == (*acc_comm_view[idx]));
+    std::vector<FF> folded_public_inputs(instances[0]->public_inputs.size(), 0);
+    size_t el_idx = 0;
+    for (auto expected_el : folded_public_inputs) {
+        size_t inst = 0;
+        for (auto& instance : instances) {
+            expected_el += instance->public_inputs[el_idx] * lagranges[inst];
+            inst++;
+        }
+        auto el = transcript.template receive_from_prover<FF>("next_public_input" + std::to_string(el_idx));
+        verified = verified & (el == expected_el);
+        el_idx++;
     }
 
-    std::vector<FF> folded_public_inputs(instances[0]->public_inputs.size());
-    auto folded_alpha = FF(0);
-    auto folded_parameters = proof_system::RelationParameters<FF>{};
+    auto expected_alpha = FF(0);
+    auto expected_parameters = proof_system::RelationParameters<FF>{};
     for (size_t inst_idx = 0; inst_idx < VerifierInstances::NUM; inst_idx++) {
         auto instance = instances[inst_idx];
-        auto inst_public_inputs = instance->public_inputs;
-        for (size_t el_idx = 0; el_idx < inst_public_inputs.size(); el_idx++) {
-            folded_public_inputs[el_idx] += inst_public_inputs[el_idx] * lagranges[inst_idx];
-        }
-        folded_alpha += instance->alpha * lagranges[inst_idx];
-        folded_parameters.eta += instance->relation_parameters.eta * lagranges[inst_idx];
-        folded_parameters.beta += instance->relation_parameters.beta * lagranges[inst_idx];
-        folded_parameters.gamma += instance->relation_parameters.gamma * lagranges[inst_idx];
-        folded_parameters.public_input_delta += instance->relation_parameters.public_input_delta * lagranges[inst_idx];
-        folded_parameters.lookup_grand_product_delta +=
+        expected_alpha += instance->alpha * lagranges[inst_idx];
+        expected_parameters.eta += instance->relation_parameters.eta * lagranges[inst_idx];
+        expected_parameters.beta += instance->relation_parameters.beta * lagranges[inst_idx];
+        expected_parameters.gamma += instance->relation_parameters.gamma * lagranges[inst_idx];
+        expected_parameters.public_input_delta +=
+            instance->relation_parameters.public_input_delta * lagranges[inst_idx];
+        expected_parameters.lookup_grand_product_delta +=
             instance->relation_parameters.lookup_grand_product_delta * lagranges[inst_idx];
     }
 
-    for (size_t idx = 0; idx < folded_public_inputs.size(); idx++) {
-        auto public_input = transcript.template receive_from_prover<FF>("next_public_input" + std::to_string(idx));
-        verified = verified & (public_input == folded_public_inputs[idx]);
-    }
-
     auto next_alpha = transcript.template receive_from_prover<FF>("next_alpha");
-    verified = verified & (next_alpha == folded_alpha);
+    verified = verified & (next_alpha == expected_alpha);
 
     auto next_eta = transcript.template receive_from_prover<FF>("next_eta");
-    verified = verified & (next_eta == folded_parameters.eta);
+    verified = verified & (next_eta == expected_parameters.eta);
 
     auto next_beta = transcript.template receive_from_prover<FF>("next_beta");
-    verified = verified & (next_beta == folded_parameters.beta);
+    verified = verified & (next_beta == expected_parameters.beta);
 
     auto next_gamma = transcript.template receive_from_prover<FF>("next_gamma");
-    verified = verified & (next_gamma == folded_parameters.gamma);
+    verified = verified & (next_gamma == expected_parameters.gamma);
 
     auto next_public_input_delta = transcript.template receive_from_prover<FF>("next_public_input_delta");
-    verified = verified & (next_public_input_delta == folded_parameters.public_input_delta);
+    verified = verified & (next_public_input_delta == expected_parameters.public_input_delta);
 
     auto next_lookup_grand_product_delta =
         transcript.template receive_from_prover<FF>("next_lookup_grand_product_delta");
-    verified = verified & (next_lookup_grand_product_delta == folded_parameters.lookup_grand_product_delta);
+    verified = verified & (next_lookup_grand_product_delta == expected_parameters.lookup_grand_product_delta);
 
     auto acc_vk = std::make_shared<VerificationKey>(instances[0]->instance_size, instances[0]->public_input_size);
-    auto acc_vk_view = acc_vk->pointer_view();
-    for (size_t inst_idx = 0; inst_idx < VerifierInstances::NUM; inst_idx++) {
-        auto inst_vk_view = instances[inst_idx]->verification_key->pointer_view();
-        for (size_t idx = 0; idx < inst_vk_view.size(); idx++) {
-            (*acc_vk_view[idx]) = (*acc_vk_view[idx]) + (*inst_vk_view[idx]) * lagranges[inst_idx];
+    auto vk_labels = commitment_labels.get_precomputed();
+    size_t vk_idx = 0;
+    for (auto expected_vk : acc_vk->pointer_view()) {
+        size_t inst = 0;
+        for (auto& instance : instances) {
+            (*expected_vk) = (*expected_vk) + (*instance->verification_key->pointer_view()[vk_idx]) * lagranges[inst];
+            inst++;
         }
-    }
-
-    auto labels = typename Flavor::CommitmentLabels().get_precomputed();
-    for (size_t idx = 0; idx < labels.size(); idx++) {
-        auto vk = transcript.template receive_from_prover<Commitment>("next" + labels[idx]);
-        verified = verified & (vk == (*acc_vk_view[idx]));
+        auto vk = transcript.template receive_from_prover<Commitment>("next_" + vk_labels[vk_idx]);
+        verified = verified & (vk == (*expected_vk));
+        vk_idx++;
     }
 
     return verified;
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
index 8dbb372962a..6cc149a87da 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
@@ -15,15 +15,18 @@ template <class VerifierInstances> class ProtoGalaxyVerifier_ {
     using Instance = typename VerifierInstances::Instance;
     using VerificationKey = typename Flavor::VerificationKey;
     using WitnessCommitments = typename Flavor::WitnessCommitments;
+    using CommitmentLabels = typename Flavor::CommitmentLabels;
 
     VerifierInstances instances;
     BaseTranscript<FF> transcript;
 
+    CommitmentLabels commitment_labels;
+
     ProtoGalaxyVerifier_(VerifierInstances insts)
         : instances(insts){};
     ~ProtoGalaxyVerifier_() = default;
     /**
-     * @brief For a new round challenge δ at each iteration of the ProtoGalaxy protocol, compute the vector
+     * @brief Given a new round challenge δ for each iteration of the full ProtoGalaxy protocol, compute the vector
      * [δ, δ^2,..., δ^t] where t = logn and n is the size of the instance.
      */
     static std::vector<FF> compute_round_challenge_pows(size_t log_instance_size, FF round_challenge)

From c5c7c614ee072dbcd4413dd2f008f7a6d648628e Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Mon, 4 Dec 2023 10:32:44 +0000
Subject: [PATCH 41/86] cleanup

---
 .../protogalaxy/protogalaxy_prover.cpp        | 211 ++++++++++--------
 .../protogalaxy/protogalaxy_prover.hpp        |   4 +
 .../protogalaxy/protogalaxy_verifier.cpp      | 192 +++++++++-------
 .../protogalaxy/protogalaxy_verifier.hpp      |   8 +-
 4 files changed, 241 insertions(+), 174 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index d2ca5f4af45..e4a19bdd6cf 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -1,103 +1,132 @@
 #include "protogalaxy_prover.hpp"
 #include "barretenberg/flavor/flavor.hpp"
 namespace proof_system::honk {
+template <class ProverInstances>
+void ProtoGalaxyProver_<ProverInstances>::finalise_and_send_instance(std::shared_ptr<Instance> instance,
+                                                                     std::string domain_separator)
+{
+    instance->initialize_prover_polynomials();
+
+    const auto instance_size = static_cast<uint32_t>(instance->instance_size);
+    const auto num_public_inputs = static_cast<uint32_t>(instance->public_inputs.size());
+    transcript.send_to_verifier(domain_separator + "_instance_size", instance_size);
+    transcript.send_to_verifier(domain_separator + "_public_input_size", num_public_inputs);
+
+    for (size_t i = 0; i < instance->public_inputs.size(); ++i) {
+        auto public_input_i = instance->public_inputs[i];
+        transcript.send_to_verifier(domain_separator + "_public_input_" + std::to_string(i), public_input_i);
+    }
+    transcript.send_to_verifier(domain_separator + "_pub_inputs_offset",
+                                static_cast<uint32_t>(instance->pub_inputs_offset));
+
+    auto& witness_commitments = instance->witness_commitments;
+
+    // Commit to the first three wire polynomials of the instance
+    // We only commit to the fourth wire polynomial after adding memory recordss
+    witness_commitments.w_l = commitment_key->commit(instance->proving_key->w_l);
+    witness_commitments.w_r = commitment_key->commit(instance->proving_key->w_r);
+    witness_commitments.w_o = commitment_key->commit(instance->proving_key->w_o);
+
+    auto wire_comms = witness_commitments.get_wires();
+    auto commitment_labels = instance->commitment_labels;
+    auto wire_labels = commitment_labels.get_wires();
+    for (size_t idx = 0; idx < 3; ++idx) {
+        transcript.send_to_verifier(domain_separator + "_" + wire_labels[idx], wire_comms[idx]);
+    }
+
+    auto eta = transcript.get_challenge(domain_separator + "_eta");
+    instance->compute_sorted_accumulator_polynomials(eta);
+
+    // Commit to the sorted withness-table accumulator and the finalized (i.e. with memory records) fourth wire
+    // polynomial
+    witness_commitments.sorted_accum = commitment_key->commit(instance->prover_polynomials.sorted_accum);
+    witness_commitments.w_4 = commitment_key->commit(instance->prover_polynomials.w_4);
+
+    transcript.send_to_verifier(domain_separator + "_" + commitment_labels.sorted_accum,
+                                witness_commitments.sorted_accum);
+    transcript.send_to_verifier(domain_separator + "_" + commitment_labels.w_4, witness_commitments.w_4);
+
+    auto [beta, gamma] = transcript.get_challenges(domain_separator + "_beta", domain_separator + "_gamma");
+    instance->compute_grand_product_polynomials(beta, gamma);
+
+    witness_commitments.z_perm = commitment_key->commit(instance->prover_polynomials.z_perm);
+    witness_commitments.z_lookup = commitment_key->commit(instance->prover_polynomials.z_lookup);
+
+    transcript.send_to_verifier(domain_separator + "_" + commitment_labels.z_perm,
+                                instance->witness_commitments.z_perm);
+    transcript.send_to_verifier(domain_separator + "_" + commitment_labels.z_lookup,
+                                instance->witness_commitments.z_lookup);
+
+    instance->alpha = transcript.get_challenge(domain_separator + "_alpha");
+
+    auto vk_view = instance->verification_key->pointer_view();
+    auto labels = instance->commitment_labels.get_precomputed();
+    for (size_t idx = 0; idx < labels.size(); idx++) {
+        transcript.send_to_verifier(domain_separator + "_" + labels[idx], (*vk_view[idx]));
+    }
+}
+
+template <class ProverInstances>
+void ProtoGalaxyProver_<ProverInstances>::send_accumulator(std::shared_ptr<Instance> instance,
+                                                           std::string domain_separator)
+{
+    const auto instance_size = static_cast<uint32_t>(instance->instance_size);
+    const auto num_public_inputs = static_cast<uint32_t>(instance->public_inputs.size());
+    transcript.send_to_verifier(domain_separator + "_instance_size", instance_size);
+    transcript.send_to_verifier(domain_separator + "_public_input_size", num_public_inputs);
+
+    for (size_t i = 0; i < instance->public_inputs.size(); ++i) {
+        auto public_input_i = instance->public_inputs[i];
+        transcript.send_to_verifier(domain_separator + "_public_input_" + std::to_string(i), public_input_i);
+    }
+
+    transcript.send_to_verifier(domain_separator + "_eta", instance->relation_parameters.eta);
+    transcript.send_to_verifier(domain_separator + "_beta", instance->relation_parameters.beta);
+    transcript.send_to_verifier(domain_separator + "_gamma", instance->relation_parameters.gamma);
+    transcript.send_to_verifier(domain_separator + "_public_input_delta",
+                                instance->relation_parameters.public_input_delta);
+    transcript.send_to_verifier(domain_separator + "_lookup_grand_product_delta",
+                                instance->relation_parameters.lookup_grand_product_delta);
+
+    transcript.send_to_verifier(domain_separator + "_alpha", instance->alpha);
+
+    auto folding_parameters = instance->folding_parameters;
+    transcript.send_to_verifier(domain_separator + "_target_sum", folding_parameters.target_sum);
+    for (size_t idx = 0; idx < folding_parameters.gate_challenges.size(); idx++) {
+        transcript.send_to_verifier(domain_separator + "_gate_challenge_" + std::to_string(idx),
+                                    folding_parameters.gate_challenges[idx]);
+    }
+
+    auto comm_view = instance->witness_commitments.pointer_view();
+    auto witness_labels = instance->commitment_labels.get_witness();
+    for (size_t idx = 0; idx < witness_labels.size(); idx++) {
+        transcript.send_to_verifier(domain_separator + "_" + witness_labels[idx], (*comm_view[idx]));
+    }
+
+    auto vk_view = instance->verification_key->pointer_view();
+    auto vk_labels = instance->commitment_labels.get_precomputed();
+    for (size_t idx = 0; idx < vk_labels.size(); idx++) {
+        transcript.send_to_verifier(domain_separator + "_" + vk_labels[idx], (*vk_view[idx]));
+    }
+}
 
 template <class ProverInstances> void ProtoGalaxyProver_<ProverInstances>::prepare_for_folding()
 {
     auto idx = 0;
-    for (auto it = instances.begin(); it != instances.end(); it++, idx++) {
+    auto instance = instances[0];
+    auto domain_separator = std::to_string(idx);
+    transcript.send_to_verifier(domain_separator + "is_accumulator", instance->is_accumulator);
+    if (instance->is_accumulator) {
+        send_accumulator(instance, domain_separator);
+    } else {
+        finalise_and_send_instance(instance, domain_separator);
+    }
+    idx++;
+
+    for (auto it = instances.begin() + 1; it != instances.end(); it++, idx++) {
         auto instance = *it;
         auto domain_separator = std::to_string(idx);
-        transcript.send_to_verifier(domain_separator + "is_accumulator", instance->is_accumulator);
-
-        if (!static_cast<bool>(instance->is_accumulator)) {
-            instance->initialize_prover_polynomials();
-        }
-
-        const auto instance_size = static_cast<uint32_t>(instance->instance_size);
-        const auto num_public_inputs = static_cast<uint32_t>(instance->public_inputs.size());
-        transcript.send_to_verifier(domain_separator + "_instance_size", instance_size);
-        transcript.send_to_verifier(domain_separator + "_public_input_size", num_public_inputs);
-
-        for (size_t i = 0; i < instance->public_inputs.size(); ++i) {
-            auto public_input_i = instance->public_inputs[i];
-            transcript.send_to_verifier(domain_separator + "_public_input_" + std::to_string(i), public_input_i);
-        }
-
-        if (!static_cast<bool>(instance->is_accumulator)) {
-            transcript.send_to_verifier(domain_separator + "_pub_inputs_offset",
-                                        static_cast<uint32_t>(instance->pub_inputs_offset));
-
-            auto& witness_commitments = instance->witness_commitments;
-
-            // Commit to the first three wire polynomials of the instance
-            // We only commit to the fourth wire polynomial after adding memory recordss
-            witness_commitments.w_l = commitment_key->commit(instance->proving_key->w_l);
-            witness_commitments.w_r = commitment_key->commit(instance->proving_key->w_r);
-            witness_commitments.w_o = commitment_key->commit(instance->proving_key->w_o);
-
-            auto wire_comms = witness_commitments.get_wires();
-            auto commitment_labels = instance->commitment_labels;
-            auto wire_labels = commitment_labels.get_wires();
-            for (size_t idx = 0; idx < 3; ++idx) {
-                transcript.send_to_verifier(domain_separator + "_" + wire_labels[idx], wire_comms[idx]);
-            }
-
-            auto eta = transcript.get_challenge(domain_separator + "_eta");
-            instance->compute_sorted_accumulator_polynomials(eta);
-
-            // Commit to the sorted withness-table accumulator and the finalized (i.e. with memory records) fourth wire
-            // polynomial
-            witness_commitments.sorted_accum = commitment_key->commit(instance->prover_polynomials.sorted_accum);
-            witness_commitments.w_4 = commitment_key->commit(instance->prover_polynomials.w_4);
-
-            transcript.send_to_verifier(domain_separator + "_" + commitment_labels.sorted_accum,
-                                        witness_commitments.sorted_accum);
-            transcript.send_to_verifier(domain_separator + "_" + commitment_labels.w_4, witness_commitments.w_4);
-
-            auto [beta, gamma] = transcript.get_challenges(domain_separator + "_beta", domain_separator + "_gamma");
-            instance->compute_grand_product_polynomials(beta, gamma);
-
-            witness_commitments.z_perm = commitment_key->commit(instance->prover_polynomials.z_perm);
-            witness_commitments.z_lookup = commitment_key->commit(instance->prover_polynomials.z_lookup);
-
-            transcript.send_to_verifier(domain_separator + "_" + commitment_labels.z_perm,
-                                        instance->witness_commitments.z_perm);
-            transcript.send_to_verifier(domain_separator + "_" + commitment_labels.z_lookup,
-                                        instance->witness_commitments.z_lookup);
-
-            instance->alpha = transcript.get_challenge(domain_separator + "_alpha");
-
-        } else {
-            transcript.send_to_verifier(domain_separator + "_eta", instance->relation_parameters.eta);
-            transcript.send_to_verifier(domain_separator + "_beta", instance->relation_parameters.beta);
-            transcript.send_to_verifier(domain_separator + "_gamma", instance->relation_parameters.gamma);
-            transcript.send_to_verifier(domain_separator + "_public_input_delta",
-                                        instance->relation_parameters.public_input_delta);
-            transcript.send_to_verifier(domain_separator + "_lookup_grand_product_delta",
-                                        instance->relation_parameters.lookup_grand_product_delta);
-
-            transcript.send_to_verifier(domain_separator + "_alpha", instance->alpha);
-
-            auto folding_parameters = instance->folding_parameters;
-            transcript.send_to_verifier(domain_separator + "_target_sum", folding_parameters.target_sum);
-            for (size_t idx = 0; idx < folding_parameters.gate_challenges.size(); idx++) {
-                transcript.send_to_verifier(domain_separator + "_gate_challenge_" + std::to_string(idx),
-                                            folding_parameters.gate_challenges[idx]);
-            }
-
-            auto comm_view = instance->witness_commitments.pointer_view();
-            auto labels = instance->commitment_labels.get_witness();
-            for (size_t idx = 0; idx < labels.size(); idx++) {
-                transcript.send_to_verifier(domain_separator + "_" + labels[idx], (*comm_view[idx]));
-            }
-        }
-
-        auto vk_view = instance->verification_key->pointer_view();
-        auto labels = instance->commitment_labels.get_precomputed();
-        for (size_t idx = 0; idx < labels.size(); idx++) {
-            transcript.send_to_verifier(domain_separator + "_" + labels[idx], (*vk_view[idx]));
-        }
+        finalise_and_send_instance(instance, domain_separator);
     }
 }
 
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index bdd29b03f83..9d53b087879 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -63,6 +63,10 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      */
     void prepare_for_folding();
 
+    void send_accumulator(std::shared_ptr<Instance>, std::string);
+
+    void finalise_and_send_instance(std::shared_ptr<Instance>, std::string);
+
     /**
      * @brief Given a vector \vec{\beta} of values, compute the pow polynomial on these values as defined in the paper.
      */
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index 16928326905..f5d218e3c79 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -2,94 +2,124 @@
 #include "barretenberg/proof_system/library/grand_product_delta.hpp"
 namespace proof_system::honk {
 
+template <class VerifierInstances>
+void ProtoGalaxyVerifier_<VerifierInstances>::receive_accumulator(std::shared_ptr<Instance> inst,
+                                                                  std::string domain_separator)
+{
+    inst->instance_size = transcript.template receive_from_prover<uint32_t>(domain_separator + "_instance_size");
+    inst->log_instance_size = static_cast<size_t>(numeric::get_msb(inst->instance_size));
+    inst->public_input_size =
+        transcript.template receive_from_prover<uint32_t>(domain_separator + "_public_input_size");
+
+    for (size_t i = 0; i < inst->public_input_size; ++i) {
+        auto public_input_i =
+            transcript.template receive_from_prover<FF>(domain_separator + "_public_input_" + std::to_string(i));
+        inst->public_inputs.emplace_back(public_input_i);
+    }
+
+    auto eta = transcript.template receive_from_prover<FF>(domain_separator + "_eta");
+    auto beta = transcript.template receive_from_prover<FF>(domain_separator + "_beta");
+    auto gamma = transcript.template receive_from_prover<FF>(domain_separator + "_gamma");
+    auto public_input_delta = transcript.template receive_from_prover<FF>(domain_separator + "_public_input_delta");
+    auto lookup_grand_product_delta =
+        transcript.template receive_from_prover<FF>(domain_separator + "_lookup_grand_product_delta");
+    inst->relation_parameters =
+        RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
+    inst->alpha = transcript.template receive_from_prover<FF>(domain_separator + "_alpha");
+
+    inst->folding_parameters.target_sum = transcript.template receive_from_prover<FF>(domain_separator + "_target_sum");
+
+    inst->folding_parameters.gate_challenges = std::vector<FF>(inst->log_instance_size);
+    for (size_t idx = 0; idx < inst->log_instance_size; idx++) {
+        inst->folding_parameters.gate_challenges[idx] =
+            transcript.template receive_from_prover<FF>(domain_separator + "_gate_challenge_" + std::to_string(idx));
+    }
+    auto comm_view = inst->witness_commitments.pointer_view();
+    auto witness_labels = inst->commitment_labels.get_witness();
+    for (size_t idx = 0; idx < witness_labels.size(); idx++) {
+        (*comm_view[idx]) =
+            transcript.template receive_from_prover<Commitment>(domain_separator + "_" + witness_labels[idx]);
+    }
+
+    inst->verification_key = std::make_shared<VerificationKey>(inst->instance_size, inst->public_input_size);
+    auto vk_view = inst->verification_key->pointer_view();
+    auto vk_labels = inst->commitment_labels.get_precomputed();
+    for (size_t idx = 0; idx < vk_labels.size(); idx++) {
+        (*vk_view[idx]) = transcript.template receive_from_prover<Commitment>(domain_separator + "_" + vk_labels[idx]);
+    }
+}
+
+template <class VerifierInstances>
+void ProtoGalaxyVerifier_<VerifierInstances>::receive_and_finalise_instance(std::shared_ptr<Instance> inst,
+                                                                            std::string domain_separator)
+{
+    inst->instance_size = transcript.template receive_from_prover<uint32_t>(domain_separator + "_instance_size");
+    inst->log_instance_size = static_cast<size_t>(numeric::get_msb(inst->instance_size));
+    inst->public_input_size =
+        transcript.template receive_from_prover<uint32_t>(domain_separator + "_public_input_size");
+
+    for (size_t i = 0; i < inst->public_input_size; ++i) {
+        auto public_input_i =
+            transcript.template receive_from_prover<FF>(domain_separator + "_public_input_" + std::to_string(i));
+        inst->public_inputs.emplace_back(public_input_i);
+    }
+
+    inst->pub_inputs_offset =
+        transcript.template receive_from_prover<uint32_t>(domain_separator + "_pub_inputs_offset");
+
+    auto labels = inst->commitment_labels;
+    auto& witness_commitments = inst->witness_commitments;
+    witness_commitments.w_l = transcript.template receive_from_prover<Commitment>(domain_separator + "_" + labels.w_l);
+    witness_commitments.w_r = transcript.template receive_from_prover<Commitment>(domain_separator + "_" + labels.w_r);
+    witness_commitments.w_o = transcript.template receive_from_prover<Commitment>(domain_separator + "_" + labels.w_o);
+
+    auto eta = transcript.get_challenge(domain_separator + "_eta");
+    witness_commitments.sorted_accum =
+        transcript.template receive_from_prover<Commitment>(domain_separator + "_" + labels.sorted_accum);
+    witness_commitments.w_4 = transcript.template receive_from_prover<Commitment>(domain_separator + "_" + labels.w_4);
+
+    auto [beta, gamma] = transcript.get_challenges(domain_separator + "_beta", domain_separator + "_gamma");
+    witness_commitments.z_perm =
+        transcript.template receive_from_prover<Commitment>(domain_separator + "_" + labels.z_perm);
+    witness_commitments.z_lookup =
+        transcript.template receive_from_prover<Commitment>(domain_separator + "_" + labels.z_lookup);
+
+    const FF public_input_delta = compute_public_input_delta<Flavor>(
+        inst->public_inputs, beta, gamma, inst->instance_size, inst->pub_inputs_offset);
+    const FF lookup_grand_product_delta = compute_lookup_grand_product_delta<FF>(beta, gamma, inst->instance_size);
+    inst->relation_parameters =
+        RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
+
+    inst->alpha = transcript.get_challenge(domain_separator + "_alpha");
+
+    inst->verification_key = std::make_shared<VerificationKey>(inst->instance_size, inst->public_input_size);
+    auto vk_view = inst->verification_key->pointer_view();
+    auto vk_labels = labels.get_precomputed();
+    for (size_t idx = 0; idx < vk_labels.size(); idx++) {
+        (*vk_view[idx]) = transcript.template receive_from_prover<Commitment>(domain_separator + "_" + vk_labels[idx]);
+    }
+}
+
+// TODO(#795)
 template <class VerifierInstances>
 void ProtoGalaxyVerifier_<VerifierInstances>::prepare_for_folding(std::vector<uint8_t> fold_data)
 {
     transcript = BaseTranscript<FF>{ fold_data };
     auto index = 0;
-    for (auto it = instances.begin(); it != instances.end(); it++, index++) {
+    auto inst = instances[0];
+    auto domain_separator = std::to_string(index);
+    inst->is_accumulator = transcript.template receive_from_prover<uint32_t>(domain_separator + "is_accumulator");
+    if (inst->is_accumulator) {
+        receive_accumulator(inst, domain_separator);
+    } else {
+        receive_and_finalise_instance(inst, domain_separator);
+    }
+    index++;
+
+    for (auto it = instances.begin() + 1; it != instances.end(); it++, index++) {
         auto inst = *it;
         auto domain_separator = std::to_string(index);
-        inst->is_accumulator = transcript.template receive_from_prover<uint32_t>(domain_separator + "is_accumulator");
-
-        inst->instance_size = transcript.template receive_from_prover<uint32_t>(domain_separator + "_instance_size");
-        inst->log_instance_size = static_cast<size_t>(numeric::get_msb(inst->instance_size));
-        inst->public_input_size =
-            transcript.template receive_from_prover<uint32_t>(domain_separator + "_public_input_size");
-
-        for (size_t i = 0; i < inst->public_input_size; ++i) {
-            auto public_input_i =
-                transcript.template receive_from_prover<FF>(domain_separator + "_public_input_" + std::to_string(i));
-            inst->public_inputs.emplace_back(public_input_i);
-        }
-
-        if (static_cast<bool>(inst->is_accumulator)) {
-            auto eta = transcript.template receive_from_prover<FF>(domain_separator + "_eta");
-            auto beta = transcript.template receive_from_prover<FF>(domain_separator + "_beta");
-            auto gamma = transcript.template receive_from_prover<FF>(domain_separator + "_gamma");
-            auto public_input_delta =
-                transcript.template receive_from_prover<FF>(domain_separator + "_public_input_delta");
-            auto lookup_grand_product_delta =
-                transcript.template receive_from_prover<FF>(domain_separator + "_lookup_grand_product_delta");
-            inst->relation_parameters =
-                RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
-            inst->alpha = transcript.template receive_from_prover<FF>(domain_separator + "_alpha");
-
-            inst->folding_parameters.target_sum =
-                transcript.template receive_from_prover<FF>(domain_separator + "_target_sum");
-
-            inst->folding_parameters.gate_challenges = std::vector<FF>(inst->log_instance_size);
-            for (size_t idx = 0; idx < inst->log_instance_size; idx++) {
-                inst->folding_parameters.gate_challenges[idx] = transcript.template receive_from_prover<FF>(
-                    domain_separator + "_gate_challenge_" + std::to_string(idx));
-            }
-            auto comm_view = inst->witness_commitments.pointer_view();
-            auto labels = inst->commitment_labels.get_witness();
-            for (size_t idx = 0; idx < labels.size(); idx++) {
-                (*comm_view[idx]) =
-                    transcript.template receive_from_prover<Commitment>(domain_separator + "_" + labels[idx]);
-            }
-        } else {
-            inst->pub_inputs_offset =
-                transcript.template receive_from_prover<uint32_t>(domain_separator + "_pub_inputs_offset");
-
-            auto labels = inst->commitment_labels;
-            auto& witness_commitments = inst->witness_commitments;
-            witness_commitments.w_l =
-                transcript.template receive_from_prover<Commitment>(domain_separator + "_" + labels.w_l);
-            witness_commitments.w_r =
-                transcript.template receive_from_prover<Commitment>(domain_separator + "_" + labels.w_r);
-            witness_commitments.w_o =
-                transcript.template receive_from_prover<Commitment>(domain_separator + "_" + labels.w_o);
-
-            auto eta = transcript.get_challenge(domain_separator + "_eta");
-            witness_commitments.sorted_accum =
-                transcript.template receive_from_prover<Commitment>(domain_separator + "_" + labels.sorted_accum);
-            witness_commitments.w_4 =
-                transcript.template receive_from_prover<Commitment>(domain_separator + "_" + labels.w_4);
-
-            auto [beta, gamma] = transcript.get_challenges(domain_separator + "_beta", domain_separator + "_gamma");
-            witness_commitments.z_perm =
-                transcript.template receive_from_prover<Commitment>(domain_separator + "_" + labels.z_perm);
-            witness_commitments.z_lookup =
-                transcript.template receive_from_prover<Commitment>(domain_separator + "_" + labels.z_lookup);
-
-            const FF public_input_delta = compute_public_input_delta<Flavor>(
-                inst->public_inputs, beta, gamma, inst->instance_size, inst->pub_inputs_offset);
-            const FF lookup_grand_product_delta =
-                compute_lookup_grand_product_delta<FF>(beta, gamma, inst->instance_size);
-            inst->relation_parameters =
-                RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
-
-            inst->alpha = transcript.get_challenge(domain_separator + "_alpha");
-        }
-
-        inst->verification_key = std::make_shared<VerificationKey>(inst->instance_size, inst->public_input_size);
-        auto vk_view = inst->verification_key->pointer_view();
-        auto labels = typename Flavor::CommitmentLabels().get_precomputed();
-        for (size_t idx = 0; idx < labels.size(); idx++) {
-            (*vk_view[idx]) = transcript.template receive_from_prover<Commitment>(domain_separator + "_" + labels[idx]);
-        }
+        receive_and_finalise_instance(inst, domain_separator);
     }
 }
 
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
index 6cc149a87da..86b1c698b4a 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
@@ -60,7 +60,11 @@ template <class VerifierInstances> class ProtoGalaxyVerifier_ {
      *
      * @param fold_data The data transmitted via the transcript by the prover.
      */
-    void prepare_for_folding(std::vector<uint8_t> fold_data);
+    void prepare_for_folding(std::vector<uint8_t>);
+
+    void receive_accumulator(std::shared_ptr<Instance>, std::string);
+
+    void receive_and_finalise_instance(std::shared_ptr<Instance>, std::string);
 
     /**
      * @brief Run the folding protocol on the verifier side to verify the public data ϕ of the new accumulator, received
@@ -68,7 +72,7 @@ template <class VerifierInstances> class ProtoGalaxyVerifier_ {
      *
      * TODO(https://github.com/AztecProtocol/barretenberg/issues/690): finalise the implementation of this function
      */
-    bool verify_folding_proof(std::vector<uint8_t> fold_data);
+    bool verify_folding_proof(std::vector<uint8_t>);
 };
 
 extern template class ProtoGalaxyVerifier_<VerifierInstances_<honk::flavor::Ultra, 2>>;

From 9796186eb91f0858be690195267cf5e2575dab58 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Mon, 4 Dec 2023 16:33:09 +0000
Subject: [PATCH 42/86] fix merge bug and cleanup

---
 .../protogalaxy/protogalaxy_prover.hpp        | 47 +++++++++++--------
 .../protogalaxy/protogalaxy_verifier.cpp      | 15 ++----
 .../protogalaxy/protogalaxy_verifier.hpp      | 14 ++++--
 3 files changed, 42 insertions(+), 34 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index 91e6ee49b1d..ed53ca72eb0 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -57,16 +57,28 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     ~ProtoGalaxyProver_() = default;
 
     /**
-     * @brief Prior to folding we need to add all the public inputs to the transcript, labelled by their corresponding
-     * instance index, compute all the instance's polynomials and record the relation parameters involved in computing
-     * these polynomials in the transcript.
-     *
+     * @brief Prior to folding, we need to finalise the given instances and add all their public data ϕ to the
+     * transcript, labelled by their corresponding instance index for domain separation.
      */
     void prepare_for_folding();
 
-    void send_accumulator(std::shared_ptr<Instance>, const std::string&);
+    /**
+     * @brief Send the public data of an accumulator, i.e. a relaxed instance, to the verifier (ϕ in the paper).
+     *
+     *  @param domain_separator separates the same type of data coming from difference instances by instance
+     * index
+     */
+    void send_accumulator(std::shared_ptr<Instance>, const std::string& domain_separator);
 
-    void finalise_and_send_instance(std::shared_ptr<Instance>, const std::string&);
+    /**
+     * @brief For each instance produced by a circuit, prior to folding, we need to complete the computation of its
+     * prover polynomials, commit to witnesses and generate the relation parameters as well as send the public data ϕ of
+     * an instance to the verifier.
+     *
+     * @param domain_separator  separates the same type of data coming from difference instances by instance
+     * index
+     */
+    void finalise_and_send_instance(std::shared_ptr<Instance>, const std::string& domain_separator);
 
     /**
      * @brief Given a vector \vec{\beta} of values, compute the pow polynomial on these values as defined in the paper.
@@ -472,23 +484,18 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         }
         next_accumulator->prover_polynomials = acc_prover_polynomials;
 
-        // Allocate space, initialised to 0, for the folded witness commitments
-        WitnessCommitments acc_witness_commitments;
-        auto acc_comm_view = acc_witness_commitments.get_all();
-        for (auto c : acc_comm_view) {
-            (c) = Commitment::infinity();
-        }
-
-        // Fold the witness commtimenets and send them to the verifier
+        // Fold the witness commtiments and send them to the verifier
         auto witness_labels = next_accumulator->commitment_labels.get_witness();
-        for (size_t comm_idx = 0; comm_idx < acc_comm_view.size(); comm_idx++) {
+        size_t comm_idx = 0;
+        for (auto& acc_comm : next_accumulator->witness_commitments.get_all()) {
+            acc_comm = Commitment::infinity();
             size_t inst_idx = 0;
             for (auto& instance : instances) {
-                (acc_comm_view[comm_idx]) = (acc_comm_view[comm_idx]) +
-                                            (instance->witness_commitments.get_all()[comm_idx]) * lagranges[inst_idx];
+                acc_comm = acc_comm + instance->witness_commitments.get_all()[comm_idx] * lagranges[inst_idx];
                 inst_idx++;
             }
-            transcript->send_to_verifier("next_" + witness_labels[comm_idx], (acc_comm_view[comm_idx]));
+            transcript->send_to_verifier("next_" + witness_labels[comm_idx], acc_comm);
+            comm_idx++;
         }
 
         // Fold the public inputs and send to the verifier
@@ -535,8 +542,8 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         for (size_t vk_idx = 0; vk_idx < acc_vk_view.size(); vk_idx++) {
             size_t inst = 0;
             for (auto& instance : instances) {
-                (acc_vk_view[vk_idx]) =
-                    (acc_vk_view[vk_idx]) + (instance->verification_key->get_all()[vk_idx]) * lagranges[inst];
+                acc_vk_view[vk_idx] =
+                    acc_vk_view[vk_idx] + (instance->verification_key->get_all()[vk_idx]) * lagranges[inst];
                 inst++;
             }
             transcript->send_to_verifier("next_" + labels[vk_idx], (acc_vk_view[vk_idx]));
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index d0509090def..8380c43b0ba 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -4,7 +4,7 @@ namespace proof_system::honk {
 
 template <class VerifierInstances>
 void ProtoGalaxyVerifier_<VerifierInstances>::receive_accumulator(std::shared_ptr<Instance> inst,
-                                                                  std::string domain_separator)
+                                                                  const std::string& domain_separator)
 {
     inst->instance_size = transcript->template receive_from_prover<uint32_t>(domain_separator + "_instance_size");
     inst->log_instance_size = static_cast<size_t>(numeric::get_msb(inst->instance_size));
@@ -52,7 +52,7 @@ void ProtoGalaxyVerifier_<VerifierInstances>::receive_accumulator(std::shared_pt
 
 template <class VerifierInstances>
 void ProtoGalaxyVerifier_<VerifierInstances>::receive_and_finalise_instance(std::shared_ptr<Instance> inst,
-                                                                            std::string domain_separator)
+                                                                            const std::string& domain_separator)
 {
     inst->instance_size = transcript->template receive_from_prover<uint32_t>(domain_separator + "_instance_size");
     inst->log_instance_size = static_cast<size_t>(numeric::get_msb(inst->instance_size));
@@ -172,7 +172,7 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
     WitnessCommitments acc_witness_commitments;
     auto witness_labels = commitment_labels.get_witness();
     size_t comm_idx = 0;
-    for (auto expected_comm : acc_witness_commitments.get_all()) {
+    for (auto& expected_comm : acc_witness_commitments.get_all()) {
         expected_comm = Commitment::infinity();
         size_t inst = 0;
         for (auto& instance : instances) {
@@ -186,7 +186,7 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
 
     std::vector<FF> folded_public_inputs(instances[0]->public_inputs.size(), 0);
     size_t el_idx = 0;
-    for (auto expected_el : folded_public_inputs) {
+    for (auto& expected_el : folded_public_inputs) {
         size_t inst = 0;
         for (auto& instance : instances) {
             expected_el += instance->public_inputs[el_idx] * lagranges[inst];
@@ -213,28 +213,22 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
 
     auto next_alpha = transcript->template receive_from_prover<FF>("next_alpha");
     verified = verified & (next_alpha == expected_alpha);
-    info(verified);
 
     auto next_eta = transcript->template receive_from_prover<FF>("next_eta");
     verified = verified & (next_eta == expected_parameters.eta);
-    info(verified);
 
     auto next_beta = transcript->template receive_from_prover<FF>("next_beta");
     verified = verified & (next_beta == expected_parameters.beta);
-    info(verified);
 
     auto next_gamma = transcript->template receive_from_prover<FF>("next_gamma");
     verified = verified & (next_gamma == expected_parameters.gamma);
-    info(verified);
 
     auto next_public_input_delta = transcript->template receive_from_prover<FF>("next_public_input_delta");
     verified = verified & (next_public_input_delta == expected_parameters.public_input_delta);
-    info(verified);
 
     auto next_lookup_grand_product_delta =
         transcript->template receive_from_prover<FF>("next_lookup_grand_product_delta");
     verified = verified & (next_lookup_grand_product_delta == expected_parameters.lookup_grand_product_delta);
-    info(verified);
 
     auto acc_vk = std::make_shared<VerificationKey>(instances[0]->instance_size, instances[0]->public_input_size);
     auto vk_labels = commitment_labels.get_precomputed();
@@ -249,7 +243,6 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
         verified = verified & (vk == expected_vk);
         vk_idx++;
     }
-    info(verified);
 
     return verified;
 }
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
index 8ea467d3a1a..0210d620a54 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
@@ -58,15 +58,23 @@ template <class VerifierInstances> class ProtoGalaxyVerifier_ {
     std::shared_ptr<Instance> get_accumulator() { return instances[0]; }
 
     /**
-     * @brief Instatiate the VerifierInstances and the VerifierTranscript.
+     * @brief Instatiate the instances and the transcript.
      *
      * @param fold_data The data transmitted via the transcript by the prover.
      */
     void prepare_for_folding(std::vector<uint8_t>);
 
-    void receive_accumulator(std::shared_ptr<Instance>, std::string);
+    /**
+     * @brief Instantiatied the accumulator (i.e. the relaxed instance) from the transcript.
+     *
+     */
+    void receive_accumulator(std::shared_ptr<Instance>, const std::string&);
 
-    void receive_and_finalise_instance(std::shared_ptr<Instance>, std::string);
+    /**
+     * @brief Process the public data ϕ for the Instances to be folded.
+     *
+     */
+    void receive_and_finalise_instance(std::shared_ptr<Instance>, const std::string&);
 
     /**
      * @brief Run the folding protocol on the verifier side to verify the public data ϕ of the new accumulator, received

From 1d2464350ce76e39e30b3f0037a5d5e044131cbe Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Mon, 4 Dec 2023 16:50:04 +0000
Subject: [PATCH 43/86] fix formatting

---
 barretenberg/cpp/src/barretenberg/relations/utils.hpp | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/barretenberg/cpp/src/barretenberg/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
index 1d9e53c97c4..a87f71855f8 100644
--- a/barretenberg/cpp/src/barretenberg/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
@@ -86,7 +86,8 @@ template <typename Flavor> class RelationUtils {
     template <typename... T>
     static constexpr void add_tuples(std::tuple<T...>& tuple_1, const std::tuple<T...>& tuple_2)
     {
-        auto add_tuples_helper = [&]<std::size_t... I>(std::index_sequence<I...>) {
+        auto add_tuples_helper = [&]<std::size_t... I>(std::index_sequence<I...>)
+        {
             ((std::get<I>(tuple_1) += std::get<I>(tuple_2)), ...);
         };
 

From a720c80b31d96c1a90086741b84e2bfba20b5cd4 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Mon, 4 Dec 2023 16:53:57 +0000
Subject: [PATCH 44/86] add todo

---
 barretenberg/barretenberg.code-workspace                        | 2 +-
 .../cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp     | 2 ++
 2 files changed, 3 insertions(+), 1 deletion(-)

diff --git a/barretenberg/barretenberg.code-workspace b/barretenberg/barretenberg.code-workspace
index 6201b7ca788..24c8a186612 100644
--- a/barretenberg/barretenberg.code-workspace
+++ b/barretenberg/barretenberg.code-workspace
@@ -5,7 +5,7 @@
         {
             "name": "barretenberg",
             "path": "./"
-        }
+        },
     ],
     // List recommended extensions for the whole workspace.
     "extensions": {
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index 0229f3b814a..e9503e53e03 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -435,6 +435,8 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      * @param combiner_quotient polynomial K in the paper
      * @param challenge
      * @param compressed_perturbator
+     *
+     * TODO(https://github.com/AztecProtocol/barretenberg/issues/796): optimise the construction of the new accumulator
      */
     std::shared_ptr<Instance> compute_next_accumulator(ProverInstances& instances,
                                                        auto& combiner_quotient,

From e6e5db373b0eadd9272c9fb1431384d5884771f6 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Mon, 4 Dec 2023 20:46:06 +0000
Subject: [PATCH 45/86] print statements

---
 .../barretenberg/protogalaxy/protogalaxy_verifier.cpp  | 10 ++++++++++
 1 file changed, 10 insertions(+)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index 8380c43b0ba..762bd8d22a2 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -167,6 +167,7 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
         auto beta_star = transcript->template receive_from_prover<FF>("next_gate_challenge_" + std::to_string(idx));
         verified = verified & (expected_betas_star[idx] == beta_star);
     }
+    info(verified);
 
     // Compute ϕ and verify against the data received from the prover
     WitnessCommitments acc_witness_commitments;
@@ -183,6 +184,7 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
         verified = verified & (comm == expected_comm);
         comm_idx++;
     }
+    info(verified);
 
     std::vector<FF> folded_public_inputs(instances[0]->public_inputs.size(), 0);
     size_t el_idx = 0;
@@ -196,6 +198,7 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
         verified = verified & (el == expected_el);
         el_idx++;
     }
+    info(verified);
 
     auto expected_alpha = FF(0);
     auto expected_parameters = proof_system::RelationParameters<FF>{};
@@ -213,22 +216,28 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
 
     auto next_alpha = transcript->template receive_from_prover<FF>("next_alpha");
     verified = verified & (next_alpha == expected_alpha);
+    info(verified);
 
     auto next_eta = transcript->template receive_from_prover<FF>("next_eta");
     verified = verified & (next_eta == expected_parameters.eta);
+    info(verified);
 
     auto next_beta = transcript->template receive_from_prover<FF>("next_beta");
     verified = verified & (next_beta == expected_parameters.beta);
+    info(verified);
 
     auto next_gamma = transcript->template receive_from_prover<FF>("next_gamma");
     verified = verified & (next_gamma == expected_parameters.gamma);
+    info(verified);
 
     auto next_public_input_delta = transcript->template receive_from_prover<FF>("next_public_input_delta");
     verified = verified & (next_public_input_delta == expected_parameters.public_input_delta);
+    info(verified);
 
     auto next_lookup_grand_product_delta =
         transcript->template receive_from_prover<FF>("next_lookup_grand_product_delta");
     verified = verified & (next_lookup_grand_product_delta == expected_parameters.lookup_grand_product_delta);
+    info(verified);
 
     auto acc_vk = std::make_shared<VerificationKey>(instances[0]->instance_size, instances[0]->public_input_size);
     auto vk_labels = commitment_labels.get_precomputed();
@@ -243,6 +252,7 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
         verified = verified & (vk == expected_vk);
         vk_idx++;
     }
+    info(verified);
 
     return verified;
 }

From ee926f8c29fe91069c9e9576f049f7cdc626c003 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Mon, 4 Dec 2023 21:01:18 +0000
Subject: [PATCH 46/86] fix bug?

---
 .../barretenberg/protogalaxy/protogalaxy_prover.hpp   | 11 ++++++-----
 .../barretenberg/protogalaxy/protogalaxy_verifier.cpp |  3 ++-
 2 files changed, 8 insertions(+), 6 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index e9503e53e03..38234b49cf3 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -543,16 +543,17 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         // Fold the verification key and send it to the verifier
         auto acc_vk = std::make_shared<VerificationKey>(instances[0]->prover_polynomials.get_polynomial_size(),
                                                         instances[0]->public_inputs.size());
-        auto acc_vk_view = acc_vk->get_all();
         auto labels = next_accumulator->commitment_labels.get_precomputed();
-        for (size_t vk_idx = 0; vk_idx < acc_vk_view.size(); vk_idx++) {
+        size_t vk_idx = 0;
+        for (auto& vk : acc_vk->get_all()) {
             size_t inst = 0;
+            vk = Commitment::infinity();
             for (auto& instance : instances) {
-                acc_vk_view[vk_idx] =
-                    acc_vk_view[vk_idx] + (instance->verification_key->get_all()[vk_idx]) * lagranges[inst];
+                vk = vk + (instance->verification_key->get_all()[vk_idx]) * lagranges[inst];
                 inst++;
             }
-            transcript->send_to_verifier("next_" + labels[vk_idx], (acc_vk_view[vk_idx]));
+            transcript->send_to_verifier("next_" + labels[vk_idx], vk);
+            vk_idx++;
         }
         next_accumulator->verification_key = acc_vk;
 
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index 762bd8d22a2..61ae8851e61 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -242,8 +242,9 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
     auto acc_vk = std::make_shared<VerificationKey>(instances[0]->instance_size, instances[0]->public_input_size);
     auto vk_labels = commitment_labels.get_precomputed();
     size_t vk_idx = 0;
-    for (auto expected_vk : acc_vk->get_all()) {
+    for (auto& expected_vk : acc_vk->get_all()) {
         size_t inst = 0;
+        expected_vk = Commitment::infinity();
         for (auto& instance : instances) {
             expected_vk = expected_vk + instance->verification_key->get_all()[vk_idx] * lagranges[inst];
             inst++;

From 89bf847064a7501cbb29fb6cfce9cce43170fad9 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Mon, 4 Dec 2023 21:10:56 +0000
Subject: [PATCH 47/86] remove prints

---
 .../barretenberg/protogalaxy/protogalaxy_verifier.cpp  | 10 ----------
 1 file changed, 10 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index 61ae8851e61..a76f93827de 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -167,7 +167,6 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
         auto beta_star = transcript->template receive_from_prover<FF>("next_gate_challenge_" + std::to_string(idx));
         verified = verified & (expected_betas_star[idx] == beta_star);
     }
-    info(verified);
 
     // Compute ϕ and verify against the data received from the prover
     WitnessCommitments acc_witness_commitments;
@@ -184,7 +183,6 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
         verified = verified & (comm == expected_comm);
         comm_idx++;
     }
-    info(verified);
 
     std::vector<FF> folded_public_inputs(instances[0]->public_inputs.size(), 0);
     size_t el_idx = 0;
@@ -198,7 +196,6 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
         verified = verified & (el == expected_el);
         el_idx++;
     }
-    info(verified);
 
     auto expected_alpha = FF(0);
     auto expected_parameters = proof_system::RelationParameters<FF>{};
@@ -216,28 +213,22 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
 
     auto next_alpha = transcript->template receive_from_prover<FF>("next_alpha");
     verified = verified & (next_alpha == expected_alpha);
-    info(verified);
 
     auto next_eta = transcript->template receive_from_prover<FF>("next_eta");
     verified = verified & (next_eta == expected_parameters.eta);
-    info(verified);
 
     auto next_beta = transcript->template receive_from_prover<FF>("next_beta");
     verified = verified & (next_beta == expected_parameters.beta);
-    info(verified);
 
     auto next_gamma = transcript->template receive_from_prover<FF>("next_gamma");
     verified = verified & (next_gamma == expected_parameters.gamma);
-    info(verified);
 
     auto next_public_input_delta = transcript->template receive_from_prover<FF>("next_public_input_delta");
     verified = verified & (next_public_input_delta == expected_parameters.public_input_delta);
-    info(verified);
 
     auto next_lookup_grand_product_delta =
         transcript->template receive_from_prover<FF>("next_lookup_grand_product_delta");
     verified = verified & (next_lookup_grand_product_delta == expected_parameters.lookup_grand_product_delta);
-    info(verified);
 
     auto acc_vk = std::make_shared<VerificationKey>(instances[0]->instance_size, instances[0]->public_input_size);
     auto vk_labels = commitment_labels.get_precomputed();
@@ -253,7 +244,6 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
         verified = verified & (vk == expected_vk);
         vk_idx++;
     }
-    info(verified);
 
     return verified;
 }

From ac8b77c74f0ee24c61d35abd1d0a279810164228 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Wed, 6 Dec 2023 10:30:02 +0000
Subject: [PATCH 48/86] remove unnecessary stuff

---
 .../src/barretenberg/eccvm/eccvm_prover.cpp   |  1 -
 .../src/barretenberg/eccvm/eccvm_verifier.cpp |  1 -
 .../honk/proof_system/power_polynomial.hpp    | 69 -------------------
 .../proof_system/power_polynomial.test.cpp    | 35 ----------
 .../proof_system/composer/composer_lib.hpp    |  4 +-
 .../verifier/ultra_recursive_verifier.cpp     |  1 -
 .../sumcheck/instance/prover_instance.hpp     |  1 -
 .../goblin_translator_prover.cpp              |  1 -
 .../goblin_translator_verifier.cpp            |  1 -
 .../barretenberg/ultra_honk/ultra_prover.cpp  |  1 -
 .../ultra_honk/ultra_verifier.cpp             |  1 -
 .../vm/generated/AvmMini_prover.cpp           |  1 -
 .../vm/generated/AvmMini_verifier.cpp         |  1 -
 .../barretenberg/vm/generated/Fib_prover.cpp  |  1 -
 .../vm/generated/Fib_verifier.cpp             |  1 -
 15 files changed, 2 insertions(+), 118 deletions(-)
 delete mode 100644 barretenberg/cpp/src/barretenberg/honk/proof_system/power_polynomial.hpp
 delete mode 100644 barretenberg/cpp/src/barretenberg/honk/proof_system/power_polynomial.test.cpp

diff --git a/barretenberg/cpp/src/barretenberg/eccvm/eccvm_prover.cpp b/barretenberg/cpp/src/barretenberg/eccvm/eccvm_prover.cpp
index 16d3f3a9cfb..5a65ec5cca2 100644
--- a/barretenberg/cpp/src/barretenberg/eccvm/eccvm_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/eccvm/eccvm_prover.cpp
@@ -4,7 +4,6 @@
 #include "barretenberg/common/ref_array.hpp"
 #include "barretenberg/honk/proof_system/lookup_library.hpp"
 #include "barretenberg/honk/proof_system/permutation_library.hpp"
-#include "barretenberg/honk/proof_system/power_polynomial.hpp"
 #include "barretenberg/polynomials/polynomial.hpp"
 #include "barretenberg/proof_system/library/grand_product_library.hpp"
 #include "barretenberg/relations/lookup_relation.hpp"
diff --git a/barretenberg/cpp/src/barretenberg/eccvm/eccvm_verifier.cpp b/barretenberg/cpp/src/barretenberg/eccvm/eccvm_verifier.cpp
index d7f26ce7299..7edc7df07e9 100644
--- a/barretenberg/cpp/src/barretenberg/eccvm/eccvm_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/eccvm/eccvm_verifier.cpp
@@ -1,7 +1,6 @@
 #include "./eccvm_verifier.hpp"
 #include "barretenberg/commitment_schemes/gemini/gemini.hpp"
 #include "barretenberg/commitment_schemes/shplonk/shplonk.hpp"
-#include "barretenberg/honk/proof_system/power_polynomial.hpp"
 #include "barretenberg/numeric/bitop/get_msb.hpp"
 #include "barretenberg/transcript/transcript.hpp"
 
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/power_polynomial.hpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/power_polynomial.hpp
deleted file mode 100644
index 5dfe7701a32..00000000000
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/power_polynomial.hpp
+++ /dev/null
@@ -1,69 +0,0 @@
-#pragma once
-#include "barretenberg/common/thread.hpp"
-#include "barretenberg/polynomials/polynomial.hpp"
-#include <span>
-
-namespace proof_system::honk::power_polynomial {
-/**
- * @brief Generate the power polynomial vector
- *
- * @details Generates a vector, where v[i]=ζⁱ
- *
- * @param zeta
- * @param vector_size
- * @return barretenberg::polynomial
- */
-template <typename Fr> barretenberg::Polynomial<Fr> generate_vector(Fr zeta, size_t vector_size)
-{
-    // We know the size from the start, so we can allocate exactly the right amount of memory
-    barretenberg::Polynomial<Fr> pow_vector(vector_size);
-
-    constexpr size_t usefulness_margin = 4;
-    size_t num_threads = get_num_cpus_pow2();
-    if (vector_size < (usefulness_margin * num_threads)) {
-        num_threads = 1;
-    }
-    // Prepare for a random number of threads. We need to handle the last thread separately
-    size_t thread_size = vector_size / num_threads;
-    size_t last_thread_size = thread_size;
-    // Check if the vector size is divided into threads cleanly
-    if ((vector_size % thread_size) != 0) {
-        thread_size += 1;
-        last_thread_size = vector_size % thread_size;
-    }
-    parallel_for(num_threads, [&](size_t i) {
-        // Exponentiate ζ to the starting power of the chunk
-        Fr starting_power = zeta.pow(i * thread_size);
-        // Set the chunk size depending ontwhether this is the last thread
-        size_t chunk_size = i != (num_threads - 1) ? thread_size : last_thread_size;
-        size_t j = 0;
-        // Go through elements and compute ζ powers
-        for (; j < chunk_size - 1; j++) {
-            pow_vector[i * thread_size + j] = starting_power;
-            starting_power *= zeta;
-        }
-        pow_vector[i * thread_size + j] = starting_power;
-    });
-    return pow_vector;
-}
-
-/**
- * @brief Evaluate the power polynomial on {x_0,..,x_d}
- *
- * @details The power polynomial can be efficiently evaluated as ∏( ( b^{2^i} - 1 ) * x_i + 1)
- *
- * @param zeta ζ
- * @param variables
- * @return barretenberg::fr
- */
-template <typename Fr> Fr evaluate(Fr zeta, std::span<const Fr> variables)
-{
-    Fr evaluation = Fr::one();
-    for (size_t i = 0; i < variables.size(); i++) {
-        // evaluation *= (b^{2^i} - 1) * x_i + 1
-        evaluation *= (zeta - 1) * variables[i] + 1;
-        zeta *= zeta;
-    }
-    return evaluation;
-}
-} // namespace proof_system::honk::power_polynomial
diff --git a/barretenberg/cpp/src/barretenberg/honk/proof_system/power_polynomial.test.cpp b/barretenberg/cpp/src/barretenberg/honk/proof_system/power_polynomial.test.cpp
deleted file mode 100644
index 748731f0146..00000000000
--- a/barretenberg/cpp/src/barretenberg/honk/proof_system/power_polynomial.test.cpp
+++ /dev/null
@@ -1,35 +0,0 @@
-#include "power_polynomial.hpp"
-#include "barretenberg/numeric/random/engine.hpp"
-#include <gtest/gtest.h>
-
-TEST(power_polynomial, test_full_polynomial_correctness)
-{
-    const size_t order = 17;
-    const size_t n = size_t(1) << order;
-    barretenberg::fr zeta = barretenberg::fr::random_element();
-    barretenberg::polynomial power_polynomial = proof_system::honk::power_polynomial::generate_vector(zeta, n);
-
-    barretenberg::fr current_power = barretenberg::fr::one();
-    for (size_t i = 0; i < n; i++) {
-        EXPECT_EQ(power_polynomial[i], current_power);
-        if (power_polynomial[i] != current_power) {
-            break;
-        }
-        current_power *= zeta;
-    }
-}
-
-TEST(power_polynomial, test_evaluation_correctness)
-{
-    const size_t order = 30;
-    const size_t n = size_t(1) << order;
-    barretenberg::fr zeta = barretenberg::fr::random_element();
-    // Not using the debug engine, because we want to test randomly
-    size_t random_index = static_cast<size_t>(numeric::random::get_engine().get_random_uint32()) % n;
-    std::vector<barretenberg::fr> variables;
-    for (size_t i = 0; i < order; i++) {
-        variables.emplace_back((random_index >> i) & 1);
-    }
-    EXPECT_EQ(zeta.pow(static_cast<size_t>(random_index)),
-              proof_system::honk::power_polynomial::evaluate<barretenberg::fr>(zeta, variables));
-}
diff --git a/barretenberg/cpp/src/barretenberg/proof_system/composer/composer_lib.hpp b/barretenberg/cpp/src/barretenberg/proof_system/composer/composer_lib.hpp
index 728a5cbfda3..1e23786a39d 100644
--- a/barretenberg/cpp/src/barretenberg/proof_system/composer/composer_lib.hpp
+++ b/barretenberg/cpp/src/barretenberg/proof_system/composer/composer_lib.hpp
@@ -1,7 +1,7 @@
 #pragma once
 #include "barretenberg/flavor/flavor.hpp"
-#include "barretenberg/plonk/proof_system/proving_key/proving_key.hpp"
-#include "barretenberg/srs/factories/crs_factory.hpp"
+#include "barretenberg/proof_system/polynomial_store/polynomial_store.hpp"
+
 #include <memory>
 
 namespace proof_system {
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp
index 785f2c8331a..1912c5db0bb 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp
@@ -1,6 +1,5 @@
 #include "barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.hpp"
 #include "barretenberg/commitment_schemes/zeromorph/zeromorph.hpp"
-#include "barretenberg/honk/proof_system/power_polynomial.hpp"
 #include "barretenberg/numeric/bitop/get_msb.hpp"
 #include "barretenberg/proof_system/library/grand_product_delta.hpp"
 #include "barretenberg/transcript/transcript.hpp"
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
index 73b699f8af0..a433c0d1863 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
@@ -4,7 +4,6 @@
 #include "barretenberg/flavor/ultra.hpp"
 #include "barretenberg/proof_system/composer/composer_lib.hpp"
 #include "barretenberg/relations/relation_parameters.hpp"
-#include "barretenberg/srs/factories/file_crs_factory.hpp"
 
 namespace proof_system::honk {
 /**
diff --git a/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_prover.cpp b/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_prover.cpp
index cd3e010b0d9..6c55e65cb1c 100644
--- a/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_prover.cpp
@@ -2,7 +2,6 @@
 #include "barretenberg/commitment_schemes/claim.hpp"
 #include "barretenberg/commitment_schemes/commitment_key.hpp"
 #include "barretenberg/commitment_schemes/zeromorph/zeromorph.hpp"
-#include "barretenberg/honk/proof_system/power_polynomial.hpp"
 #include "barretenberg/proof_system/library/grand_product_library.hpp"
 #include "barretenberg/sumcheck/sumcheck.hpp"
 
diff --git a/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_verifier.cpp b/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_verifier.cpp
index 089c2c37451..b7fdf82cc11 100644
--- a/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_verifier.cpp
@@ -1,6 +1,5 @@
 #include "./goblin_translator_verifier.hpp"
 #include "barretenberg/commitment_schemes/zeromorph/zeromorph.hpp"
-#include "barretenberg/honk/proof_system/power_polynomial.hpp"
 #include "barretenberg/sumcheck/sumcheck.hpp"
 #include "barretenberg/transcript/transcript.hpp"
 
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp
index fe3031181ef..eb95099ef4c 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp
@@ -1,5 +1,4 @@
 #include "ultra_prover.hpp"
-#include "barretenberg/honk/proof_system/power_polynomial.hpp"
 #include "barretenberg/sumcheck/sumcheck.hpp"
 
 namespace proof_system::honk {
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp
index df70e5ac12c..f6900dd2a53 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp
@@ -1,6 +1,5 @@
 #include "./ultra_verifier.hpp"
 #include "barretenberg/commitment_schemes/zeromorph/zeromorph.hpp"
-#include "barretenberg/honk/proof_system/power_polynomial.hpp"
 #include "barretenberg/numeric/bitop/get_msb.hpp"
 #include "barretenberg/transcript/transcript.hpp"
 
diff --git a/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_prover.cpp b/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_prover.cpp
index 913dc121988..a9e8862b1a1 100644
--- a/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_prover.cpp
@@ -5,7 +5,6 @@
 #include "barretenberg/commitment_schemes/commitment_key.hpp"
 #include "barretenberg/honk/proof_system/lookup_library.hpp"
 #include "barretenberg/honk/proof_system/permutation_library.hpp"
-#include "barretenberg/honk/proof_system/power_polynomial.hpp"
 #include "barretenberg/polynomials/polynomial.hpp"
 #include "barretenberg/proof_system/library/grand_product_library.hpp"
 #include "barretenberg/relations/lookup_relation.hpp"
diff --git a/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_verifier.cpp b/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_verifier.cpp
index b0b9da4ec44..deab7e7c55d 100644
--- a/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_verifier.cpp
@@ -2,7 +2,6 @@
 
 #include "./AvmMini_verifier.hpp"
 #include "barretenberg/commitment_schemes/zeromorph/zeromorph.hpp"
-#include "barretenberg/honk/proof_system/power_polynomial.hpp"
 #include "barretenberg/numeric/bitop/get_msb.hpp"
 #include "barretenberg/transcript/transcript.hpp"
 
diff --git a/barretenberg/cpp/src/barretenberg/vm/generated/Fib_prover.cpp b/barretenberg/cpp/src/barretenberg/vm/generated/Fib_prover.cpp
index b8cd3fe8907..4bf90e78780 100644
--- a/barretenberg/cpp/src/barretenberg/vm/generated/Fib_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/vm/generated/Fib_prover.cpp
@@ -5,7 +5,6 @@
 #include "barretenberg/commitment_schemes/commitment_key.hpp"
 #include "barretenberg/honk/proof_system/lookup_library.hpp"
 #include "barretenberg/honk/proof_system/permutation_library.hpp"
-#include "barretenberg/honk/proof_system/power_polynomial.hpp"
 #include "barretenberg/polynomials/polynomial.hpp"
 #include "barretenberg/proof_system/library/grand_product_library.hpp"
 #include "barretenberg/relations/lookup_relation.hpp"
diff --git a/barretenberg/cpp/src/barretenberg/vm/generated/Fib_verifier.cpp b/barretenberg/cpp/src/barretenberg/vm/generated/Fib_verifier.cpp
index a47e2c0fdf0..648ca8caefc 100644
--- a/barretenberg/cpp/src/barretenberg/vm/generated/Fib_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/vm/generated/Fib_verifier.cpp
@@ -2,7 +2,6 @@
 
 #include "./Fib_verifier.hpp"
 #include "barretenberg/commitment_schemes/zeromorph/zeromorph.hpp"
-#include "barretenberg/honk/proof_system/power_polynomial.hpp"
 #include "barretenberg/numeric/bitop/get_msb.hpp"
 #include "barretenberg/transcript/transcript.hpp"
 

From 4cc7742e710c30cc78387d69a2ca72d8d8283edd Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Wed, 6 Dec 2023 16:27:41 +0000
Subject: [PATCH 49/86] stuff

---
 .../cpp/src/barretenberg/polynomials/pow.hpp  | 80 +++++++++++++------
 .../src/barretenberg/polynomials/pow.test.cpp |  2 +-
 .../cpp/src/barretenberg/relations/utils.hpp  | 13 ++-
 .../src/barretenberg/sumcheck/sumcheck.hpp    | 18 +++--
 .../barretenberg/sumcheck/sumcheck_round.hpp  | 35 ++++----
 5 files changed, 93 insertions(+), 55 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/polynomials/pow.hpp b/barretenberg/cpp/src/barretenberg/polynomials/pow.hpp
index 9039266cac0..383244ea3a4 100644
--- a/barretenberg/cpp/src/barretenberg/polynomials/pow.hpp
+++ b/barretenberg/cpp/src/barretenberg/polynomials/pow.hpp
@@ -1,32 +1,34 @@
 #pragma once
 
+#include <cstddef>
+#include <vector>
 namespace barretenberg {
 
 /**
  * @brief Succinct representation of the `pow` polynomial that can be partially evaluated variable-by-variable.
- * pow(X) = ∏_{0≤l<d} ((1−X_l) + X_l⋅ζ_l)
+ * pow(X_0,X_1,..,X_d) = \prod_{0≤l<d} ((1−X_l) + X_l⋅ζ_l)
  *
  * @details Let
  * - d be the number of variables
  * - l be the current Sumcheck round ( l ∈ {0, …, d-1} )
- * - u_{0}, ..., u_{l-1} the challenges sent by the verifier in rounds 0 to l-1.
+ * - u_0, ..., u_l-1 the challenges sent by the verifier in rounds 0 to l-1.
  *
  * We define
  *
- * - ζ_{0}, ..., ζ_{d-1}, as ζ_{l} = ζ^{ 2^l }.
- *   When 0 ≤ i < 2^d is represented in bits [i_{0}, ..., i_{d-1}] where i_{0} is the MSB, we have
+ * - ζ_0, ..., ζ_{d-1}, as ζ_l = ζ^{ 2^l }.
+ *   When 0 ≤ i < 2^d is represented in bits [i_0, ..., i_{d-1}] where i_0 is the MSB, we have
  *   ζ^{i} = ζ^{ ∑_{0≤l<d} i_l ⋅ 2^l }
  *         =     ∏_{0≤l<d} ζ^{ i_l ⋅ 2^l }
  *         =     ∏_{0≤l<d} ζ_l^{ i_l }
- *         =     ∏_{0≤l<d} { ( 1-i_l ) + i_l ⋅ ζ_l } // As i_{l} \in \{0, 1\}
+ *         =     ∏_{0≤l<d} { ( 1-i_l ) + i_l ⋅ ζ_l } // As i_l \in \{0, 1\}
  *   Note that
  *   - ζ_{0} = ζ,
  *   - ζ_{l+1} = ζ_{l}^2,
  *   - ζ_{d-1}   = ζ^{ 2^{d-1} }
  *
- * - pow(X) = ∏_{0≤l<d} ((1−X_l) + X_l⋅ζ_l) is the multi-linear polynomial whose evaluation at the i-th index
+ * - pow(X) = ∏_{0≤l<d} ((1−X_l) + X_l⋅ζ_l) is the multilinear polynomial whose evaluation at the i-th index
  *   of the full hypercube, equals ζⁱ.
- *   We can also see it as the multi-linear extension of the vector (1, ζ, ζ², ...).
+ *   We can also see it as the multi-linear extension of the vector (ζ, ζ^2, ζ^4, ..., ζ^{2^{d-1}}).
  *
  * - At round l, we iterate over all remaining vertices (i_{l+1}, ..., i_{d-1}) ∈ {0,1}^{d-l-1}.
  *   Let i = ∑_{l<k<d} i_k ⋅ 2^{k-(l+1)} be the index of the current edge over which we are evaluating the relation.
@@ -43,9 +45,8 @@ namespace barretenberg {
  *                  = c_l ⋅ ( (1−X_l) + X_l⋅ζ^{2^l} ) ⋅(ζ^2^{l+1})^{ ∑_{l<k<d} i_k ⋅ 2^{k-(l+1)} }
  *                  = c_l ⋅ ( (1−X_l) + X_l⋅ζ^{2^l} ) ⋅ζ_{l+1}^{i}
  *
- *   This is the pow polynomial, partially evaluated in
- *     (X_{0}, ..., X_{l-1}) = (u_{0}, ..., u_{l-1}),
- *   at the index 0 ≤ i < 2^{d-l-1} of the dimension-{d-l-1} hypercube.
+ *   This is the pow polynomial, partially evaluated in the first l-1 variables as
+ *     (X_{0}, ..., X_{l-1}) = (u_{0}, ..., u_{l-1}).
  *
  * - Sˡᵢ( X_l ) is the univariate of the full relation at edge pair i
  * i.e. it is the alpha-linear-combination of the relations evaluated in the edge at index i.
@@ -73,9 +74,9 @@ namespace barretenberg {
  * claimed sum, the procedure is modified as follows
  *
  * Init:
- * - σ_{ 0 } <-- 0 // Claimed Sumcheck sum
- * - c_{ 0 } <-- 1 // Partial evaluation constant, before any evaluation
- * - ζ_{ 0 } <-- ζ // Initial power of ζ
+ * - σ_0 <-- 0 // Claimed Sumcheck sum
+ * - c_0  <-- 1 // Partial evaluation constant, before any evaluation
+ * - ζ_0 <-- ζ // Initial power of ζ
  *
  * Round 0≤l<d-1:
  * - σ_{ l } =?= S'ˡ(0) + S'ˡ(1) = Tˡ(0) + ζ_{l}⋅Tˡ(1)  // Check partial sum
@@ -92,16 +93,15 @@ namespace barretenberg {
  */
 template <typename FF> struct PowUnivariate {
     // ζ_{l}, initialized as ζ_{0} = ζ
-    // At round l, equals ζ^{ 2^l }
+    // At round l, this equals ζ^{ 2^l }
     FF zeta_pow;
     // ζ_{l+1}, initialized as ζ_{1} = ζ^2
-    // Always equal to zeta_pow^2
-    // At round l, equals ζ^{ 2^{l+1} }
+    // At round l,this  equals ζ^{ 2^{l+1 } }
     FF zeta_pow_sqr;
-    // c_{l}, initialized as c_{0} = 1
-    // c_{l} = ∏_{0 ≤ k < l-1} ( (1-u_{k}) + u_{k}⋅ζ_{k} )
-    // At round d-1, equals pow(u_{0}, ..., u_{d-1}).
-    FF partial_evaluation_constant = FF(1);
+    // The constant c_{l} obtained by partially evaluating one variable in the power polynomial at each round. At the
+    // end of round l in the sumcheck protocol, variable X_l is replaced by a verifier challenge u_l. The partial
+    // evaluation constant is updated to represent pow(u_0,.., u_l) = \prod_{0 ≤ k < l} ( (1-u_{k}) + u_{k}⋅ζ_{k}).
+    FF partial_evaluation_result = FF(1);
 
     // Initialize with the random zeta
     explicit PowUnivariate(FF zeta_pow)
@@ -109,11 +109,13 @@ template <typename FF> struct PowUnivariate {
         , zeta_pow_sqr(zeta_pow.sqr())
     {}
 
-    // Evaluate the monomial ((1−X_{l}) + X_{l}⋅ζ_{l}) in the challenge point X_{l}=u_{l}.
+    /**
+     * @brief Evaluate the monomial ((1−X_l) + X_l⋅ζ_l) in the challenge point X_l=u_l.
+     */
     FF univariate_eval(FF challenge) const { return (FF(1) + (challenge * (zeta_pow - FF(1)))); };
 
     /**
-     * @brief Parially evaluate the polynomial in the new challenge, by updating the constant c_{l} -> c_{l+1}.
+     * @brief Parially evaluate the pow polynomial in the new challenge, by updating the constant c_{l} -> c_{l+1}.
      * Also update (ζ_{l}, ζ_{l+1}) -> (ζ_{l+1}, ζ_{l+1}^2)
      *
      * @param challenge l-th verifier challenge u_{l}
@@ -125,7 +127,39 @@ template <typename FF> struct PowUnivariate {
         // TODO(luke): for native FF, this could be self_sqr()
         zeta_pow_sqr = zeta_pow_sqr.sqr();
 
-        partial_evaluation_constant *= current_univariate_eval;
+        partial_evaluation_result *= current_univariate_eval;
+    }
+};
+
+template <typename FF> struct PowPolynomial {
+    std::vector<FF> scalars;
+    FF current_element;
+    size_t current_element_idx = 0;
+    FF partial_evaluation_result = FF(1);
+
+    explicit PowPolynomial(const std::vector<FF>& scalars)
+        : scalars(scalars)
+        , current_element(scalars[0])
+    {}
+
+    /**
+     * @brief Evaluate the monomial ((1−X_l) + X_l⋅ζ_l) in the challenge point X_l=u_l.
+     */
+    FF univariate_eval(FF challenge) const { return (FF(1) + (challenge * (current_element - FF(1)))); };
+
+    /**
+     * @brief Parially evaluate the pow polynomial in the new challenge, by updating the constant c_{l} -> c_{l+1}.
+     * Also update (ζ_{l}, ζ_{l+1}) -> (ζ_{l+1}, ζ_{l+1}^2)
+     *
+     * @param challenge l-th verifier challenge u_{l}
+     */
+    void partially_evaluate(FF challenge)
+    {
+        FF current_univariate_eval = univariate_eval(challenge);
+        partial_evaluation_result *= current_univariate_eval;
+
+        current_element = scalars[current_element_idx + 1];
+        current_element_idx++;
     }
 };
 } // namespace barretenberg
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/polynomials/pow.test.cpp b/barretenberg/cpp/src/barretenberg/polynomials/pow.test.cpp
index ad75285fce7..1768efef28f 100644
--- a/barretenberg/cpp/src/barretenberg/polynomials/pow.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/polynomials/pow.test.cpp
@@ -26,6 +26,6 @@ TEST(SumcheckPow, FullPowConsistency)
         zeta_power *= zeta_power;
     }
 
-    EXPECT_EQ(pow_univariate.partial_evaluation_constant, expected_eval);
+    EXPECT_EQ(pow_univariate.partial_evaluation_result, expected_eval);
 }
 } // namespace barretenberg::test_pow
diff --git a/barretenberg/cpp/src/barretenberg/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
index a87f71855f8..b9d1c1c5273 100644
--- a/barretenberg/cpp/src/barretenberg/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
@@ -86,8 +86,7 @@ template <typename Flavor> class RelationUtils {
     template <typename... T>
     static constexpr void add_tuples(std::tuple<T...>& tuple_1, const std::tuple<T...>& tuple_2)
     {
-        auto add_tuples_helper = [&]<std::size_t... I>(std::index_sequence<I...>)
-        {
+        auto add_tuples_helper = [&]<std::size_t... I>(std::index_sequence<I...>) {
             ((std::get<I>(tuple_1) += std::get<I>(tuple_2)), ...);
         };
 
@@ -127,18 +126,16 @@ template <typename Flavor> class RelationUtils {
     inline static void accumulate_relation_evaluations(PolynomialEvaluations evaluations,
                                                        RelationEvaluations& relation_evaluations,
                                                        const Parameters& relation_parameters,
-                                                       const FF& partial_evaluation_constant)
+                                                       const FF& partial_evaluation_result)
     {
         using Relation = std::tuple_element_t<relation_idx, Relations>;
-        Relation::accumulate(std::get<relation_idx>(relation_evaluations),
-                             evaluations,
-                             relation_parameters,
-                             partial_evaluation_constant);
+        Relation::accumulate(
+            std::get<relation_idx>(relation_evaluations), evaluations, relation_parameters, partial_evaluation_result);
 
         // Repeat for the next relation.
         if constexpr (relation_idx + 1 < NUM_RELATIONS) {
             accumulate_relation_evaluations<Parameters, relation_idx + 1>(
-                evaluations, relation_evaluations, relation_parameters, partial_evaluation_constant);
+                evaluations, relation_evaluations, relation_parameters, partial_evaluation_result);
         }
     }
 
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
index bfd9c2d7f54..9382e19aeaf 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
@@ -74,9 +74,13 @@ template <typename Flavor> class SumcheckProver {
                                  const proof_system::RelationParameters<FF>& relation_parameters,
                                  FF alpha) // pass by value, not by reference
     {
-        FF zeta = transcript->get_challenge("Sumcheck:zeta");
+        auto betas = std::vector<FF>(multivariate_d, 0);
+        size_t beta_idx = 0;
+        for (auto& beta : betas) {
+            beta = transcript->get_challenge("Sumcheck:beta_" + std::to_string(beta_idx));
+        }
 
-        barretenberg::PowUnivariate<FF> pow_univariate(zeta);
+        barretenberg::PowPolynomial<FF> pow_univariate(betas);
 
         std::vector<FF> multivariate_challenge;
         multivariate_challenge.reserve(multivariate_d);
@@ -110,7 +114,7 @@ template <typename Flavor> class SumcheckProver {
         ClaimedEvaluations multivariate_evaluations;
         for (auto [eval, poly] :
              zip_view(multivariate_evaluations.get_all(), partially_evaluated_polynomials.get_all())) {
-            eval = (poly)[0];
+            eval = poly[0];
         }
         transcript->send_to_verifier("Sumcheck:evaluations", multivariate_evaluations);
 
@@ -203,9 +207,13 @@ template <typename Flavor> class SumcheckVerifier {
     {
         bool verified(true);
 
-        FF zeta = transcript->get_challenge("Sumcheck:zeta");
+        auto betas = std::vector<FF>(multivariate_d, 0);
+        size_t beta_idx = 0;
+        for (auto& beta : betas) {
+            beta = transcript->get_challenge("Sumcheck:beta_" + std::to_string(beta_idx));
+        }
 
-        barretenberg::PowUnivariate<FF> pow_univariate(zeta);
+        barretenberg::PowPolynomial<FF> pow_univariate(betas);
         // All but final round.
         // target_total_sum is initialized to zero then mutated in place.
 
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
index b5c366be750..c3601d7d37c 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
@@ -103,16 +103,16 @@ template <typename Flavor> class SumcheckProverRound {
     barretenberg::Univariate<FF, BATCHED_RELATION_PARTIAL_LENGTH> compute_univariate(
         ProverPolynomialsOrPartiallyEvaluatedMultivariates& polynomials,
         const proof_system::RelationParameters<FF>& relation_parameters,
-        const barretenberg::PowUnivariate<FF>& pow_univariate,
+        const barretenberg::PowPolynomial<FF>& pow_polynomial,
         const FF alpha)
     {
         // Precompute the vector of required powers of zeta
         // TODO(luke): Parallelize this
-        std::vector<FF> pow_challenges(round_size >> 1);
-        pow_challenges[0] = pow_univariate.partial_evaluation_constant;
-        for (size_t i = 1; i < (round_size >> 1); ++i) {
-            pow_challenges[i] = pow_challenges[i - 1] * pow_univariate.zeta_pow_sqr;
-        }
+        // std::vector<FF> pow_challenges(round_size >> 1);
+        // pow_challenges[0] = Ahuit;
+        // for (size_t i = 1; i < (round_size >> 1); ++i) {
+        //     pow_challenges[i] = pow_challenges[i - 1] * pow_polyresult;
+        // }
 
         // Determine number of threads for multithreading.
         // Note: Multithreading is "on" for every round but we reduce the number of threads from the max available based
@@ -144,7 +144,8 @@ template <typename Flavor> class SumcheckProverRound {
                 extend_edges(extended_edges[thread_idx], polynomials, edge_idx);
 
                 // Update the pow polynomial's contribution c_l ⋅ ζ_{l+1}ⁱ for the next edge.
-                FF pow_challenge = pow_challenges[edge_idx >> 1];
+                // THIS IS PROBABLY WRONG
+                FF pow_challenge = pow_polynomial.scalars[edge_idx >> 1];
 
                 // Compute the i-th edge's univariate contribution,
                 // scale it by the pow polynomial's constant and zeta power "c_l ⋅ ζ_{l+1}ⁱ"
@@ -162,7 +163,7 @@ template <typename Flavor> class SumcheckProverRound {
         }
         // Batch the univariate contributions from each sub-relation to obtain the round univariate
         return batch_over_relations<barretenberg::Univariate<FF, BATCHED_RELATION_PARTIAL_LENGTH>>(
-            univariate_accumulators, alpha, pow_univariate);
+            univariate_accumulators, alpha, pow_polynomial);
     }
 
     /**
@@ -172,13 +173,13 @@ template <typename Flavor> class SumcheckProverRound {
     template <typename ExtendedUnivariate, typename ContainerOverSubrelations>
     static ExtendedUnivariate batch_over_relations(ContainerOverSubrelations& univariate_accumulators,
                                                    const FF& challenge,
-                                                   const barretenberg::PowUnivariate<FF>& pow_univariate)
+                                                   const barretenberg::PowPolynomial<FF>& pow_polynomial)
     {
         auto running_challenge = FF(1);
         Utils::scale_univariates(univariate_accumulators, challenge, running_challenge);
 
         auto result = ExtendedUnivariate(0);
-        extend_and_batch_univariates(univariate_accumulators, result, pow_univariate);
+        extend_and_batch_univariates(univariate_accumulators, result, pow_polynomial);
 
         // Reset all univariate accumulators to 0 before beginning accumulation in the next round
         Utils::zero_univariates(univariate_accumulators);
@@ -191,16 +192,16 @@ template <typename Flavor> class SumcheckProverRound {
      * @tparam extended_size Size after extension
      * @param tuple A tuple of tuples of Univariates
      * @param result A Univariate of length extended_size
-     * @param pow_univariate Power polynomial univariate
+     * @param pow_polynomial Power polynomial univariate
      */
     template <typename ExtendedUnivariate, typename TupleOfTuplesOfUnivariates>
     static void extend_and_batch_univariates(const TupleOfTuplesOfUnivariates& tuple,
                                              ExtendedUnivariate& result,
-                                             const barretenberg::PowUnivariate<FF>& pow_univariate)
+                                             const barretenberg::PowPolynomial<FF>& pow_polynomial)
     {
         ExtendedUnivariate extended_random_polynomial;
         // Random poly R(X) = (1-X) + X.zeta_pow
-        auto random_polynomial = barretenberg::Univariate<FF, 2>({ 1, pow_univariate.zeta_pow });
+        auto random_polynomial = barretenberg::Univariate<FF, 2>({ 1, pow_polynomial.current_element });
         extended_random_polynomial = random_polynomial.template extend_to<ExtendedUnivariate::LENGTH>();
 
         auto extend_and_sum = [&]<size_t relation_idx, size_t subrelation_idx, typename Element>(Element& element) {
@@ -323,13 +324,11 @@ template <typename Flavor> class SumcheckVerifierRound {
     // kill the pow_univariat
     FF compute_full_honk_relation_purported_value(ClaimedEvaluations purported_evaluations,
                                                   const proof_system::RelationParameters<FF>& relation_parameters,
-                                                  const barretenberg::PowUnivariate<FF>& pow_univariate,
+                                                  const barretenberg::PowPolynomial<FF>& pow_polynomial,
                                                   const FF alpha)
     {
-        Utils::template accumulate_relation_evaluations<>(purported_evaluations,
-                                                          relation_evaluations,
-                                                          relation_parameters,
-                                                          pow_univariate.partial_evaluation_constant);
+        Utils::template accumulate_relation_evaluations<>(
+            purported_evaluations, relation_evaluations, relation_parameters, pow_polynomial.partial_evaluation_result);
 
         auto running_challenge = FF(1);
         auto output = FF(0);

From 3e9671294342a3a3c4572684b4bb7c3121dcaae6 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Mon, 11 Dec 2023 12:24:48 +0000
Subject: [PATCH 50/86] pow poly works

---
 .../eccvm/eccvm_transcript.test.cpp           |  7 ++--
 .../cpp/src/barretenberg/polynomials/pow.hpp  | 34 +++++++++++++++----
 .../src/barretenberg/sumcheck/sumcheck.hpp    |  9 +++--
 .../barretenberg/sumcheck/sumcheck_round.hpp  | 18 +++++++---
 .../sumcheck/sumcheck_round.test.cpp          |  4 +--
 .../goblin_ultra_transcript.test.cpp          |  7 ++--
 .../ultra_honk/ultra_transcript.test.cpp      |  7 ++--
 7 files changed, 64 insertions(+), 22 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/eccvm/eccvm_transcript.test.cpp b/barretenberg/cpp/src/barretenberg/eccvm/eccvm_transcript.test.cpp
index 8df69c80e44..19044c4e45b 100644
--- a/barretenberg/cpp/src/barretenberg/eccvm/eccvm_transcript.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/eccvm/eccvm_transcript.test.cpp
@@ -128,8 +128,11 @@ template <typename Flavor> class ECCVMTranscriptTests : public ::testing::Test {
         manifest_expected.add_entry(round, "Z_PERM", size_G);
         manifest_expected.add_challenge(round, "alpha");
 
-        round++;
-        manifest_expected.add_challenge(round, "Sumcheck:zeta");
+        for (size_t i = 0; i < log_n; i++) {
+            round++;
+            std::string label = "Sumcheck:beta_" + std::to_string(i);
+            manifest_expected.add_challenge(round, label);
+        }
 
         for (size_t i = 0; i < log_n; ++i) {
             round++;
diff --git a/barretenberg/cpp/src/barretenberg/polynomials/pow.hpp b/barretenberg/cpp/src/barretenberg/polynomials/pow.hpp
index 383244ea3a4..874987b5e4f 100644
--- a/barretenberg/cpp/src/barretenberg/polynomials/pow.hpp
+++ b/barretenberg/cpp/src/barretenberg/polynomials/pow.hpp
@@ -132,15 +132,18 @@ template <typename FF> struct PowUnivariate {
 };
 
 template <typename FF> struct PowPolynomial {
-    std::vector<FF> scalars;
+    std::vector<FF> values;
+    std::vector<FF> pow_values;
     FF current_element;
     size_t current_element_idx = 0;
     FF partial_evaluation_result = FF(1);
 
-    explicit PowPolynomial(const std::vector<FF>& scalars)
-        : scalars(scalars)
-        , current_element(scalars[0])
-    {}
+    explicit PowPolynomial(const std::vector<FF>& values)
+        : values(values)
+        , current_element(values[0])
+    {
+        compute_pow_polynomial_at_values();
+    }
 
     /**
      * @brief Evaluate the monomial ((1−X_l) + X_l⋅ζ_l) in the challenge point X_l=u_l.
@@ -158,8 +161,25 @@ template <typename FF> struct PowPolynomial {
         FF current_univariate_eval = univariate_eval(challenge);
         partial_evaluation_result *= current_univariate_eval;
 
-        current_element = scalars[current_element_idx + 1];
-        current_element_idx++;
+        if (current_element_idx != values.size() - 1) {
+            current_element_idx++;
+            current_element = values[current_element_idx];
+        }
+    }
+
+    void compute_pow_polynomial_at_values()
+    {
+        size_t pow_size = 1 << values.size();
+        pow_values = std::vector<FF>(1 << values.size(), 0);
+        for (size_t i = 0; i < pow_size; i++) {
+            auto res = FF(1);
+            for (size_t j = i, beta_idx = 0; j > 0; j >>= 1, beta_idx++) {
+                if ((j & 1) == 1) {
+                    res *= values[beta_idx];
+                }
+            }
+            pow_values[i] = res;
+        }
     }
 };
 } // namespace barretenberg
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
index 9382e19aeaf..17c0f75a4c8 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
@@ -61,7 +61,7 @@ template <typename Flavor> class SumcheckProver {
         : transcript(transcript)
         , multivariate_n(multivariate_n)
         , multivariate_d(numeric::get_msb(multivariate_n))
-        , round(multivariate_n)
+        , round(multivariate_n, 2)
         , partially_evaluated_polynomials(multivariate_n){};
 
     /**
@@ -74,10 +74,11 @@ template <typename Flavor> class SumcheckProver {
                                  const proof_system::RelationParameters<FF>& relation_parameters,
                                  FF alpha) // pass by value, not by reference
     {
-        auto betas = std::vector<FF>(multivariate_d, 0);
+        std::vector<FF> betas(multivariate_d, 0);
         size_t beta_idx = 0;
         for (auto& beta : betas) {
             beta = transcript->get_challenge("Sumcheck:beta_" + std::to_string(beta_idx));
+            beta_idx++;
         }
 
         barretenberg::PowPolynomial<FF> pow_univariate(betas);
@@ -95,7 +96,7 @@ template <typename Flavor> class SumcheckProver {
         pow_univariate.partially_evaluate(round_challenge);
         round.round_size =
             round.round_size >> 1; // TODO(#224)(Cody): Maybe partially_evaluate should do this and release memory?
-
+        round.hop_size = round.hop_size << 1;
         // All but final round
         // We operate on partially_evaluated_polynomials in place.
         for (size_t round_idx = 1; round_idx < multivariate_d; round_idx++) {
@@ -108,6 +109,7 @@ template <typename Flavor> class SumcheckProver {
             partially_evaluate(partially_evaluated_polynomials, round.round_size, round_challenge);
             pow_univariate.partially_evaluate(round_challenge);
             round.round_size = round.round_size >> 1;
+            round.hop_size = round.hop_size << 1;
         }
 
         // Final round: Extract multivariate evaluations from partially_evaluated_polynomials and add to transcript
@@ -211,6 +213,7 @@ template <typename Flavor> class SumcheckVerifier {
         size_t beta_idx = 0;
         for (auto& beta : betas) {
             beta = transcript->get_challenge("Sumcheck:beta_" + std::to_string(beta_idx));
+            beta_idx++;
         }
 
         barretenberg::PowPolynomial<FF> pow_univariate(betas);
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
index c3601d7d37c..2f91a24e9d2 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
@@ -61,6 +61,7 @@ template <typename Flavor> class SumcheckProverRound {
     using ExtendedEdges = typename Flavor::ExtendedEdges;
 
     size_t round_size; // a power of 2
+    size_t hop_size;
 
     static constexpr size_t NUM_RELATIONS = Flavor::NUM_RELATIONS;
     static constexpr size_t MAX_PARTIAL_RELATION_LENGTH = Flavor::MAX_PARTIAL_RELATION_LENGTH;
@@ -69,8 +70,9 @@ template <typename Flavor> class SumcheckProverRound {
     SumcheckTupleOfTuplesOfUnivariates univariate_accumulators;
 
     // Prover constructor
-    SumcheckProverRound(size_t initial_round_size)
+    SumcheckProverRound(size_t initial_round_size, size_t initial_hop_size)
         : round_size(initial_round_size)
+        , hop_size(initial_hop_size)
     {
         // Initialize univariate accumulators to 0
         Utils::zero_univariates(univariate_accumulators);
@@ -109,11 +111,19 @@ template <typename Flavor> class SumcheckProverRound {
         // Precompute the vector of required powers of zeta
         // TODO(luke): Parallelize this
         // std::vector<FF> pow_challenges(round_size >> 1);
-        // pow_challenges[0] = Ahuit;
+        // pow_challenges[0] = pow_univariate.partial_evaluation_constant;
         // for (size_t i = 1; i < (round_size >> 1); ++i) {
-        //     pow_challenges[i] = pow_challenges[i - 1] * pow_polyresult;
+        //     pow_challenges[i] = pow_challenges[i - 1] * pow_univariate.zeta_pow_sqr;
         // }
 
+        std::vector<FF> pow_challenges(round_size >> 1);
+        pow_challenges[0] = pow_polynomial.partial_evaluation_result;
+        for (size_t i = 1; i < (round_size >> 1); ++i) {
+            info(hop_size);
+            info(i * hop_size);
+            pow_challenges[i] = pow_challenges[0] * pow_polynomial.pow_values[i * hop_size];
+        }
+
         // Determine number of threads for multithreading.
         // Note: Multithreading is "on" for every round but we reduce the number of threads from the max available based
         // on a specified minimum number of iterations per thread. This eventually leads to the use of a single thread.
@@ -145,7 +155,7 @@ template <typename Flavor> class SumcheckProverRound {
 
                 // Update the pow polynomial's contribution c_l ⋅ ζ_{l+1}ⁱ for the next edge.
                 // THIS IS PROBABLY WRONG
-                FF pow_challenge = pow_polynomial.scalars[edge_idx >> 1];
+                FF pow_challenge = pow_challenges[edge_idx >> 1];
 
                 // Compute the i-th edge's univariate contribution,
                 // scale it by the pow polynomial's constant and zeta power "c_l ⋅ ζ_{l+1}ⁱ"
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.test.cpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.test.cpp
index 2856012b163..dd26b11678a 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.test.cpp
@@ -41,9 +41,9 @@ TEST(SumcheckRound, SumcheckTupleOfTuplesOfUnivariates)
     Utils::scale_univariates(tuple_of_tuples, challenge, running_challenge);
 
     // Use extend_and_batch_univariates to extend to MAX_LENGTH then accumulate
-    PowUnivariate<FF> pow_univariate(1);
+    barretenberg::PowPolynomial<FF> pow_polynomial({ 1 });
     auto result = Univariate<FF, MAX_LENGTH>();
-    SumcheckProverRound<Flavor>::extend_and_batch_univariates(tuple_of_tuples, result, pow_univariate);
+    SumcheckProverRound<Flavor>::extend_and_batch_univariates(tuple_of_tuples, result, pow_polynomial);
 
     // Repeat the batching process manually
     auto result_expected = univariate_1.template extend_to<MAX_LENGTH>() * 1 +
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/goblin_ultra_transcript.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/goblin_ultra_transcript.test.cpp
index c1e174f0d62..89044843a74 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/goblin_ultra_transcript.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/goblin_ultra_transcript.test.cpp
@@ -66,8 +66,11 @@ class GoblinUltraTranscriptTests : public ::testing::Test {
         manifest_expected.add_entry(round, "Z_LOOKUP", size_G);
         manifest_expected.add_challenge(round, "alpha");
 
-        round++;
-        manifest_expected.add_challenge(round, "Sumcheck:zeta");
+        for (size_t i = 0; i < log_n; i++) {
+            round++;
+            std::string label = "Sumcheck:beta_" + std::to_string(i);
+            manifest_expected.add_challenge(round, label);
+        }
 
         for (size_t i = 0; i < log_n; ++i) {
             round++;
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_transcript.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_transcript.test.cpp
index c5564e8fd10..38cef0e5a1d 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_transcript.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_transcript.test.cpp
@@ -59,8 +59,11 @@ class UltraTranscriptTests : public ::testing::Test {
         manifest_expected.add_entry(round, "Z_LOOKUP", size_G);
         manifest_expected.add_challenge(round, "alpha");
 
-        round++;
-        manifest_expected.add_challenge(round, "Sumcheck:zeta");
+        for (size_t i = 0; i < log_n; i++) {
+            round++;
+            std::string label = "Sumcheck:beta_" + std::to_string(i);
+            manifest_expected.add_challenge(round, label);
+        }
 
         for (size_t i = 0; i < log_n; ++i) {
             round++;

From c8909d09e9ca821374b14a7ca6aa78e61afe2d7b Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Mon, 11 Dec 2023 16:48:00 +0000
Subject: [PATCH 51/86] wip that works

---
 .../commitment_schemes/commitment_key.hpp     |  3 -
 .../src/barretenberg/eccvm/eccvm_prover.cpp   |  6 +-
 .../eccvm/eccvm_transcript.test.cpp           |  2 +-
 .../src/barretenberg/eccvm/eccvm_verifier.cpp | 10 ++-
 .../src/barretenberg/flavor/goblin_ultra.hpp  |  5 --
 .../cpp/src/barretenberg/flavor/ultra.hpp     |  6 --
 .../protogalaxy/protogalaxy_prover.cpp        | 11 ++--
 .../protogalaxy/protogalaxy_prover.hpp        |  5 +-
 .../protogalaxy/protogalaxy_verifier.cpp      | 14 ++--
 .../verifier/ultra_recursive_verifier.cpp     | 16 +++--
 .../verifier/ultra_recursive_verifier.hpp     |  2 -
 .../sumcheck/instance/instances.hpp           |  1 -
 .../sumcheck/instance/prover_instance.hpp     | 11 ++--
 .../sumcheck/instance/verifier_instance.hpp   |  9 ++-
 .../src/barretenberg/sumcheck/sumcheck.hpp    | 65 +++++++++----------
 .../barretenberg/sumcheck/sumcheck.test.cpp   | 29 +++++++--
 .../barretenberg/sumcheck/sumcheck_round.hpp  |  6 +-
 .../goblin_translator_prover.cpp              |  7 +-
 .../goblin_translator_verifier.cpp            | 11 +++-
 .../goblin_ultra_transcript.test.cpp          |  2 +-
 .../ultra_honk/protogalaxy.test.cpp           |  6 +-
 .../barretenberg/ultra_honk/sumcheck.test.cpp | 17 ++++-
 .../barretenberg/ultra_honk/ultra_prover.cpp  |  9 ++-
 .../ultra_honk/ultra_transcript.test.cpp      |  2 +-
 .../ultra_honk/ultra_verifier.cpp             |  9 ++-
 .../vm/generated/AvmMini_prover.cpp           |  9 ++-
 .../vm/generated/AvmMini_verifier.cpp         | 12 ++--
 27 files changed, 171 insertions(+), 114 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/commitment_schemes/commitment_key.hpp b/barretenberg/cpp/src/barretenberg/commitment_schemes/commitment_key.hpp
index 2e9c65864a0..b7d4e29c65c 100644
--- a/barretenberg/cpp/src/barretenberg/commitment_schemes/commitment_key.hpp
+++ b/barretenberg/cpp/src/barretenberg/commitment_schemes/commitment_key.hpp
@@ -7,9 +7,6 @@
  * simplify the codebase.
  */
 
-#include "barretenberg/ecc/curves/bn254/bn254.hpp"
-#include "barretenberg/ecc/curves/bn254/pairing.hpp"
-#include "barretenberg/ecc/curves/grumpkin/grumpkin.hpp"
 #include "barretenberg/ecc/scalar_multiplication/scalar_multiplication.hpp"
 #include "barretenberg/numeric/bitop/pow.hpp"
 #include "barretenberg/polynomials/polynomial.hpp"
diff --git a/barretenberg/cpp/src/barretenberg/eccvm/eccvm_prover.cpp b/barretenberg/cpp/src/barretenberg/eccvm/eccvm_prover.cpp
index 24cef733e0c..4bfb7137357 100644
--- a/barretenberg/cpp/src/barretenberg/eccvm/eccvm_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/eccvm/eccvm_prover.cpp
@@ -210,7 +210,11 @@ template <ECCVMFlavor Flavor> void ECCVMProver_<Flavor>::execute_relation_check_
 
     auto sumcheck = Sumcheck(key->circuit_size, transcript);
     FF alpha = transcript->get_challenge("alpha");
-    sumcheck_output = sumcheck.prove(prover_polynomials, relation_parameters, alpha);
+    std::vector<FF> gate_challenges(numeric::get_msb(key->circuit_size));
+    for (size_t idx = 0; idx < gate_challenges.size(); idx++) {
+        gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
+    }
+    sumcheck_output = sumcheck.prove(prover_polynomials, relation_parameters, alpha, gate_challenges);
 }
 
 /**
diff --git a/barretenberg/cpp/src/barretenberg/eccvm/eccvm_transcript.test.cpp b/barretenberg/cpp/src/barretenberg/eccvm/eccvm_transcript.test.cpp
index 19044c4e45b..a9295fb42b6 100644
--- a/barretenberg/cpp/src/barretenberg/eccvm/eccvm_transcript.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/eccvm/eccvm_transcript.test.cpp
@@ -130,7 +130,7 @@ template <typename Flavor> class ECCVMTranscriptTests : public ::testing::Test {
 
         for (size_t i = 0; i < log_n; i++) {
             round++;
-            std::string label = "Sumcheck:beta_" + std::to_string(i);
+            std::string label = "Sumcheck:gate_challenge_" + std::to_string(i);
             manifest_expected.add_challenge(round, label);
         }
 
diff --git a/barretenberg/cpp/src/barretenberg/eccvm/eccvm_verifier.cpp b/barretenberg/cpp/src/barretenberg/eccvm/eccvm_verifier.cpp
index 29640b40266..6a43f5409ff 100644
--- a/barretenberg/cpp/src/barretenberg/eccvm/eccvm_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/eccvm/eccvm_verifier.cpp
@@ -157,10 +157,16 @@ template <typename Flavor> bool ECCVMVerifier_<Flavor>::verify_proof(const plonk
     commitments.z_perm = receive_commitment(commitment_labels.z_perm);
 
     // Execute Sumcheck Verifier
-    auto sumcheck = SumcheckVerifier<Flavor>(circuit_size);
+    const size_t log_circuit_size = numeric::get_msb(circuit_size);
+    auto sumcheck = SumcheckVerifier<Flavor>(log_circuit_size, transcript);
     FF alpha = transcript->get_challenge("alpha");
+    std::vector<FF> gate_challenges(numeric::get_msb(key->circuit_size));
+    for (size_t idx = 0; idx < gate_challenges.size(); idx++) {
+        gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
+    }
+
     auto [multivariate_challenge, purported_evaluations, sumcheck_verified] =
-        sumcheck.verify(relation_parameters, alpha, transcript);
+        sumcheck.verify(relation_parameters, alpha, gate_challenges);
 
     // If Sumcheck did not verify, return false
     if (sumcheck_verified.has_value() && !sumcheck_verified.value()) {
diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
index 34ed6c536d4..ad10b56139b 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
@@ -530,11 +530,6 @@ class GoblinUltra {
     };
     // Specialize for GoblinUltra (general case used in GoblinUltraRecursive).
     using VerifierCommitments = VerifierCommitments_<Commitment, VerificationKey>;
-    class FoldingParameters {
-      public:
-        std::vector<FF> gate_challenges;
-        FF target_sum;
-    };
 
     /**
      * @brief Derived class that defines proof structure for GoblinUltra proofs, as well as supporting functions.
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
index 799b10f6bb2..3f9b14aa4ba 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
@@ -395,12 +395,6 @@ class Ultra {
         }
     };
 
-    class FoldingParameters {
-      public:
-        std::vector<FF> gate_challenges;
-        FF target_sum;
-    };
-
     /**
      * @brief Derived class that defines proof structure for Ultra proofs, as well as supporting functions.
      *
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index 8aa77e78612..2d8d20abeab 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -90,11 +90,10 @@ void ProtoGalaxyProver_<ProverInstances>::send_accumulator(std::shared_ptr<Insta
 
     transcript->send_to_verifier(domain_separator + "_alpha", instance->alpha);
 
-    auto folding_parameters = instance->folding_parameters;
-    transcript->send_to_verifier(domain_separator + "_target_sum", folding_parameters.target_sum);
-    for (size_t idx = 0; idx < folding_parameters.gate_challenges.size(); idx++) {
+    transcript->send_to_verifier(domain_separator + "_target_sum", instance->target_sum);
+    for (size_t idx = 0; idx < instance->gate_challenges.size(); idx++) {
         transcript->send_to_verifier(domain_separator + "_gate_challenge_" + std::to_string(idx),
-                                     folding_parameters.gate_challenges[idx]);
+                                     instance->gate_challenges[idx]);
     }
 
     auto comm_view = instance->witness_commitments.get_all();
@@ -146,10 +145,10 @@ FoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverInstanc
     for (size_t idx = 0; idx <= accumulator->log_instance_size; idx++) {
         transcript->send_to_verifier("perturbator_" + std::to_string(idx), perturbator[idx]);
     }
-    assert(perturbator[0] == accumulator->folding_parameters.target_sum);
+    assert(perturbator[0] == accumulator->target_sum);
     auto perturbator_challenge = transcript->get_challenge("perturbator_challenge");
     instances.next_gate_challenges =
-        update_gate_challenges(perturbator_challenge, accumulator->folding_parameters.gate_challenges, deltas);
+        update_gate_challenges(perturbator_challenge, accumulator->gate_challenges, deltas);
     const auto pow_betas_star =
         compute_pow_polynomial_at_values(instances.next_gate_challenges, accumulator->instance_size);
 
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index 99c3b3c82fb..1844a1374b8 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -235,7 +235,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     {
         auto full_honk_evaluations = compute_full_honk_evaluations(
             accumulator->prover_polynomials, accumulator->alpha, accumulator->relation_parameters);
-        const auto betas = accumulator->folding_parameters.gate_challenges;
+        const auto betas = accumulator->gate_challenges;
         assert(betas.size() == deltas.size());
         auto coeffs = construct_perturbator_coefficients(betas, deltas, full_honk_evaluations);
         return Polynomial<FF>(coeffs);
@@ -456,12 +456,13 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         // Compute the next target sum and send the next folding parameters to the verifier
         auto next_target_sum =
             compressed_perturbator * lagranges[0] + vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
-        next_accumulator->folding_parameters = { instances.next_gate_challenges, next_target_sum };
         transcript->send_to_verifier("next_target_sum", next_target_sum);
+        next_accumulator->target_sum = next_target_sum;
         for (size_t idx = 0; idx < instances.next_gate_challenges.size(); idx++) {
             transcript->send_to_verifier("next_gate_challenge_" + std::to_string(idx),
                                          instances.next_gate_challenges[idx]);
         }
+        next_accumulator->gate_challenges = instances.next_gate_challenges;
 
         // Allocate space, initialised to 0, for the prover polynomials of the next accumulator
         std::array<barretenberg::Polynomial<FF>, Flavor::NUM_ALL_ENTITIES> storage;
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index a76f93827de..d43639ec66b 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -27,12 +27,11 @@ void ProtoGalaxyVerifier_<VerifierInstances>::receive_accumulator(std::shared_pt
         RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
     inst->alpha = transcript->template receive_from_prover<FF>(domain_separator + "_alpha");
 
-    inst->folding_parameters.target_sum =
-        transcript->template receive_from_prover<FF>(domain_separator + "_target_sum");
+    inst->target_sum = transcript->template receive_from_prover<FF>(domain_separator + "_target_sum");
 
-    inst->folding_parameters.gate_challenges = std::vector<FF>(inst->log_instance_size);
+    inst->gate_challenges = std::vector<FF>(inst->log_instance_size);
     for (size_t idx = 0; idx < inst->log_instance_size; idx++) {
-        inst->folding_parameters.gate_challenges[idx] =
+        inst->gate_challenges[idx] =
             transcript->template receive_from_prover<FF>(domain_separator + "_gate_challenge_" + std::to_string(idx));
     }
     auto comm_view = inst->witness_commitments.get_all();
@@ -109,7 +108,7 @@ void ProtoGalaxyVerifier_<VerifierInstances>::prepare_for_folding(std::vector<ui
     auto index = 0;
     auto inst = instances[0];
     auto domain_separator = std::to_string(index);
-    inst->is_accumulator = transcript->template receive_from_prover<uint32_t>(domain_separator + "is_accumulator");
+    inst->is_accumulator = transcript->template receive_from_prover<bool>(domain_separator + "is_accumulator");
     if (inst->is_accumulator) {
         receive_accumulator(inst, domain_separator);
     } else {
@@ -137,7 +136,7 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
     for (size_t idx = 0; idx <= accumulator->log_instance_size; idx++) {
         perturbator_coeffs[idx] = transcript->template receive_from_prover<FF>("perturbator_" + std::to_string(idx));
     }
-    assert(perturbator_coeffs[0] == accumulator->folding_parameters.target_sum);
+    assert(perturbator_coeffs[0] == accumulator->target_sum);
     auto perturbator = Polynomial<FF>(perturbator_coeffs);
     FF perturbator_challenge = transcript->get_challenge("perturbator_challenge");
     auto perturbator_at_challenge = perturbator.evaluate(perturbator_challenge);
@@ -161,8 +160,7 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
         perturbator_at_challenge * lagranges[0] + vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
     auto next_target_sum = transcript->template receive_from_prover<FF>("next_target_sum");
     bool verified = (expected_next_target_sum == next_target_sum);
-    auto expected_betas_star =
-        update_gate_challenges(perturbator_challenge, accumulator->folding_parameters.gate_challenges, deltas);
+    auto expected_betas_star = update_gate_challenges(perturbator_challenge, accumulator->gate_challenges, deltas);
     for (size_t idx = 0; idx < accumulator->log_instance_size; idx++) {
         auto beta_star = transcript->template receive_from_prover<FF>("next_gate_challenge_" + std::to_string(idx));
         verified = verified & (expected_betas_star[idx] == beta_star);
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp
index ae4b057ae5a..3154a456e31 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp
@@ -102,11 +102,17 @@ std::array<typename Flavor::GroupElement, 2> UltraRecursiveVerifier_<Flavor>::ve
 
     // Execute Sumcheck Verifier and extract multivariate opening point u = (u_0, ..., u_{d-1}) and purported
     // multivariate evaluations at u
-    auto sumcheck = Sumcheck(key->circuit_size);
-    auto alpha = transcript->get_challenge("alpha");
-    auto [multivariate_challenge, claimed_evaluations, verified] =
-        sumcheck.verify(relation_parameters, alpha, transcript);
-
+    const size_t log_circuit_size = numeric::get_msb(static_cast<uint32_t>(circuit_size.get_value()));
+    auto sumcheck = Sumcheck(log_circuit_size, transcript);
+    FF alpha = transcript->get_challenge("alpha");
+    auto gate_challenges = std::vector<FF>(log_circuit_size);
+    for (size_t idx = 0; idx < log_circuit_size; idx++) {
+        gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
+    }
+    info("here");
+    auto [multivariate_challenge, claimed_evaluations, sumcheck_verified] =
+        sumcheck.verify(relation_parameters, alpha, gate_challenges);
+    info("out of sumcheck");
     // Execute ZeroMorph multilinear PCS evaluation verifier
     auto pairing_points = ZeroMorph::verify(commitments.get_unshifted(),
                                             commitments.get_to_be_shifted(),
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.hpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.hpp
index 62803746d5d..5b9f371f88e 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.hpp
@@ -1,7 +1,5 @@
 #pragma once
-#include "barretenberg/flavor/goblin_ultra.hpp"
 #include "barretenberg/flavor/goblin_ultra_recursive.hpp"
-#include "barretenberg/flavor/ultra.hpp"
 #include "barretenberg/flavor/ultra_recursive.hpp"
 #include "barretenberg/plonk/proof_system/types/proof.hpp"
 #include "barretenberg/stdlib/recursion/honk/transcript/transcript.hpp"
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
index 1df270fbfe2..eb2fd35739f 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
@@ -8,7 +8,6 @@ template <typename Flavor_, size_t NUM_> struct ProverInstances_ {
   public:
     static_assert(NUM_ > 0, "Must have at least one prover instance");
     using Flavor = Flavor_;
-    using FoldingParameters = typename Flavor::FoldingParameters;
     using FF = typename Flavor::FF;
     static constexpr size_t NUM = NUM_;
     using Instance = ProverInstance_<Flavor>;
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
index a433c0d1863..616c5f5ae21 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
@@ -22,7 +22,6 @@ template <class Flavor> class ProverInstance_ {
     using VerificationKey = typename Flavor::VerificationKey;
     using CommitmentKey = typename Flavor::CommitmentKey;
     using FF = typename Flavor::FF;
-    using FoldingParameters = typename Flavor::FoldingParameters;
     using ProverPolynomials = typename Flavor::ProverPolynomials;
     using Polynomial = typename Flavor::Polynomial;
     using WitnessCommitments = typename Flavor::WitnessCommitments;
@@ -48,9 +47,13 @@ template <class Flavor> class ProverInstance_ {
     FF alpha;
     proof_system::RelationParameters<FF> relation_parameters;
     std::vector<uint32_t> recursive_proof_public_input_indices;
-    // non-empty for the accumulated instances
-    FoldingParameters folding_parameters;
-    uint32_t is_accumulator = 0;
+
+    bool is_accumulator = false;
+
+    // The folding parameters (\vec{β}, e) which are set for accumulators (i.e. relaxed instances).
+    std::vector<FF> gate_challenges;
+    FF target_sum;
+
     size_t instance_size;
     size_t log_instance_size;
 
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
index 2ce29717e6f..39ce8371c80 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
@@ -7,7 +7,6 @@ template <class Flavor> class VerifierInstance_ {
   public:
     using FF = typename Flavor::FF;
     using VerificationKey = typename Flavor::VerificationKey;
-    using FoldingParameters = typename Flavor::FoldingParameters;
     using WitnessCommitments = typename Flavor::WitnessCommitments;
     using CommitmentLabels = typename Flavor::CommitmentLabels;
 
@@ -19,8 +18,12 @@ template <class Flavor> class VerifierInstance_ {
     size_t log_instance_size;
     RelationParameters<FF> relation_parameters;
     FF alpha;
-    uint32_t is_accumulator = 0;
-    FoldingParameters folding_parameters;
+    bool is_accumulator = false;
+
+    // The folding parameters (\vec{β}, e) which are set for accumulators (i.e. relaxed instances).
+    std::vector<FF> gate_challenges;
+    FF target_sum;
+
     WitnessCommitments witness_commitments;
     CommitmentLabels commitment_labels;
 };
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
index 17c0f75a4c8..5d873b0d688 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
@@ -17,10 +17,10 @@ template <typename Flavor> class SumcheckProver {
     using Transcript = typename Flavor::Transcript;
     using Instance = ProverInstance_<Flavor>;
 
-    std::shared_ptr<Transcript> transcript;
-
     const size_t multivariate_n;
     const size_t multivariate_d;
+
+    std::shared_ptr<Transcript> transcript;
     SumcheckProverRound<Flavor> round;
 
     /**
@@ -58,12 +58,23 @@ template <typename Flavor> class SumcheckProver {
 
     // prover instantiates sumcheck with circuit size and a prover transcript
     SumcheckProver(size_t multivariate_n, const std::shared_ptr<Transcript>& transcript)
-        : transcript(transcript)
-        , multivariate_n(multivariate_n)
+        : multivariate_n(multivariate_n)
         , multivariate_d(numeric::get_msb(multivariate_n))
+        , transcript(transcript)
         , round(multivariate_n, 2)
         , partially_evaluated_polynomials(multivariate_n){};
 
+    // WORKTODO delete this
+    /**
+     * @brief Compute univariate restriction place in transcript, generate challenge, partially evaluate,... repeat
+     * until final round, then compute multivariate evaluations and place in transcript.
+     */
+    SumcheckOutput<Flavor> prove(std::shared_ptr<Instance> instance)
+    {
+        return prove(
+            instance->prover_polynomials, instance->relation_parameters, instance->alpha, instance->gate_challenges);
+    };
+
     /**
      * @brief Compute univariate restriction place in transcript, generate challenge, partially evaluate,... repeat
      * until final round, then compute multivariate evaluations and place in transcript.
@@ -72,16 +83,11 @@ template <typename Flavor> class SumcheckProver {
      */
     SumcheckOutput<Flavor> prove(ProverPolynomials full_polynomials,
                                  const proof_system::RelationParameters<FF>& relation_parameters,
-                                 FF alpha) // pass by value, not by reference
+                                 const FF alpha,
+                                 const std::vector<FF>& gate_challenges)
     {
-        std::vector<FF> betas(multivariate_d, 0);
-        size_t beta_idx = 0;
-        for (auto& beta : betas) {
-            beta = transcript->get_challenge("Sumcheck:beta_" + std::to_string(beta_idx));
-            beta_idx++;
-        }
 
-        barretenberg::PowPolynomial<FF> pow_univariate(betas);
+        barretenberg::PowPolynomial<FF> pow_univariate(gate_challenges);
 
         std::vector<FF> multivariate_challenge;
         multivariate_challenge.reserve(multivariate_d);
@@ -123,15 +129,6 @@ template <typename Flavor> class SumcheckProver {
         return { multivariate_challenge, multivariate_evaluations };
     };
 
-    /**
-     * @brief Compute univariate restriction place in transcript, generate challenge, partially evaluate,... repeat
-     * until final round, then compute multivariate evaluations and place in transcript.
-     */
-    SumcheckOutput<Flavor> prove(std::shared_ptr<Instance> instance)
-    {
-        return prove(instance->prover_polynomials, instance->relation_parameters, instance->alpha);
-    };
-
     /**
      * @brief Evaluate at the round challenge and prepare class for next round.
      * Illustration of layout in example of first round when d==3 (showing just one Honk polynomial,
@@ -187,17 +184,20 @@ template <typename Flavor> class SumcheckVerifier {
     static constexpr size_t NUM_POLYNOMIALS = Flavor::NUM_ALL_ENTITIES;
 
     const size_t multivariate_d;
+    FF targegt_sum = 0;
+    std::shared_ptr<Transcript> transcript;
     SumcheckVerifierRound<Flavor> round;
 
     // verifier instantiates sumcheck with circuit size
-    explicit SumcheckVerifier(size_t multivariate_n)
-        : multivariate_d(numeric::get_msb(multivariate_n))
-        , round(){};
+    explicit SumcheckVerifier(size_t multivariate_d, std::shared_ptr<Transcript> transcript, FF target_sum = 0)
+        : multivariate_d(multivariate_d)
+        , transcript(transcript)
+        , round(target_sum){};
 
     /**
-     * @brief Extract round univariate, check sum, generate challenge, compute next target sum..., repeat until final
-     * round, then use purported evaluations to generate purported full Honk relation value and check against final
-     * target sum.
+     * @brief Extract round univariate, check sum, generate challenge, compute next target sum..., repeat until
+     * final round, then use purported evaluations to generate purported full Honk relation value and check against
+     * final target sum.
      *
      * @details If verification fails, returns std::nullopt, otherwise returns SumcheckOutput
      * @param relation_parameters
@@ -205,18 +205,11 @@ template <typename Flavor> class SumcheckVerifier {
      */
     SumcheckOutput<Flavor> verify(const proof_system::RelationParameters<FF>& relation_parameters,
                                   FF alpha,
-                                  const std::shared_ptr<Transcript>& transcript)
+                                  const std::vector<FF>& gate_challenges)
     {
         bool verified(true);
 
-        auto betas = std::vector<FF>(multivariate_d, 0);
-        size_t beta_idx = 0;
-        for (auto& beta : betas) {
-            beta = transcript->get_challenge("Sumcheck:beta_" + std::to_string(beta_idx));
-            beta_idx++;
-        }
-
-        barretenberg::PowPolynomial<FF> pow_univariate(betas);
+        barretenberg::PowPolynomial<FF> pow_univariate(gate_challenges);
         // All but final round.
         // target_total_sum is initialized to zero then mutated in place.
 
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp
index 2eaa93341a8..b58b19c7b18 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp
@@ -108,7 +108,12 @@ TEST_F(SumcheckTests, PolynomialNormalization)
 
     auto sumcheck = SumcheckProver<Flavor>(multivariate_n, transcript);
     FF alpha = transcript->get_challenge("alpha");
-    auto output = sumcheck.prove(full_polynomials, {}, alpha);
+    std::vector<FF> gate_challenges(multivariate_d);
+    for (size_t idx = 0; idx < multivariate_d; idx++) {
+        gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
+    }
+    info("here");
+    auto output = sumcheck.prove(full_polynomials, {}, alpha, gate_challenges);
 
     FF u_0 = output.challenge[0];
     FF u_1 = output.challenge[1];
@@ -172,7 +177,11 @@ TEST_F(SumcheckTests, Prover)
     auto sumcheck = SumcheckProver<Flavor>(multivariate_n, transcript);
 
     FF alpha = transcript->get_challenge("alpha");
-    auto output = sumcheck.prove(full_polynomials, {}, alpha);
+    std::vector<FF> gate_challenges(multivariate_d);
+    for (size_t idx = 0; idx < gate_challenges.size(); idx++) {
+        gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
+    }
+    auto output = sumcheck.prove(full_polynomials, {}, alpha, gate_challenges);
     FF u_0 = output.challenge[0];
     FF u_1 = output.challenge[1];
     std::vector<FF> expected_values;
@@ -247,13 +256,23 @@ TEST_F(SumcheckTests, ProverAndVerifierSimple)
         auto sumcheck_prover = SumcheckProver<Flavor>(multivariate_n, prover_transcript);
 
         FF prover_alpha = prover_transcript->get_challenge("alpha");
-        auto output = sumcheck_prover.prove(full_polynomials, {}, prover_alpha);
+        std::vector<FF> prover_gate_challenges(multivariate_d);
+        for (size_t idx = 0; idx < multivariate_d; idx++) {
+            prover_gate_challenges[idx] =
+                prover_transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
+        }
+        auto output = sumcheck_prover.prove(full_polynomials, {}, prover_alpha, prover_gate_challenges);
 
         auto verifier_transcript = Flavor::Transcript::verifier_init_empty(prover_transcript);
 
-        auto sumcheck_verifier = SumcheckVerifier<Flavor>(multivariate_n);
+        auto sumcheck_verifier = SumcheckVerifier<Flavor>(multivariate_d, verifier_transcript);
         FF verifier_alpha = verifier_transcript->get_challenge("alpha");
-        auto verifier_output = sumcheck_verifier.verify(relation_parameters, verifier_alpha, verifier_transcript);
+        std::vector<FF> verifier_gate_challenges(multivariate_d);
+        for (size_t idx = 0; idx < multivariate_d; idx++) {
+            verifier_gate_challenges[idx] =
+                verifier_transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
+        }
+        auto verifier_output = sumcheck_verifier.verify(relation_parameters, verifier_alpha, verifier_gate_challenges);
 
         auto verified = verifier_output.verified.value();
 
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
index 8f3f0e72da2..339a0d0a18d 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
@@ -285,7 +285,11 @@ template <typename Flavor> class SumcheckVerifierRound {
     TupleOfArraysOfValues relation_evaluations;
 
     // Verifier constructor
-    explicit SumcheckVerifierRound() { Utils::zero_elements(relation_evaluations); };
+    explicit SumcheckVerifierRound(FF target_total_sum = 0)
+        : target_total_sum(target_total_sum)
+    {
+        Utils::zero_elements(relation_evaluations);
+    };
 
     bool check_sum(barretenberg::Univariate<FF, BATCHED_RELATION_PARTIAL_LENGTH>& univariate)
     {
diff --git a/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_prover.cpp b/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_prover.cpp
index 6626632ee2b..4ac37285d5c 100644
--- a/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_prover.cpp
@@ -337,9 +337,12 @@ void GoblinTranslatorProver::execute_relation_check_rounds()
     using Sumcheck = sumcheck::SumcheckProver<Flavor>;
 
     auto sumcheck = Sumcheck(key->circuit_size, transcript);
-
     FF alpha = transcript->get_challenge("alpha");
-    sumcheck_output = sumcheck.prove(prover_polynomials, relation_parameters, alpha);
+    std::vector<FF> gate_challenges(numeric::get_msb(key->circuit_size));
+    for (size_t idx = 0; idx < gate_challenges.size(); idx++) {
+        gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
+    }
+    sumcheck_output = sumcheck.prove(prover_polynomials, relation_parameters, alpha, gate_challenges);
 }
 
 /**
diff --git a/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_verifier.cpp b/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_verifier.cpp
index e090efe7c4b..c2ecf1dec68 100644
--- a/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_verifier.cpp
@@ -243,11 +243,16 @@ bool GoblinTranslatorVerifier::verify_proof(const plonk::proof& proof)
     commitments.z_perm = receive_commitment(commitment_labels.z_perm);
 
     // Execute Sumcheck Verifier
-    auto sumcheck = SumcheckVerifier<Flavor>(circuit_size);
-
+    const size_t log_circuit_size = numeric::get_msb(circuit_size);
+    auto sumcheck = SumcheckVerifier<Flavor>(log_circuit_size, transcript);
     FF alpha = transcript->get_challenge("alpha");
+    std::vector<FF> gate_challenges(numeric::get_msb(key->circuit_size));
+    for (size_t idx = 0; idx < gate_challenges.size(); idx++) {
+        gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
+    }
+
     auto [multivariate_challenge, claimed_evaluations, sumcheck_verified] =
-        sumcheck.verify(relation_parameters, alpha, transcript);
+        sumcheck.verify(relation_parameters, alpha, gate_challenges);
 
     // If Sumcheck did not verify, return false
     if (sumcheck_verified.has_value() && !sumcheck_verified.value()) {
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/goblin_ultra_transcript.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/goblin_ultra_transcript.test.cpp
index 89044843a74..af57281bdd8 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/goblin_ultra_transcript.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/goblin_ultra_transcript.test.cpp
@@ -68,7 +68,7 @@ class GoblinUltraTranscriptTests : public ::testing::Test {
 
         for (size_t i = 0; i < log_n; i++) {
             round++;
-            std::string label = "Sumcheck:beta_" + std::to_string(i);
+            std::string label = "Sumcheck:gate_challenge_" + std::to_string(i);
             manifest_expected.add_challenge(round, label);
         }
 
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
index 2746c5410f0..86ee52976be 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
@@ -185,7 +185,8 @@ TEST_F(ProtoGalaxyTests, PerturbatorPolynomial)
 
     auto accumulator = std::make_shared<Instance>();
     accumulator->prover_polynomials = full_polynomials;
-    accumulator->folding_parameters = { betas, target_sum };
+    accumulator->gate_challenges = betas;
+    accumulator->target_sum = target_sum;
     accumulator->relation_parameters = relation_parameters;
     accumulator->alpha = alpha;
 
@@ -308,7 +309,8 @@ TEST_F(ProtoGalaxyTests, ComputeNewAccumulator)
     accumulator->instance_size = instance_size;
     accumulator->log_instance_size = log_instance_size;
     accumulator->prover_polynomials = full_polynomials;
-    accumulator->folding_parameters = { betas, target_sum };
+    accumulator->gate_challenges = betas;
+    accumulator->target_sum = target_sum;
     accumulator->relation_parameters = relation_parameters;
     accumulator->alpha = alpha;
     accumulator->is_accumulator = 1;
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/sumcheck.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/sumcheck.test.cpp
index c0a1c6b1f61..2f8aad66079 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/sumcheck.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/sumcheck.test.cpp
@@ -162,16 +162,27 @@ TEST_F(SumcheckTestsRealCircuit, Ultra)
 
     auto prover_transcript = Transcript::prover_init_empty();
     auto circuit_size = instance->proving_key->circuit_size;
+    auto log_circuit_size = numeric::get_msb(circuit_size);
     instance->alpha = prover_transcript->get_challenge("alpha");
     auto sumcheck_prover = SumcheckProver<Flavor>(circuit_size, prover_transcript);
-
+    std::vector<FF> prover_gate_challenges(log_circuit_size);
+    for (size_t idx = 0; idx < log_circuit_size; idx++) {
+        prover_gate_challenges[idx] =
+            prover_transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
+    }
+    instance->gate_challenges = prover_gate_challenges;
     auto prover_output = sumcheck_prover.prove(instance);
 
     auto verifier_transcript = Transcript::verifier_init_empty(prover_transcript);
 
-    auto sumcheck_verifier = SumcheckVerifier<Flavor>(circuit_size);
+    auto sumcheck_verifier = SumcheckVerifier<Flavor>(log_circuit_size, verifier_transcript);
     FF alpha = verifier_transcript->get_challenge("alpha");
-    auto verifier_output = sumcheck_verifier.verify(instance->relation_parameters, alpha, verifier_transcript);
+    std::vector<FF> verifier_gate_challenges(log_circuit_size);
+    for (size_t idx = 0; idx < log_circuit_size; idx++) {
+        verifier_gate_challenges[idx] =
+            verifier_transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
+    }
+    auto verifier_output = sumcheck_verifier.verify(instance->relation_parameters, alpha, verifier_gate_challenges);
 
     auto verified = verifier_output.verified.value();
 
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp
index eb95099ef4c..c57453d1932 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp
@@ -147,9 +147,14 @@ template <UltraFlavor Flavor> void UltraProver_<Flavor>::execute_grand_product_c
 template <UltraFlavor Flavor> void UltraProver_<Flavor>::execute_relation_check_rounds()
 {
     using Sumcheck = sumcheck::SumcheckProver<Flavor>;
-
-    auto sumcheck = Sumcheck(instance->proving_key->circuit_size, transcript);
+    auto circuit_size = instance->proving_key->circuit_size;
+    auto sumcheck = Sumcheck(circuit_size, transcript);
     instance->alpha = transcript->get_challenge("alpha");
+    std::vector<FF> gate_challenges(numeric::get_msb(circuit_size));
+    for (size_t idx = 0; idx < gate_challenges.size(); idx++) {
+        gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
+    }
+    instance->gate_challenges = gate_challenges;
     sumcheck_output = sumcheck.prove(instance);
 }
 
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_transcript.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_transcript.test.cpp
index 38cef0e5a1d..4d642f18893 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_transcript.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_transcript.test.cpp
@@ -61,7 +61,7 @@ class UltraTranscriptTests : public ::testing::Test {
 
         for (size_t i = 0; i < log_n; i++) {
             round++;
-            std::string label = "Sumcheck:beta_" + std::to_string(i);
+            std::string label = "Sumcheck:gate_challenge_" + std::to_string(i);
             manifest_expected.add_challenge(round, label);
         }
 
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp
index f6900dd2a53..47abdb900f5 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp
@@ -117,10 +117,15 @@ template <typename Flavor> bool UltraVerifier_<Flavor>::verify_proof(const plonk
     commitments.z_lookup = transcript->template receive_from_prover<Commitment>(commitment_labels.z_lookup);
 
     // Execute Sumcheck Verifier
-    auto sumcheck = SumcheckVerifier<Flavor>(circuit_size);
+    const size_t log_circuit_size = numeric::get_msb(circuit_size);
+    auto sumcheck = SumcheckVerifier<Flavor>(log_circuit_size, transcript);
     FF alpha = transcript->get_challenge("alpha");
+    auto gate_challenges = std::vector<FF>(log_circuit_size);
+    for (size_t idx = 0; idx < log_circuit_size; idx++) {
+        gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
+    }
     auto [multivariate_challenge, claimed_evaluations, sumcheck_verified] =
-        sumcheck.verify(relation_parameters, alpha, transcript);
+        sumcheck.verify(relation_parameters, alpha, gate_challenges);
 
     // If Sumcheck did not verify, return false
     if (sumcheck_verified.has_value() && !sumcheck_verified.value()) {
diff --git a/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_prover.cpp b/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_prover.cpp
index 696511fa104..79b3b7ea450 100644
--- a/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_prover.cpp
@@ -100,9 +100,12 @@ void AvmMiniProver::execute_relation_check_rounds()
     using Sumcheck = sumcheck::SumcheckProver<Flavor>;
 
     auto sumcheck = Sumcheck(key->circuit_size, transcript);
-    auto alpha = transcript->get_challenge("alpha");
-
-    sumcheck_output = sumcheck.prove(prover_polynomials, relation_parameters, alpha);
+    FF alpha = transcript->get_challenge("alpha");
+    std::vector<FF> gate_challenges(numeric::get_msb(key->circuit_size));
+    for (size_t idx = 0; idx < gate_challenges.size(); idx++) {
+        gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
+    }
+    sumcheck_output = sumcheck.prove(prover_polynomials, relation_parameters, alpha, gate_challenges);
 }
 
 /**
diff --git a/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_verifier.cpp b/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_verifier.cpp
index deab7e7c55d..fba5a203a1a 100644
--- a/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_verifier.cpp
@@ -85,11 +85,15 @@ bool AvmMiniVerifier::verify_proof(const plonk::proof& proof)
     commitments.avmMini_last = transcript->template receive_from_prover<Commitment>(commitment_labels.avmMini_last);
 
     // Execute Sumcheck Verifier
-    auto sumcheck = SumcheckVerifier<Flavor>(circuit_size);
-
-    auto alpha = transcript->get_challenge("alpha");
+    const size_t log_circuit_size = numeric::get_msb(circuit_size);
+    auto sumcheck = SumcheckVerifier<Flavor>(log_circuit_size, transcript);
+    FF alpha = transcript->get_challenge("alpha");
+    auto gate_challenges = std::vector<FF>(log_circuit_size);
+    for (size_t idx = 0; idx < log_circuit_size; idx++) {
+        gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
+    }
     auto [multivariate_challenge, claimed_evaluations, sumcheck_verified] =
-        sumcheck.verify(relation_parameters, alpha, transcript);
+        sumcheck.verify(relation_parameters, alpha, gate_challenges);
 
     // If Sumcheck did not verify, return false
     if (sumcheck_verified.has_value() && !sumcheck_verified.value()) {

From 98a7b6cfa7232a9bf009f948efb76232b6df0a41 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Tue, 12 Dec 2023 14:07:38 +0000
Subject: [PATCH 52/86] wip wip

---
 .../protogalaxy/decider_prover.cpp            | 118 ++++++++++++++++++
 .../protogalaxy/decider_prover.hpp            |  62 +++++++++
 .../protogalaxy/decider_verifier.cpp          |  98 +++++++++++++++
 .../protogalaxy/decider_verifier.hpp          |  32 +++++
 .../protogalaxy/protogalaxy_prover.cpp        |   1 +
 .../sumcheck/instance/prover_instance.hpp     |   1 +
 6 files changed, 312 insertions(+)
 create mode 100644 barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
 create mode 100644 barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.hpp
 create mode 100644 barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
 create mode 100644 barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.hpp

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
new file mode 100644
index 00000000000..69777063090
--- /dev/null
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
@@ -0,0 +1,118 @@
+#include "decider_prover.hpp"
+#include "barretenberg/sumcheck/sumcheck.hpp"
+
+namespace proof_system::honk {
+
+/**
+ * Create DeciderProver_ from an accumulator.
+ *
+ * @param accumulator Instance whose proof we want to generate.
+ *
+ * @tparam a type of UltraFlavor
+ * */
+template <UltraFlavor Flavor>
+DeciderProver_<Flavor>::DeciderProver_(const std::shared_ptr<Instance>& inst,
+                                       const std::shared_ptr<CommitmentKey>& commitment_key,
+                                       const std::shared_ptr<Transcript>& transcript)
+    : accumulator(std::move(inst))
+    , transcript(transcript)
+    , commitment_key(commitment_key)
+{}
+
+/**
+ * @brief Add circuit size, public input size, and public inputs to transcript
+ *
+ */
+template <UltraFlavor Flavor> void DeciderProver_<Flavor>::execute_preamble_round()
+{
+    const auto accumulator_size = static_cast<uint32_t>(accumulator->instance_size);
+    const auto num_public_inputs = static_cast<uint32_t>(accumulator->public_inputs.size());
+    transcript->send_to_verifier("accumulator_size", accumulator_size);
+    transcript->send_to_verifier("public_input_size", num_public_inputs);
+
+    for (size_t i = 0; i < accumulator->public_inputs.size(); ++i) {
+        auto public_input_i = accumulator->public_inputs[i];
+        transcript->send_to_verifier("public_input_" + std::to_string(i), public_input_i);
+    }
+
+    transcript->send_to_verifier("eta", accumulator->relation_parameters.eta);
+    transcript->send_to_verifier("beta", accumulator->relation_parameters.beta);
+    transcript->send_to_verifier("gamma", accumulator->relation_parameters.gamma);
+    transcript->send_to_verifier("public_input_delta", accumulator->relation_parameters.public_input_delta);
+    transcript->send_to_verifier("lookup_grand_product_delta",
+                                 accumulator->relation_parameters.lookup_grand_product_delta);
+
+    transcript->send_to_verifier("alpha", accumulator->alpha);
+
+    transcript->send_to_verifier("target_sum", accumulator->target_sum);
+    for (size_t idx = 0; idx < accumulator->gate_challenges.size(); idx++) {
+        transcript->send_to_verifier("gate_challenge_" + std::to_string(idx), accumulator->gate_challenges[idx]);
+    }
+
+    auto comm_view = accumulator->witness_commitments.get_all();
+    auto witness_labels = accumulator->commitment_labels.get_witness();
+    for (size_t idx = 0; idx < witness_labels.size(); idx++) {
+        transcript->send_to_verifier(witness_labels[idx], comm_view[idx]);
+    }
+
+    auto vk_view = accumulator->verification_key->get_all();
+    auto vk_labels = accumulator->commitment_labels.get_precomputed();
+    for (size_t idx = 0; idx < vk_labels.size(); idx++) {
+        transcript->send_to_verifier(vk_labels[idx], vk_view[idx]);
+    }
+}
+
+/**
+ * @brief Run Sumcheck resulting in u = (u_1,...,u_d) challenges and all evaluations at u being calculated.
+ *
+ */
+template <UltraFlavor Flavor> void DeciderProver_<Flavor>::execute_relation_check_rounds()
+{
+    using Sumcheck = sumcheck::SumcheckProver<Flavor>;
+    auto circuit_size = accumulator->proving_key->circuit_size;
+    auto sumcheck = Sumcheck(circuit_size, transcript);
+    sumcheck_output = sumcheck.prove(accumulator);
+}
+
+/**
+ * @brief Execute the ZeroMorph protocol to prove the multilinear evaluations produced by Sumcheck
+ * @details See https://hackmd.io/dlf9xEwhTQyE3hiGbq4FsA?view for a complete description of the unrolled protocol.
+ *
+ * */
+template <UltraFlavor Flavor> void DeciderProver_<Flavor>::execute_zeromorph_rounds()
+{
+    ZeroMorph::prove(accumulator->prover_polynomials.get_unshifted(),
+                     accumulator->prover_polynomials.get_to_be_shifted(),
+                     sumcheck_output.claimed_evaluations.get_unshifted(),
+                     sumcheck_output.claimed_evaluations.get_shifted(),
+                     sumcheck_output.challenge,
+                     commitment_key,
+                     transcript);
+}
+
+template <UltraFlavor Flavor> plonk::proof& DeciderProver_<Flavor>::export_proof()
+{
+    proof.proof_data = transcript->proof_data;
+    return proof;
+}
+
+template <UltraFlavor Flavor> plonk::proof& DeciderProver_<Flavor>::construct_proof()
+{
+    // Add circuit size public input size and public inputs to transcript->
+    execute_preamble_round();
+
+    // Fiat-Shamir: alpha
+    // Run sumcheck subprotocol.
+    execute_relation_check_rounds();
+
+    // Fiat-Shamir: rho, y, x, z
+    // Execute Zeromorph multilinear PCS
+    execute_zeromorph_rounds();
+
+    return export_proof();
+}
+
+template class DeciderProver_<honk::flavor::Ultra>;
+template class DeciderProver_<honk::flavor::GoblinUltra>;
+
+} // namespace proof_system::honk
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.hpp
new file mode 100644
index 00000000000..fd60707292d
--- /dev/null
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.hpp
@@ -0,0 +1,62 @@
+#pragma once
+#include "barretenberg/commitment_schemes/zeromorph/zeromorph.hpp"
+#include "barretenberg/flavor/goblin_ultra.hpp"
+#include "barretenberg/flavor/ultra.hpp"
+#include "barretenberg/plonk/proof_system/types/proof.hpp"
+#include "barretenberg/relations/relation_parameters.hpp"
+#include "barretenberg/sumcheck/instance/prover_instance.hpp"
+#include "barretenberg/sumcheck/sumcheck_output.hpp"
+#include "barretenberg/transcript/transcript.hpp"
+
+namespace proof_system::honk {
+
+template <UltraFlavor Flavor> class DeciderProver_ {
+    using FF = typename Flavor::FF;
+    using Commitment = typename Flavor::Commitment;
+    using CommitmentKey = typename Flavor::CommitmentKey;
+    using ProvingKey = typename Flavor::ProvingKey;
+    using Polynomial = typename Flavor::Polynomial;
+    using ProverPolynomials = typename Flavor::ProverPolynomials;
+    using CommitmentLabels = typename Flavor::CommitmentLabels;
+    using Curve = typename Flavor::Curve;
+    using Instance = ProverInstance_<Flavor>;
+    using Transcript = typename Flavor::Transcript;
+
+  public:
+    explicit DeciderProver_(const std::shared_ptr<Instance>&,
+                            const std::shared_ptr<CommitmentKey>&,
+                            const std::shared_ptr<Transcript>& transcript = std::make_shared<Transcript>());
+
+    BBERG_PROFILE void execute_preamble_round();
+    BBERG_PROFILE void execute_relation_check_rounds();
+    BBERG_PROFILE void execute_zeromorph_rounds();
+
+    plonk::proof& export_proof();
+    plonk::proof& construct_proof();
+
+    std::shared_ptr<Instance> accumulator;
+
+    std::shared_ptr<Transcript> transcript;
+
+    proof_system::RelationParameters<FF> relation_parameters;
+
+    CommitmentLabels commitment_labels;
+
+    Polynomial quotient_W;
+
+    sumcheck::SumcheckOutput<Flavor> sumcheck_output;
+
+    std::shared_ptr<CommitmentKey> commitment_key;
+
+    using ZeroMorph = pcs::zeromorph::ZeroMorphProver_<Curve>;
+
+  private:
+    plonk::proof proof;
+};
+
+extern template class DeciderProver_<honk::flavor::Ultra>;
+extern template class DeciderProver_<honk::flavor::GoblinUltra>;
+
+using DeciderProver = DeciderProver_<honk::flavor::Ultra>;
+
+} // namespace proof_system::honk
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
new file mode 100644
index 00000000000..98313616c3e
--- /dev/null
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
@@ -0,0 +1,98 @@
+#include "decider_verifier.hpp"
+#include "barretenberg/commitment_schemes/zeromorph/zeromorph.hpp"
+#include "barretenberg/numeric/bitop/get_msb.hpp"
+#include "barretenberg/sumcheck/instance/verifier_instance.hpp"
+#include "barretenberg/transcript/transcript.hpp"
+
+using namespace barretenberg;
+using namespace proof_system::honk::sumcheck;
+
+namespace proof_system::honk {
+
+template <typename Flavor>
+DeciderVerifier_<Flavor>::DeciderVerifier_()
+    : pcs_verification_key(std::make_unique<VerifierCommitmentKey>(0, barretenberg::srs::get_crs_factory()))
+    , transcript(std::make_shared<Transcript>())
+{}
+
+/**
+ * @brief This function verifies an Ultra Honk proof for a given Flavor.
+ *
+ */
+template <typename Flavor> bool DeciderVerifier_<Flavor>::verify_proof(const plonk::proof& proof)
+{
+    using FF = typename Flavor::FF;
+    using Commitment = typename Flavor::Commitment;
+    using Curve = typename Flavor::Curve;
+    using ZeroMorph = pcs::zeromorph::ZeroMorphVerifier_<Curve>;
+    using Instance = VerifierInstance_<Flavor>;
+
+    transcript = std::make_shared<Transcript>(proof.proof_data);
+    auto inst = std::make_unique<Instance>();
+
+    inst->instance_size = transcript->template receive_from_prover<uint32_t>("instance_size");
+    inst->log_instance_size = static_cast<size_t>(numeric::get_msb(inst->instance_size));
+    inst->public_input_size = transcript->template receive_from_prover<uint32_t>("public_input_size");
+
+    for (size_t i = 0; i < inst->public_input_size; ++i) {
+        auto public_input_i = transcript->template receive_from_prover<FF>("public_input_" + std::to_string(i));
+        inst->public_inputs.emplace_back(public_input_i);
+    }
+
+    auto eta = transcript->template receive_from_prover<FF>("eta");
+    auto beta = transcript->template receive_from_prover<FF>("beta");
+    auto gamma = transcript->template receive_from_prover<FF>("gamma");
+    auto public_input_delta = transcript->template receive_from_prover<FF>("public_input_delta");
+    auto lookup_grand_product_delta = transcript->template receive_from_prover<FF>("lookup_grand_product_delta");
+    inst->relation_parameters =
+        RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
+    inst->alpha = transcript->template receive_from_prover<FF>("alpha");
+
+    inst->target_sum = transcript->template receive_from_prover<FF>("target_sum");
+
+    inst->gate_challenges = std::vector<FF>(inst->log_instance_size);
+    for (size_t idx = 0; idx < inst->log_instance_size; idx++) {
+        inst->gate_challenges[idx] =
+            transcript->template receive_from_prover<FF>("gate_challenge_" + std::to_string(idx));
+    }
+    auto comm_view = inst->witness_commitments.get_all();
+    auto witness_labels = inst->commitment_labels.get_witness();
+    for (size_t idx = 0; idx < witness_labels.size(); idx++) {
+        comm_view[idx] = transcript->template receive_from_prover<Commitment>(witness_labels[idx]);
+    }
+
+    inst->verification_key = std::make_shared<VerificationKey>(inst->instance_size, inst->public_input_size);
+    auto vk_view = inst->verification_key->get_all();
+    auto vk_labels = inst->commitment_labels.get_precomputed();
+    for (size_t idx = 0; idx < vk_labels.size(); idx++) {
+        vk_view[idx] = transcript->template receive_from_prover<Commitment>(vk_labels[idx]);
+    }
+
+    auto sumcheck = SumcheckVerifier<Flavor>(inst->log_instance_size, transcript);
+
+    auto [multivariate_challenge, claimed_evaluations, sumcheck_verified] =
+        sumcheck.verify(inst->relation_parameters, inst->alpha, inst->gate_challenges);
+
+    // If Sumcheck did not verify, return false
+    if (sumcheck_verified.has_value() && !sumcheck_verified.value()) {
+        return false;
+    }
+
+    // Execute ZeroMorph rounds. See https://hackmd.io/dlf9xEwhTQyE3hiGbq4FsA?view for a complete description of the
+    // unrolled protocol.
+    auto pairing_points = ZeroMorph::verify(inst->commitments.get_unshifted(),
+                                            inst->commitments.get_to_be_shifted(),
+                                            claimed_evaluations.get_unshifted(),
+                                            claimed_evaluations.get_shifted(),
+                                            multivariate_challenge,
+                                            transcript);
+
+    auto verified = pcs_verification_key->pairing_check(pairing_points[0], pairing_points[1]);
+
+    return sumcheck_verified.value() && verified;
+}
+
+template class DeciderVerifier_<honk::flavor::Ultra>;
+template class DeciderVerifier_<honk::flavor::GoblinUltra>;
+
+} // namespace proof_system::honk
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.hpp
new file mode 100644
index 00000000000..46d7a3514e8
--- /dev/null
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.hpp
@@ -0,0 +1,32 @@
+#pragma once
+#include "barretenberg/flavor/goblin_ultra.hpp"
+#include "barretenberg/flavor/ultra.hpp"
+#include "barretenberg/plonk/proof_system/types/proof.hpp"
+#include "barretenberg/srs/global_crs.hpp"
+#include "barretenberg/sumcheck/sumcheck.hpp"
+
+namespace proof_system::honk {
+template <typename Flavor> class DeciderVerifier_ {
+    using FF = typename Flavor::FF;
+    using Commitment = typename Flavor::Commitment;
+    using VerificationKey = typename Flavor::VerificationKey;
+    using VerifierCommitmentKey = typename Flavor::VerifierCommitmentKey;
+    using Transcript = typename Flavor::Transcript;
+
+  public:
+    explicit DeciderVerifier_();
+
+    bool verify_proof(const plonk::proof& proof);
+
+    std::shared_ptr<VerificationKey> key;
+    std::map<std::string, Commitment> commitments;
+    std::shared_ptr<VerifierCommitmentKey> pcs_verification_key;
+    std::shared_ptr<Transcript> transcript;
+};
+
+extern template class DeciderVerifier_<honk::flavor::Ultra>;
+extern template class DeciderVerifier_<honk::flavor::GoblinUltra>;
+
+using DeciderVerifier = DeciderVerifier_<honk::flavor::Ultra>;
+
+} // namespace proof_system::honk
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index 5dcca5d0ea3..221330e07e1 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -213,6 +213,7 @@ std::shared_ptr<typename ProverInstances::Instance> ProtoGalaxyProver_<ProverIns
         prover_polynomial_pointers[poly_idx] = polynomial;
         poly_idx++;
     }
+    // WORKTODO return storage
 
     // Fold the prover polynomials
     auto acc_poly_views = acc_prover_polynomials.get_all();
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
index 616c5f5ae21..67a005cb593 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
@@ -35,6 +35,7 @@ template <class Flavor> class ProverInstance_ {
     WitnessCommitments witness_commitments;
     CommitmentLabels commitment_labels;
 
+    // This is meh here
     std::array<Polynomial, 4> sorted_polynomials;
 
     // The number of public inputs has to be the same for all instances because they are

From 9b49bab7e921c2e6cf8d7ecbdde4a8805ef51be2 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Wed, 13 Dec 2023 11:38:55 +0000
Subject: [PATCH 53/86] doesn't work :(

---
 .../cpp/src/barretenberg/flavor/ultra.hpp     |  5 +++
 .../protogalaxy/decider_prover.cpp            |  8 ++--
 .../protogalaxy/decider_verifier.cpp          | 37 ++++++++++++++-----
 .../protogalaxy/decider_verifier.hpp          |  2 +
 .../protogalaxy/folding_result.hpp            |  2 +
 .../protogalaxy/protogalaxy_prover.cpp        | 11 ++++--
 .../protogalaxy/protogalaxy_prover.hpp        |  2 +-
 .../verifier/ultra_recursive_verifier.cpp     |  2 -
 .../ultra_honk/protogalaxy.test.cpp           |  9 +++++
 .../ultra_honk/ultra_composer.cpp             | 21 +++++++++++
 .../ultra_honk/ultra_composer.hpp             | 10 +++++
 11 files changed, 89 insertions(+), 20 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
index 12adf196713..74374130907 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
@@ -386,6 +386,11 @@ class Ultra {
         };
     };
 
+    /**
+     * @brief A container encapsulating all the commitments that the verifier receives (to precomputed polynomials and
+     * witness polynomials).
+     *
+     */
     class VerifierCommitments : public AllEntities<Commitment> {
       public:
         VerifierCommitments(const std::shared_ptr<VerificationKey>& verification_key)
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
index 69777063090..3e2033117e0 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
@@ -49,6 +49,8 @@ template <UltraFlavor Flavor> void DeciderProver_<Flavor>::execute_preamble_roun
         transcript->send_to_verifier("gate_challenge_" + std::to_string(idx), accumulator->gate_challenges[idx]);
     }
 
+    // send as verifier commitments
+    // should protogalaxy work on verifier commitments?
     auto comm_view = accumulator->witness_commitments.get_all();
     auto witness_labels = accumulator->commitment_labels.get_witness();
     for (size_t idx = 0; idx < witness_labels.size(); idx++) {
@@ -69,8 +71,8 @@ template <UltraFlavor Flavor> void DeciderProver_<Flavor>::execute_preamble_roun
 template <UltraFlavor Flavor> void DeciderProver_<Flavor>::execute_relation_check_rounds()
 {
     using Sumcheck = sumcheck::SumcheckProver<Flavor>;
-    auto circuit_size = accumulator->proving_key->circuit_size;
-    auto sumcheck = Sumcheck(circuit_size, transcript);
+    auto instance_size = accumulator->instance_size;
+    auto sumcheck = Sumcheck(instance_size, transcript);
     sumcheck_output = sumcheck.prove(accumulator);
 }
 
@@ -107,7 +109,7 @@ template <UltraFlavor Flavor> plonk::proof& DeciderProver_<Flavor>::construct_pr
 
     // Fiat-Shamir: rho, y, x, z
     // Execute Zeromorph multilinear PCS
-    execute_zeromorph_rounds();
+    // execute_zeromorph_rounds();
 
     return export_proof();
 }
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
index 98313616c3e..a5324047913 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
@@ -9,6 +9,12 @@ using namespace proof_system::honk::sumcheck;
 
 namespace proof_system::honk {
 
+template <typename Flavor>
+DeciderVerifier_<Flavor>::DeciderVerifier_(const std::shared_ptr<Transcript>& transcript,
+                                           const std::shared_ptr<VerificationKey>& verifier_key)
+    : key(verifier_key)
+    , transcript(transcript)
+{}
 template <typename Flavor>
 DeciderVerifier_<Flavor>::DeciderVerifier_()
     : pcs_verification_key(std::make_unique<VerifierCommitmentKey>(0, barretenberg::srs::get_crs_factory()))
@@ -23,9 +29,10 @@ template <typename Flavor> bool DeciderVerifier_<Flavor>::verify_proof(const plo
 {
     using FF = typename Flavor::FF;
     using Commitment = typename Flavor::Commitment;
-    using Curve = typename Flavor::Curve;
-    using ZeroMorph = pcs::zeromorph::ZeroMorphVerifier_<Curve>;
+    // using Curve = typename Flavor::Curve;
+    // using ZeroMorph = pcs::zeromorph::ZeroMorphVerifier_<Curve>;
     using Instance = VerifierInstance_<Flavor>;
+    using VerifierCommitments = typename Flavor::VerifierCommitments;
 
     transcript = std::make_shared<Transcript>(proof.proof_data);
     auto inst = std::make_unique<Instance>();
@@ -68,6 +75,16 @@ template <typename Flavor> bool DeciderVerifier_<Flavor>::verify_proof(const plo
         vk_view[idx] = transcript->template receive_from_prover<Commitment>(vk_labels[idx]);
     }
 
+    VerifierCommitments commitments{ inst->verification_key };
+    auto& witness_commitments = inst->witness_commitments;
+    commitments.w_l = witness_commitments.w_l;
+    commitments.w_r = witness_commitments.w_r;
+    commitments.w_o = witness_commitments.w_o;
+    commitments.sorted_accum = witness_commitments.sorted_accum;
+    commitments.w_4 = witness_commitments.w_4;
+    commitments.z_perm = witness_commitments.z_perm;
+    commitments.z_lookup = witness_commitments.z_lookup;
+
     auto sumcheck = SumcheckVerifier<Flavor>(inst->log_instance_size, transcript);
 
     auto [multivariate_challenge, claimed_evaluations, sumcheck_verified] =
@@ -80,16 +97,16 @@ template <typename Flavor> bool DeciderVerifier_<Flavor>::verify_proof(const plo
 
     // Execute ZeroMorph rounds. See https://hackmd.io/dlf9xEwhTQyE3hiGbq4FsA?view for a complete description of the
     // unrolled protocol.
-    auto pairing_points = ZeroMorph::verify(inst->commitments.get_unshifted(),
-                                            inst->commitments.get_to_be_shifted(),
-                                            claimed_evaluations.get_unshifted(),
-                                            claimed_evaluations.get_shifted(),
-                                            multivariate_challenge,
-                                            transcript);
+    // auto pairing_points = ZeroMorph::verify(commitments.get_unshifted(),
+    //                                         commitments.get_to_be_shifted(),
+    //                                         claimed_evaluations.get_unshifted(),
+    //                                         claimed_evaluations.get_shifted(),
+    //                                         multivariate_challenge,
+    //                                         transcript);
 
-    auto verified = pcs_verification_key->pairing_check(pairing_points[0], pairing_points[1]);
+    // auto verified = pcs_verification_key->pairing_check(pairing_points[0], pairing_points[1]);
 
-    return sumcheck_verified.value() && verified;
+    return sumcheck_verified.value(); // && verified;
 }
 
 template class DeciderVerifier_<honk::flavor::Ultra>;
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.hpp
index 46d7a3514e8..eb2e15a5ce0 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.hpp
@@ -15,6 +15,8 @@ template <typename Flavor> class DeciderVerifier_ {
 
   public:
     explicit DeciderVerifier_();
+    explicit DeciderVerifier_(const std::shared_ptr<Transcript>& transcript,
+                              const std::shared_ptr<VerificationKey>& verifier_key = nullptr);
 
     bool verify_proof(const plonk::proof& proof);
 
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/folding_result.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/folding_result.hpp
index 171f9b38a78..3447bcab83d 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/folding_result.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/folding_result.hpp
@@ -11,7 +11,9 @@ namespace proof_system::honk {
  */
 template <class Flavor> struct FoldingResult {
   public:
+    using AllPolynomials = typename Flavor::AllPolynomials;
     std::shared_ptr<ProverInstance_<Flavor>> accumulator;
+    AllPolynomials storage;
     // TODO(https://github.com/AztecProtocol/barretenberg/issues/656): turn folding data into a struct
     std::vector<uint8_t> folding_data;
 };
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index 221330e07e1..e0a56ccdaf7 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -165,14 +165,16 @@ FoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverInstanc
     auto combiner_challenge = transcript->get_challenge("combiner_quotient_challenge");
 
     FoldingResult<Flavor> res;
-    res.accumulator =
+    auto [next_accumulator, storage] =
         compute_next_accumulator(instances, combiner_quotient, combiner_challenge, compressed_perturbator);
     res.folding_data = transcript->proof_data;
-
+    res.accumulator = next_accumulator;
+    res.storage = storage;
     return res;
 }
 template <class ProverInstances>
-std::shared_ptr<typename ProverInstances::Instance> ProtoGalaxyProver_<ProverInstances>::compute_next_accumulator(
+std::pair<std::shared_ptr<typename ProverInstances::Instance>, typename ProverInstances::Flavor::AllPolynomials>
+ProtoGalaxyProver_<ProverInstances>::compute_next_accumulator(
     ProverInstances& instances,
     Univariate<FF, ProverInstances::BATCHED_EXTENDED_LENGTH, ProverInstances::NUM>& combiner_quotient,
     const FF& challenge,
@@ -187,6 +189,7 @@ std::shared_ptr<typename ProverInstances::Instance> ProtoGalaxyProver_<ProverIns
     std::vector<FF> lagranges{ FF(1) - challenge, challenge };
 
     auto next_accumulator = std::make_shared<Instance>();
+    next_accumulator->instance_size = instances[0]->instance_size;
 
     // Compute the next target sum and send the next folding parameters to the verifier
     FF next_target_sum =
@@ -299,7 +302,7 @@ std::shared_ptr<typename ProverInstances::Instance> ProtoGalaxyProver_<ProverIns
     }
     next_accumulator->verification_key = acc_vk;
 
-    return next_accumulator;
+    return std::pair(next_accumulator, std::move(storage));
 }
 
 template class ProtoGalaxyProver_<ProverInstances_<honk::flavor::Ultra, 2>>;
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index d87346fcd01..61f77a73160 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -442,7 +442,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      *
      * TODO(https://github.com/AztecProtocol/barretenberg/issues/796): optimise the construction of the new accumulator
      */
-    std::shared_ptr<Instance> compute_next_accumulator(
+    std::pair<std::shared_ptr<Instance>, AllPolynomials> compute_next_accumulator(
         ProverInstances& instances,
         Univariate<FF, ProverInstances::BATCHED_EXTENDED_LENGTH, ProverInstances::NUM>& combiner_quotient,
         const FF& challenge,
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp
index 3154a456e31..ea75d4a1c5f 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp
@@ -109,10 +109,8 @@ std::array<typename Flavor::GroupElement, 2> UltraRecursiveVerifier_<Flavor>::ve
     for (size_t idx = 0; idx < log_circuit_size; idx++) {
         gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
     }
-    info("here");
     auto [multivariate_challenge, claimed_evaluations, sumcheck_verified] =
         sumcheck.verify(relation_parameters, alpha, gate_challenges);
-    info("out of sumcheck");
     // Execute ZeroMorph multilinear PCS evaluation verifier
     auto pairing_points = ZeroMorph::verify(commitments.get_unshifted(),
                                             commitments.get_to_be_shifted(),
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
index 23d598e0bda..e2dca6ea55c 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
@@ -329,8 +329,17 @@ TEST_F(ProtoGalaxyTests, ComputeNewAccumulator)
     auto folding_verifier = composer.create_folding_verifier();
 
     auto proof = folding_prover.fold_instances();
+    auto next_accumulator = proof.accumulator;
+    auto storage = proof.storage;
     auto res = folding_verifier.verify_folding_proof(proof.folding_data);
     EXPECT_EQ(res, true);
+    info("folding done");
+
+    auto decider_prover = composer.create_decider_prover(next_accumulator);
+    auto decider_verifier = composer.create_decider_verifier(next_accumulator);
+    auto decision = decider_prover.construct_proof();
+    auto verified = decider_verifier.verify_proof(decision);
+    EXPECT_EQ(verified, true);
 }
 
 } // namespace protogalaxy_tests
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.cpp
index 7f318aa98a3..e2536c4c08a 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.cpp
@@ -97,6 +97,27 @@ UltraVerifier_<Flavor> UltraComposer_<Flavor>::create_verifier(const std::shared
     return output_state;
 }
 
+template <UltraFlavor Flavor>
+DeciderProver_<Flavor> UltraComposer_<Flavor>::create_decider_prover(const std::shared_ptr<Instance>& accumulator,
+                                                                     const std::shared_ptr<Transcript>& transcript)
+{
+    DeciderProver_<Flavor> output_state(accumulator, commitment_key, transcript);
+
+    return output_state;
+}
+
+template <UltraFlavor Flavor>
+DeciderVerifier_<Flavor> UltraComposer_<Flavor>::create_decider_verifier(const std::shared_ptr<Instance>& accumulator,
+                                                                         const std::shared_ptr<Transcript>& transcript)
+{
+    auto& verification_key = accumulator->verification_key;
+    DeciderVerifier_<Flavor> output_state(transcript, verification_key);
+    auto pcs_verification_key = std::make_unique<VerifierCommitmentKey>(verification_key->circuit_size, crs_factory_);
+    output_state.pcs_verification_key = std::move(pcs_verification_key);
+
+    return output_state;
+}
+
 template class UltraComposer_<honk::flavor::Ultra>;
 template class UltraComposer_<honk::flavor::GoblinUltra>;
 } // namespace proof_system::honk
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp
index 7daaab2e1d3..aaa45f3ef28 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp
@@ -1,6 +1,8 @@
 #pragma once
 #include "barretenberg/flavor/flavor.hpp"
 #include "barretenberg/proof_system/composer/composer_lib.hpp"
+#include "barretenberg/protogalaxy/decider_prover.hpp"
+#include "barretenberg/protogalaxy/decider_verifier.hpp"
 #include "barretenberg/protogalaxy/protogalaxy_prover.hpp"
 #include "barretenberg/protogalaxy/protogalaxy_verifier.hpp"
 #include "barretenberg/srs/global_crs.hpp"
@@ -63,6 +65,14 @@ template <UltraFlavor Flavor> class UltraComposer_ {
         const std::shared_ptr<Instance>&,
         const std::shared_ptr<Transcript>& transcript = std::make_shared<Transcript>());
 
+    DeciderProver_<Flavor> create_decider_prover(
+        const std::shared_ptr<Instance>&,
+        const std::shared_ptr<Transcript>& transcript = std::make_shared<Transcript>());
+
+    DeciderVerifier_<Flavor> create_decider_verifier(
+        const std::shared_ptr<Instance>&,
+        const std::shared_ptr<Transcript>& transcript = std::make_shared<Transcript>());
+
     /**
      * @brief Create Prover for Goblin ECC op queue merge protocol
      *

From 81845c99e42f4ceb951f350ae801e28557f555c8 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Thu, 14 Dec 2023 13:10:00 +0000
Subject: [PATCH 54/86] shit

---
 .../protogalaxy/decider_prover.cpp            |   3 +-
 .../protogalaxy/decider_verifier.cpp          |  29 ++--
 .../protogalaxy/protogalaxy_prover.cpp        |   9 +-
 .../protogalaxy/protogalaxy_prover.hpp        |   5 +-
 .../protogalaxy/protogalaxy_verifier.hpp      |   5 +-
 .../src/barretenberg/sumcheck/sumcheck.hpp    |   4 +
 .../ultra_honk/protogalaxy.test.cpp           | 129 ++++++++++++++++--
 .../ultra_honk/ultra_composer.cpp             |  14 +-
 .../ultra_honk/ultra_composer.hpp             |   4 +
 9 files changed, 168 insertions(+), 34 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
index 3e2033117e0..3548f7244af 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
@@ -54,6 +54,7 @@ template <UltraFlavor Flavor> void DeciderProver_<Flavor>::execute_preamble_roun
     auto comm_view = accumulator->witness_commitments.get_all();
     auto witness_labels = accumulator->commitment_labels.get_witness();
     for (size_t idx = 0; idx < witness_labels.size(); idx++) {
+        info(comm_view[idx]);
         transcript->send_to_verifier(witness_labels[idx], comm_view[idx]);
     }
 
@@ -109,7 +110,7 @@ template <UltraFlavor Flavor> plonk::proof& DeciderProver_<Flavor>::construct_pr
 
     // Fiat-Shamir: rho, y, x, z
     // Execute Zeromorph multilinear PCS
-    // execute_zeromorph_rounds();
+    execute_zeromorph_rounds();
 
     return export_proof();
 }
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
index a5324047913..62a7fd2081e 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
@@ -29,8 +29,8 @@ template <typename Flavor> bool DeciderVerifier_<Flavor>::verify_proof(const plo
 {
     using FF = typename Flavor::FF;
     using Commitment = typename Flavor::Commitment;
-    // using Curve = typename Flavor::Curve;
-    // using ZeroMorph = pcs::zeromorph::ZeroMorphVerifier_<Curve>;
+    using Curve = typename Flavor::Curve;
+    using ZeroMorph = pcs::zeromorph::ZeroMorphVerifier_<Curve>;
     using Instance = VerifierInstance_<Flavor>;
     using VerifierCommitments = typename Flavor::VerifierCommitments;
 
@@ -66,6 +66,7 @@ template <typename Flavor> bool DeciderVerifier_<Flavor>::verify_proof(const plo
     auto witness_labels = inst->commitment_labels.get_witness();
     for (size_t idx = 0; idx < witness_labels.size(); idx++) {
         comm_view[idx] = transcript->template receive_from_prover<Commitment>(witness_labels[idx]);
+        info(comm_view[idx]);
     }
 
     inst->verification_key = std::make_shared<VerificationKey>(inst->instance_size, inst->public_input_size);
@@ -84,8 +85,9 @@ template <typename Flavor> bool DeciderVerifier_<Flavor>::verify_proof(const plo
     commitments.w_4 = witness_commitments.w_4;
     commitments.z_perm = witness_commitments.z_perm;
     commitments.z_lookup = witness_commitments.z_lookup;
+    info(commitments.z_lookup);
 
-    auto sumcheck = SumcheckVerifier<Flavor>(inst->log_instance_size, transcript);
+    auto sumcheck = SumcheckVerifier<Flavor>(inst->log_instance_size, transcript, inst->target_sum);
 
     auto [multivariate_challenge, claimed_evaluations, sumcheck_verified] =
         sumcheck.verify(inst->relation_parameters, inst->alpha, inst->gate_challenges);
@@ -95,18 +97,19 @@ template <typename Flavor> bool DeciderVerifier_<Flavor>::verify_proof(const plo
         return false;
     }
 
+    info("here");
     // Execute ZeroMorph rounds. See https://hackmd.io/dlf9xEwhTQyE3hiGbq4FsA?view for a complete description of the
     // unrolled protocol.
-    // auto pairing_points = ZeroMorph::verify(commitments.get_unshifted(),
-    //                                         commitments.get_to_be_shifted(),
-    //                                         claimed_evaluations.get_unshifted(),
-    //                                         claimed_evaluations.get_shifted(),
-    //                                         multivariate_challenge,
-    //                                         transcript);
-
-    // auto verified = pcs_verification_key->pairing_check(pairing_points[0], pairing_points[1]);
-
-    return sumcheck_verified.value(); // && verified;
+    auto pairing_points = ZeroMorph::verify(commitments.get_unshifted(),
+                                            commitments.get_to_be_shifted(),
+                                            claimed_evaluations.get_unshifted(),
+                                            claimed_evaluations.get_shifted(),
+                                            multivariate_challenge,
+                                            transcript);
+
+    auto verified = pcs_verification_key->pairing_check(pairing_points[0], pairing_points[1]);
+    info(verified);
+    return sumcheck_verified.value() && verified;
 }
 
 template class DeciderVerifier_<honk::flavor::Ultra>;
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index e0a56ccdaf7..70a76fe5a66 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -169,7 +169,7 @@ FoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverInstanc
         compute_next_accumulator(instances, combiner_quotient, combiner_challenge, compressed_perturbator);
     res.folding_data = transcript->proof_data;
     res.accumulator = next_accumulator;
-    res.storage = storage;
+    res.storage = std::move(storage);
     return res;
 }
 template <class ProverInstances>
@@ -190,6 +190,7 @@ ProtoGalaxyProver_<ProverInstances>::compute_next_accumulator(
 
     auto next_accumulator = std::make_shared<Instance>();
     next_accumulator->instance_size = instances[0]->instance_size;
+    next_accumulator->log_instance_size = instances[0]->log_instance_size;
 
     // Compute the next target sum and send the next folding parameters to the verifier
     FF next_target_sum =
@@ -222,13 +223,17 @@ ProtoGalaxyProver_<ProverInstances>::compute_next_accumulator(
     auto acc_poly_views = acc_prover_polynomials.get_all();
     for (size_t inst_idx = 0; inst_idx < ProverInstances::NUM; inst_idx++) {
         auto inst_poly_views = instances[inst_idx]->prover_polynomials.get_all();
+        info(inst_poly_views[0][1]);
         for (auto [acc_poly_view, inst_poly_view] : zip_view(acc_poly_views, inst_poly_views)) {
             for (size_t poly_idx = 0; poly_idx < inst_poly_view.size(); poly_idx++) {
-                (acc_poly_view)[poly_idx] += (inst_poly_view)[poly_idx] * lagranges[inst_idx];
+                acc_poly_view[poly_idx] += inst_poly_view[poly_idx] * lagranges[inst_idx];
             }
         }
     }
     next_accumulator->prover_polynomials = acc_prover_polynomials;
+    info("look at this");
+    info(next_accumulator->prover_polynomials.get_all()[0][0]);
+    info(next_accumulator->prover_polynomials.get_all()[0][1]);
 
     // Fold the witness commtiments and send them to the verifier
     auto witness_labels = next_accumulator->commitment_labels.get_witness();
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index 61f77a73160..193a9bf7ac9 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -129,10 +129,9 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     {
         auto log_instance_size = gate_challenges.size();
         std::vector<FF> next_gate_challenges(log_instance_size);
-        next_gate_challenges[0] = 1;
 
-        for (size_t idx = 1; idx < log_instance_size; idx++) {
-            next_gate_challenges[idx] = gate_challenges[idx] + perturbator_challenge * round_challenges[idx - 1];
+        for (size_t idx = 0; idx < log_instance_size; idx++) {
+            next_gate_challenges[idx] = gate_challenges[idx] + perturbator_challenge * round_challenges[idx];
         }
         return next_gate_challenges;
     }
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
index c723532a5b9..fe4505cc348 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
@@ -47,10 +47,9 @@ template <class VerifierInstances> class ProtoGalaxyVerifier_ {
     {
         auto log_instance_size = gate_challenges.size();
         std::vector<FF> next_gate_challenges(log_instance_size);
-        next_gate_challenges[0] = 1;
 
-        for (size_t idx = 1; idx < log_instance_size; idx++) {
-            next_gate_challenges[idx] = gate_challenges[idx] + perturbator_challenge * round_challenges[idx - 1];
+        for (size_t idx = 0; idx < log_instance_size; idx++) {
+            next_gate_challenges[idx] = gate_challenges[idx] + perturbator_challenge * round_challenges[idx];
         }
         return next_gate_challenges;
     }
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
index 5d873b0d688..c97d892580b 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
@@ -71,6 +71,10 @@ template <typename Flavor> class SumcheckProver {
      */
     SumcheckOutput<Flavor> prove(std::shared_ptr<Instance> instance)
     {
+        info("look at this");
+        info(instance->prover_polynomials.get_all()[0][0]);
+        info(instance->prover_polynomials.get_all()[0][1]);
+
         return prove(
             instance->prover_polynomials, instance->relation_parameters, instance->alpha, instance->gate_challenges);
     };
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
index e2dca6ea55c..5afec51759d 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
@@ -282,16 +282,30 @@ TEST_F(ProtoGalaxyTests, ComputeNewAccumulator)
     const size_t log_instance_size(4);
     const size_t instance_size(1 << log_instance_size);
 
+    auto builder = typename Flavor::CircuitBuilder();
+    auto composer = UltraComposer();
+    builder.add_public_variable(FF(1));
+
+    auto instance = composer.create_instance(builder);
+    auto commitment_key = composer.commitment_key;
+
     std::array<barretenberg::Polynomial<FF>, NUM_POLYNOMIALS> random_polynomials;
     for (auto& poly : random_polynomials) {
         poly = get_random_polynomial(instance_size);
     }
-    auto full_polynomials = construct_ultra_full_polynomials(random_polynomials);
+    for (auto& poly : random_polynomials) {
+        poly[0] = FF::zero();
+    }
+    info(random_polynomials[0]);
+    ProverPolynomials polynomials;
+    auto poly_views = polynomials.get_all();
+    for (size_t idx = 0; idx < poly_views.size(); idx++) {
+        poly_views[idx] = random_polynomials[idx];
+    }
     auto relation_parameters = proof_system::RelationParameters<FF>::get_random();
     auto alpha = FF::random_element();
 
-    auto full_honk_evals =
-        ProtoGalaxyProver::compute_full_honk_evaluations(full_polynomials, alpha, relation_parameters);
+    auto full_honk_evals = ProtoGalaxyProver::compute_full_honk_evaluations(polynomials, alpha, relation_parameters);
     std::vector<FF> betas(log_instance_size);
     for (size_t idx = 0; idx < log_instance_size; idx++) {
         betas[idx] = FF::random_element();
@@ -307,23 +321,30 @@ TEST_F(ProtoGalaxyTests, ComputeNewAccumulator)
     }
 
     auto accumulator = std::make_shared<Instance>();
-    accumulator->witness_commitments = construct_witness_commitments();
+    accumulator->prover_polynomials = polynomials;
+    auto witness_poly = polynomials.get_witness();
+    size_t idx = 0;
+    for (auto& comm : accumulator->witness_commitments.get_all()) {
+        comm = commitment_key->commit(witness_poly[idx]);
+        idx++;
+    }
+
+    idx = 0;
+    auto selector_poly = polynomials.get_precomputed();
+    accumulator->verification_key = std::make_shared<typename Flavor::VerificationKey>(instance_size, 1);
+    for (auto& vk : accumulator->verification_key->get_all()) {
+        vk = commitment_key->commit(selector_poly[idx]);
+        idx++;
+    }
     accumulator->instance_size = instance_size;
     accumulator->log_instance_size = log_instance_size;
-    accumulator->prover_polynomials = full_polynomials;
     accumulator->gate_challenges = betas;
     accumulator->target_sum = target_sum;
     accumulator->relation_parameters = relation_parameters;
     accumulator->alpha = alpha;
     accumulator->is_accumulator = true;
     accumulator->public_inputs = std::vector<FF>{ FF::random_element() };
-    accumulator->verification_key = construct_ultra_verification_key(instance_size, 1);
-
-    auto builder = typename Flavor::CircuitBuilder();
-    auto composer = UltraComposer();
-    builder.add_public_variable(FF(1));
 
-    auto instance = composer.create_instance(builder);
     auto instances = std::vector<std::shared_ptr<Instance>>{ accumulator, instance };
     auto folding_prover = composer.create_folding_prover(instances, composer.commitment_key);
     auto folding_verifier = composer.create_folding_verifier();
@@ -335,6 +356,22 @@ TEST_F(ProtoGalaxyTests, ComputeNewAccumulator)
     EXPECT_EQ(res, true);
     info("folding done");
 
+    auto next_honk_evals = ProtoGalaxyProver::compute_full_honk_evaluations(
+        next_accumulator->prover_polynomials, next_accumulator->alpha, next_accumulator->relation_parameters);
+
+    // Construct pow(\vec{betas}) as in the paper
+    auto next_pow_beta =
+        ProtoGalaxyProver::compute_pow_polynomial_at_values(next_accumulator->gate_challenges, instance_size);
+
+    // Compute the corresponding target sum and create a dummy accumulator
+    auto next_target_sum = FF(0);
+    for (size_t i = 0; i < instance_size; i++) {
+        next_target_sum += next_honk_evals[i] * next_pow_beta[i];
+    }
+    info(next_target_sum);
+    info(next_accumulator->target_sum);
+    next_accumulator->target_sum = next_target_sum;
+
     auto decider_prover = composer.create_decider_prover(next_accumulator);
     auto decider_verifier = composer.create_decider_verifier(next_accumulator);
     auto decision = decider_prover.construct_proof();
@@ -342,4 +379,74 @@ TEST_F(ProtoGalaxyTests, ComputeNewAccumulator)
     EXPECT_EQ(verified, true);
 }
 
+TEST_F(ProtoGalaxyTests, DecideDummyAccumulator)
+{
+    const size_t log_instance_size(4);
+    const size_t instance_size(1 << log_instance_size);
+    auto composer = UltraComposer();
+    auto commitment_key = composer.compute_commitment_key(instance_size);
+
+    std::array<barretenberg::Polynomial<FF>, NUM_POLYNOMIALS> random_polynomials;
+    for (auto& poly : random_polynomials) {
+        poly = get_random_polynomial(instance_size);
+    }
+
+    for (auto& poly : random_polynomials) {
+        poly[0] = FF::zero();
+    }
+    ProverPolynomials polynomials;
+    auto poly_views = polynomials.get_all();
+    for (size_t idx = 0; idx < poly_views.size(); idx++) {
+        poly_views[idx] = random_polynomials[idx];
+    }
+    auto relation_parameters = proof_system::RelationParameters<FF>::get_random();
+    auto alpha = FF::random_element();
+
+    auto full_honk_evals = ProtoGalaxyProver::compute_full_honk_evaluations(polynomials, alpha, relation_parameters);
+    std::vector<FF> betas(log_instance_size);
+    for (size_t idx = 0; idx < log_instance_size; idx++) {
+        betas[idx] = FF::random_element();
+    }
+
+    // Construct pow(\vec{betas}) as in the paper
+    auto pow_beta = ProtoGalaxyProver::compute_pow_polynomial_at_values(betas, instance_size);
+
+    // Compute the corresponding target sum and create a dummy accumulator
+    auto target_sum = FF(0);
+    for (size_t i = 0; i < instance_size; i++) {
+        target_sum += full_honk_evals[i] * pow_beta[i];
+    }
+    auto accumulator = std::make_shared<Instance>();
+    accumulator->prover_polynomials = polynomials;
+    auto witness_poly = polynomials.get_witness();
+    size_t idx = 0;
+    for (auto& comm : accumulator->witness_commitments.get_all()) {
+        comm = commitment_key->commit(witness_poly[idx]);
+        idx++;
+    }
+
+    idx = 0;
+    auto selector_poly = polynomials.get_precomputed();
+    accumulator->verification_key = std::make_shared<typename Flavor::VerificationKey>(instance_size, 1);
+    for (auto& vk : accumulator->verification_key->get_all()) {
+        vk = commitment_key->commit(selector_poly[idx]);
+        idx++;
+    }
+
+    accumulator->instance_size = instance_size;
+    accumulator->log_instance_size = log_instance_size;
+    accumulator->gate_challenges = betas;
+    accumulator->target_sum = target_sum;
+    accumulator->relation_parameters = relation_parameters;
+    accumulator->alpha = alpha;
+    accumulator->is_accumulator = true;
+    accumulator->public_inputs = std::vector<FF>{ FF::random_element() };
+
+    auto decider_prover = composer.create_decider_prover(accumulator, commitment_key);
+    auto decider_verifier = composer.create_decider_verifier(accumulator);
+    auto decision = decider_prover.construct_proof();
+    auto verified = decider_verifier.verify_proof(decision);
+    EXPECT_EQ(verified, true);
+}
+
 } // namespace protogalaxy_tests
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.cpp
index e2536c4c08a..359c58b7949 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.cpp
@@ -100,6 +100,18 @@ UltraVerifier_<Flavor> UltraComposer_<Flavor>::create_verifier(const std::shared
 template <UltraFlavor Flavor>
 DeciderProver_<Flavor> UltraComposer_<Flavor>::create_decider_prover(const std::shared_ptr<Instance>& accumulator,
                                                                      const std::shared_ptr<Transcript>& transcript)
+{
+    commitment_key = compute_commitment_key(accumulator->instance_size);
+    DeciderProver_<Flavor> output_state(accumulator, commitment_key, transcript);
+
+    return output_state;
+}
+
+template <UltraFlavor Flavor>
+DeciderProver_<Flavor> UltraComposer_<Flavor>::create_decider_prover(
+    const std::shared_ptr<Instance>& accumulator,
+    const std::shared_ptr<CommitmentKey>& commitment_key,
+    const std::shared_ptr<Transcript>& transcript)
 {
     DeciderProver_<Flavor> output_state(accumulator, commitment_key, transcript);
 
@@ -112,7 +124,7 @@ DeciderVerifier_<Flavor> UltraComposer_<Flavor>::create_decider_verifier(const s
 {
     auto& verification_key = accumulator->verification_key;
     DeciderVerifier_<Flavor> output_state(transcript, verification_key);
-    auto pcs_verification_key = std::make_unique<VerifierCommitmentKey>(verification_key->circuit_size, crs_factory_);
+    auto pcs_verification_key = std::make_unique<VerifierCommitmentKey>(accumulator->instance_size, crs_factory_);
     output_state.pcs_verification_key = std::move(pcs_verification_key);
 
     return output_state;
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp
index aaa45f3ef28..d1f91542d61 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp
@@ -68,6 +68,10 @@ template <UltraFlavor Flavor> class UltraComposer_ {
     DeciderProver_<Flavor> create_decider_prover(
         const std::shared_ptr<Instance>&,
         const std::shared_ptr<Transcript>& transcript = std::make_shared<Transcript>());
+    DeciderProver_<Flavor> create_decider_prover(
+        const std::shared_ptr<Instance>&,
+        const std::shared_ptr<CommitmentKey>&,
+        const std::shared_ptr<Transcript>& transcript = std::make_shared<Transcript>());
 
     DeciderVerifier_<Flavor> create_decider_verifier(
         const std::shared_ptr<Instance>&,

From 4bce6e81102d5973eeff5d10de83590ef65929b2 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Fri, 15 Dec 2023 14:50:03 +0000
Subject: [PATCH 55/86] everything but beta

---
 .../protogalaxy/protogalaxy_prover.cpp        | 104 +++++++++---------
 .../protogalaxy/protogalaxy_prover.hpp        |  29 ++++-
 .../protogalaxy/protogalaxy_verifier.cpp      |   3 +-
 .../relations/relation_parameters.hpp         |   2 +-
 .../sumcheck/instance/instances.hpp           |   2 +-
 .../sumcheck/instance/prover_instance.cpp     |   2 -
 .../src/barretenberg/sumcheck/sumcheck.hpp    |   4 -
 .../ultra_honk/protogalaxy.test.cpp           |  62 ++++++++---
 8 files changed, 126 insertions(+), 82 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index 70a76fe5a66..c6bb5dc0abb 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -47,6 +47,7 @@ void ProtoGalaxyProver_<ProverInstances>::finalise_and_send_instance(std::shared
     transcript->send_to_verifier(domain_separator + "_" + commitment_labels.w_4, witness_commitments.w_4);
 
     auto [beta, gamma] = transcript->get_challenges(domain_separator + "_beta", domain_separator + "_gamma");
+    beta = FF(0);
     instance->compute_grand_product_polynomials(beta, gamma);
 
     witness_commitments.z_perm = commitment_key->commit(instance->prover_polynomials.z_perm);
@@ -58,7 +59,6 @@ void ProtoGalaxyProver_<ProverInstances>::finalise_and_send_instance(std::shared
                                  instance->witness_commitments.z_lookup);
 
     instance->alpha = transcript->get_challenge(domain_separator + "_alpha");
-
     auto vk_view = instance->verification_key->get_all();
     auto labels = instance->commitment_labels.get_precomputed();
     for (size_t idx = 0; idx < labels.size(); idx++) {
@@ -129,59 +129,15 @@ template <class ProverInstances> void ProtoGalaxyProver_<ProverInstances>::prepa
     }
 }
 
-// TODO(#https://github.com/AztecProtocol/barretenberg/issues/689): finalise implementation this function
-template <class ProverInstances>
-FoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverInstances>::fold_instances()
-{
-    prepare_for_folding();
-
-    // TODO(#https://github.com/AztecProtocol/barretenberg/issues/740): Handle the case where we are folding for the
-    // first time and accumulator is 0
-    FF delta = transcript->get_challenge("delta");
-    auto accumulator = get_accumulator();
-    auto deltas = compute_round_challenge_pows(accumulator->log_instance_size, delta);
-
-    auto perturbator = compute_perturbator(accumulator, deltas);
-    for (size_t idx = 0; idx <= accumulator->log_instance_size; idx++) {
-        transcript->send_to_verifier("perturbator_" + std::to_string(idx), perturbator[idx]);
-    }
-    assert(perturbator[0] == accumulator->target_sum);
-    auto perturbator_challenge = transcript->get_challenge("perturbator_challenge");
-    instances.next_gate_challenges =
-        update_gate_challenges(perturbator_challenge, accumulator->gate_challenges, deltas);
-    const auto pow_betas_star =
-        compute_pow_polynomial_at_values(instances.next_gate_challenges, accumulator->instance_size);
-
-    combine_relation_parameters(instances);
-    combine_alpha(instances);
-    auto combiner = compute_combiner(instances, pow_betas_star);
-
-    auto compressed_perturbator = perturbator.evaluate(perturbator_challenge);
-    auto combiner_quotient = compute_combiner_quotient(compressed_perturbator, combiner);
-
-    for (size_t idx = ProverInstances::NUM; idx < ProverInstances::BATCHED_EXTENDED_LENGTH; idx++) {
-        transcript->send_to_verifier("combiner_quotient_" + std::to_string(idx), combiner_quotient.value_at(idx));
-    }
-    auto combiner_challenge = transcript->get_challenge("combiner_quotient_challenge");
-
-    FoldingResult<Flavor> res;
-    auto [next_accumulator, storage] =
-        compute_next_accumulator(instances, combiner_quotient, combiner_challenge, compressed_perturbator);
-    res.folding_data = transcript->proof_data;
-    res.accumulator = next_accumulator;
-    res.storage = std::move(storage);
-    return res;
-}
 template <class ProverInstances>
 std::pair<std::shared_ptr<typename ProverInstances::Instance>, typename ProverInstances::Flavor::AllPolynomials>
 ProtoGalaxyProver_<ProverInstances>::compute_next_accumulator(
     ProverInstances& instances,
     Univariate<FF, ProverInstances::BATCHED_EXTENDED_LENGTH, ProverInstances::NUM>& combiner_quotient,
-    const FF& challenge,
+    FF& challenge,
     const FF& compressed_perturbator)
 {
     auto combiner_quotient_at_challenge = combiner_quotient.evaluate(challenge);
-
     // Given the challenge \gamma, compute Z(\gamma) and {L_0(\gamma),L_1(\gamma)}
     // TODO(https://github.com/AztecProtocol/barretenberg/issues/764): Generalize the vanishing polynomial formula
     // and the computation of Lagrange basis for k instances
@@ -195,6 +151,7 @@ ProtoGalaxyProver_<ProverInstances>::compute_next_accumulator(
     // Compute the next target sum and send the next folding parameters to the verifier
     FF next_target_sum =
         compressed_perturbator * lagranges[0] + vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
+
     transcript->send_to_verifier("next_target_sum", next_target_sum);
     for (size_t idx = 0; idx < instances.next_gate_challenges.size(); idx++) {
         transcript->send_to_verifier("next_gate_challenge_" + std::to_string(idx), instances.next_gate_challenges[idx]);
@@ -223,7 +180,6 @@ ProtoGalaxyProver_<ProverInstances>::compute_next_accumulator(
     auto acc_poly_views = acc_prover_polynomials.get_all();
     for (size_t inst_idx = 0; inst_idx < ProverInstances::NUM; inst_idx++) {
         auto inst_poly_views = instances[inst_idx]->prover_polynomials.get_all();
-        info(inst_poly_views[0][1]);
         for (auto [acc_poly_view, inst_poly_view] : zip_view(acc_poly_views, inst_poly_views)) {
             for (size_t poly_idx = 0; poly_idx < inst_poly_view.size(); poly_idx++) {
                 acc_poly_view[poly_idx] += inst_poly_view[poly_idx] * lagranges[inst_idx];
@@ -231,9 +187,6 @@ ProtoGalaxyProver_<ProverInstances>::compute_next_accumulator(
         }
     }
     next_accumulator->prover_polynomials = acc_prover_polynomials;
-    info("look at this");
-    info(next_accumulator->prover_polynomials.get_all()[0][0]);
-    info(next_accumulator->prover_polynomials.get_all()[0][1]);
 
     // Fold the witness commtiments and send them to the verifier
     auto witness_labels = next_accumulator->commitment_labels.get_witness();
@@ -274,7 +227,7 @@ ProtoGalaxyProver_<ProverInstances>::compute_next_accumulator(
     // the verifier
     auto& combined_relation_parameters = instances.relation_parameters;
     auto folded_relation_parameters = proof_system::RelationParameters<FF>{
-        combined_relation_parameters.eta.evaluate(challenge),
+        combined_relation_parameters.eta.evaluate(challenge), // WORKTODO change all of these to evaluate(challenge)
         combined_relation_parameters.beta.evaluate(challenge),
         combined_relation_parameters.gamma.evaluate(challenge),
         combined_relation_parameters.public_input_delta.evaluate(challenge),
@@ -310,6 +263,55 @@ ProtoGalaxyProver_<ProverInstances>::compute_next_accumulator(
     return std::pair(next_accumulator, std::move(storage));
 }
 
+// TODO(#https://github.com/AztecProtocol/barretenberg/issues/689): finalise implementation this function
+template <class ProverInstances>
+FoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverInstances>::fold_instances()
+{
+    prepare_for_folding();
+
+    // TODO(#https://github.com/AztecProtocol/barretenberg/issues/740): Handle the case where we are folding for the
+    // first time and accumulator is 0
+    FF delta = transcript->get_challenge("delta");
+
+    auto accumulator = get_accumulator();
+    auto deltas = compute_round_challenge_pows(accumulator->log_instance_size, delta);
+
+    auto perturbator = compute_perturbator(accumulator, deltas);
+    for (size_t idx = 0; idx <= accumulator->log_instance_size; idx++) {
+        transcript->send_to_verifier("perturbator_" + std::to_string(idx), perturbator[idx]);
+    }
+    assert(perturbator[0] == accumulator->target_sum);
+    auto perturbator_challenge = transcript->get_challenge("perturbator_challenge"); // alpha
+    info("F(alpha) ", perturbator.evaluate(perturbator_challenge));                  // F(\alpha)
+    instances.next_gate_challenges =
+        update_gate_challenges(perturbator_challenge, accumulator->gate_challenges, deltas);
+    const auto pow_betas_star =
+        compute_pow_polynomial_at_values(instances.next_gate_challenges, accumulator->instance_size);
+    combine_relation_parameters(instances);
+    combine_alpha(instances);
+    auto combiner = compute_combiner(instances, pow_betas_star);
+    info("G(0) ", combiner.evaluations[0]);
+
+    auto compressed_perturbator = perturbator.evaluate(perturbator_challenge);
+    auto combiner_quotient = compute_combiner_quotient(compressed_perturbator, combiner);
+
+    for (size_t idx = ProverInstances::NUM; idx < ProverInstances::BATCHED_EXTENDED_LENGTH; idx++) {
+        transcript->send_to_verifier("combiner_quotient_" + std::to_string(idx), combiner_quotient.value_at(idx));
+    }
+    FF combiner_challenge = transcript->get_challenge("combiner_quotient_challenge"); // gamma
+    info("G(gamma)", combiner.evaluate(combiner_challenge));
+    // auto combiner_challenge = 0; // gamma
+
+    FoldingResult<Flavor> res;
+    auto [next_accumulator, storage] =
+        compute_next_accumulator(instances, combiner_quotient, combiner_challenge, compressed_perturbator);
+    res.folding_data = transcript->proof_data;
+    res.accumulator = next_accumulator;
+
+    res.storage = std::move(storage);
+    return res;
+}
+
 template class ProtoGalaxyProver_<ProverInstances_<honk::flavor::Ultra, 2>>;
 template class ProtoGalaxyProver_<ProverInstances_<honk::flavor::GoblinUltra, 2>>;
 } // namespace proof_system::honk
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index 193a9bf7ac9..d96fed3a4a3 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -166,12 +166,31 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
 
             auto running_challenge = FF(1);
             auto output = FF(0);
-            Utils::scale_and_batch_elements(relation_evaluations, alpha, running_challenge, output);
+            scale_and_batch_elements(relation_evaluations, alpha, running_challenge, output);
+
             full_honk_evaluations[row] = output;
         }
         return full_honk_evaluations;
     }
 
+    /**
+     * @brief Scale elements by consecutive powers of the challenge then sum
+     * @param result Batched result
+     */
+    static void scale_and_batch_elements(auto& tuple, const FF& challenge, FF& current_scalar, FF& result)
+    {
+        auto scale_by_challenge_and_accumulate = [&](auto& element) {
+            for (auto& entry : element) {
+                result += entry * current_scalar;
+                if (current_scalar != challenge) {
+
+                    current_scalar *= challenge;
+                }
+            }
+        };
+        Utils::apply_to_tuple_of_arrays(scale_by_challenge_and_accumulate, tuple);
+    }
+
     /**
      * @brief  Recursively compute the parent nodes of each level in there, starting from the leaves. Note that at each
      * level, the resulting parent nodes will be polynomials of degree (level + 1) because we multiply by an additional
@@ -236,6 +255,8 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     static Polynomial<FF> compute_perturbator(const std::shared_ptr<Instance> accumulator,
                                               const std::vector<FF>& deltas)
     {
+        info("HERE");
+        info(accumulator->alpha);
         auto full_honk_evaluations = compute_full_honk_evaluations(
             accumulator->prover_polynomials, accumulator->alpha, accumulator->relation_parameters);
         const auto betas = accumulator->gate_challenges;
@@ -300,7 +321,6 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         size_t num_threads = std::min(desired_num_threads, max_num_threads); // fewer than max if justified
         num_threads = num_threads > 0 ? num_threads : 1;                     // ensure num threads is >= 1
         size_t iterations_per_thread = common_instance_size / num_threads;   // actual iterations per thread
-
         // Construct univariate accumulator containers; one per thread
         std::vector<TupleOfTuplesOfUnivariates> thread_univariate_accumulators(num_threads);
         for (auto& accum : thread_univariate_accumulators) {
@@ -318,6 +338,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
             size_t end = (thread_idx + 1) * iterations_per_thread;
 
             for (size_t idx = start; idx < end; idx++) {
+                // No need to initialise extended_univariates to 0, it's assigned to
                 extend_univariates(extended_univariates[thread_idx], instances, idx);
 
                 FF pow_challenge = pow_betas_star[idx];
@@ -327,7 +348,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
                 accumulate_relation_univariates(
                     thread_univariate_accumulators[thread_idx],
                     extended_univariates[thread_idx],
-                    instances.relation_parameters, // these parameters have already been folded
+                    instances.relation_parameters, // these para meters have already been folded
                     pow_challenge);
             }
         });
@@ -444,7 +465,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     std::pair<std::shared_ptr<Instance>, AllPolynomials> compute_next_accumulator(
         ProverInstances& instances,
         Univariate<FF, ProverInstances::BATCHED_EXTENDED_LENGTH, ProverInstances::NUM>& combiner_quotient,
-        const FF& challenge,
+        FF& challenge,
         const FF& compressed_perturbator);
 };
 
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index 4106d298ce0..54ee1002630 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -129,7 +129,8 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
 {
     prepare_for_folding(fold_data);
 
-    auto delta = transcript->get_challenge("delta");
+    // auto delta = transcript->get_challenge("delta");
+    FF delta = 0;
     auto accumulator = get_accumulator();
     auto deltas = compute_round_challenge_pows(accumulator->log_instance_size, delta);
 
diff --git a/barretenberg/cpp/src/barretenberg/relations/relation_parameters.hpp b/barretenberg/cpp/src/barretenberg/relations/relation_parameters.hpp
index 096f5b79603..d1905d407d7 100644
--- a/barretenberg/cpp/src/barretenberg/relations/relation_parameters.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/relation_parameters.hpp
@@ -46,9 +46,9 @@ template <typename T> struct RelationParameters {
     {
         RelationParameters result;
         result.eta = T::random_element();
+        result.beta = T::random_element();
         result.beta_sqr = result.beta * result.beta;
         result.beta_cube = result.beta_sqr * result.beta;
-        result.beta = T::random_element();
         result.gamma = T::random_element();
         result.public_input_delta = T::random_element();
         result.lookup_grand_product_delta = T::random_element();
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
index 8246c6104ae..f883bdf44c5 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
@@ -64,7 +64,7 @@ template <typename Flavor_, size_t NUM_> struct ProverInstances_ {
         for (auto& get_all : insts_prover_polynomials_views) {
             // Iterate over all columns in the trace execution of an instance and extract their value at row_idx.
             for (auto [result, poly_ptr] : zip_view(results, get_all)) {
-                result.evaluations[instance_idx] = (poly_ptr)[row_idx];
+                result.evaluations[instance_idx] = poly_ptr[row_idx];
             }
             instance_idx++;
         }
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.cpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.cpp
index 4e5a5d66123..2d891e8abef 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.cpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.cpp
@@ -464,11 +464,9 @@ template <class Flavor> void ProverInstance_<Flavor>::compute_grand_product_poly
 {
     auto public_input_delta =
         compute_public_input_delta<Flavor>(public_inputs, beta, gamma, proving_key->circuit_size, pub_inputs_offset);
-
     relation_parameters.beta = beta;
     relation_parameters.gamma = gamma;
     relation_parameters.public_input_delta = public_input_delta;
-
     auto lookup_grand_product_delta = compute_lookup_grand_product_delta(beta, gamma, proving_key->circuit_size);
     relation_parameters.lookup_grand_product_delta = lookup_grand_product_delta;
 
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
index c97d892580b..5d873b0d688 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
@@ -71,10 +71,6 @@ template <typename Flavor> class SumcheckProver {
      */
     SumcheckOutput<Flavor> prove(std::shared_ptr<Instance> instance)
     {
-        info("look at this");
-        info(instance->prover_polynomials.get_all()[0][0]);
-        info(instance->prover_polynomials.get_all()[0][1]);
-
         return prove(
             instance->prover_polynomials, instance->relation_parameters, instance->alpha, instance->gate_challenges);
     };
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
index 5afec51759d..a131c59cdb5 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
@@ -275,6 +275,29 @@ TEST_F(ProtoGalaxyTests, FoldAlpha)
     EXPECT_EQ(instances.alpha, expected_alpha);
 }
 
+TEST_F(ProtoGalaxyTests, FoldAlphaShouldBeZero)
+{
+    using Instances = ProverInstances_<Flavor, 2>;
+    using Instance = typename Instances::Instance;
+
+    Builder builder1;
+    auto instance1 = std::make_shared<Instance>(builder1);
+    instance1->alpha = FF(0);
+
+    Builder builder2;
+    builder2.add_variable(3);
+    auto instance2 = std::make_shared<Instance>(builder2);
+    instance2->alpha = FF(0);
+
+    Instances instances{ { instance1, instance2 } };
+    ProtoGalaxyProver::combine_alpha(instances);
+
+    Univariate<FF, 13> expected_alpha{ { 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 } };
+    auto challenge = FF::random_element();
+    EXPECT_EQ(instances.alpha, expected_alpha);
+    EXPECT_EQ(instances.alpha.evaluate(challenge), FF(0));
+}
+
 // TODO(https://github.com/AztecProtocol/barretenberg/issues/807): Have proper full folding testing (both failing and
 // passing) and move creating a test accumulator in a separate function.
 TEST_F(ProtoGalaxyTests, ComputeNewAccumulator)
@@ -296,13 +319,14 @@ TEST_F(ProtoGalaxyTests, ComputeNewAccumulator)
     for (auto& poly : random_polynomials) {
         poly[0] = FF::zero();
     }
-    info(random_polynomials[0]);
     ProverPolynomials polynomials;
     auto poly_views = polynomials.get_all();
     for (size_t idx = 0; idx < poly_views.size(); idx++) {
         poly_views[idx] = random_polynomials[idx];
     }
     auto relation_parameters = proof_system::RelationParameters<FF>::get_random();
+    relation_parameters.beta = FF(0);
+
     auto alpha = FF::random_element();
 
     auto full_honk_evals = ProtoGalaxyProver::compute_full_honk_evaluations(polynomials, alpha, relation_parameters);
@@ -313,7 +337,6 @@ TEST_F(ProtoGalaxyTests, ComputeNewAccumulator)
 
     // Construct pow(\vec{betas}) as in the paper
     auto pow_beta = ProtoGalaxyProver::compute_pow_polynomial_at_values(betas, instance_size);
-
     // Compute the corresponding target sum and create a dummy accumulator
     auto target_sum = FF(0);
     for (size_t i = 0; i < instance_size; i++) {
@@ -352,14 +375,13 @@ TEST_F(ProtoGalaxyTests, ComputeNewAccumulator)
     auto proof = folding_prover.fold_instances();
     auto next_accumulator = proof.accumulator;
     auto storage = proof.storage;
-    auto res = folding_verifier.verify_folding_proof(proof.folding_data);
-    EXPECT_EQ(res, true);
-    info("folding done");
-
+    // auto res = folding_verifier.verify_folding_proof(proof.folding_data);
+    // EXPECT_EQ(res, true);
+    // i got omega* \vec{\beta*} e*
+    // i want to check that \sum_i ∈ {0, n} pow_i(\vec{\beta*}) * f_i(\omega*) = e*
     auto next_honk_evals = ProtoGalaxyProver::compute_full_honk_evaluations(
         next_accumulator->prover_polynomials, next_accumulator->alpha, next_accumulator->relation_parameters);
-
-    // Construct pow(\vec{betas}) as in the paper
+    // Construct pow(\vec{betas*}) as in the paper
     auto next_pow_beta =
         ProtoGalaxyProver::compute_pow_polynomial_at_values(next_accumulator->gate_challenges, instance_size);
 
@@ -368,15 +390,19 @@ TEST_F(ProtoGalaxyTests, ComputeNewAccumulator)
     for (size_t i = 0; i < instance_size; i++) {
         next_target_sum += next_honk_evals[i] * next_pow_beta[i];
     }
-    info(next_target_sum);
-    info(next_accumulator->target_sum);
-    next_accumulator->target_sum = next_target_sum;
-
-    auto decider_prover = composer.create_decider_prover(next_accumulator);
-    auto decider_verifier = composer.create_decider_verifier(next_accumulator);
-    auto decision = decider_prover.construct_proof();
-    auto verified = decider_verifier.verify_proof(decision);
-    EXPECT_EQ(verified, true);
+    info("e* ", next_target_sum);
+
+    // for (size_t i = 0; i < log_instance_size; i++) {
+    //     EXPECT_EQ(accumulator->gate_challenges[i], next_accumulator->gate_challenges[i]);
+    // }
+    // EXPECT_EQ(next_pow_beta, pow_beta);
+
+    EXPECT_EQ(next_accumulator->target_sum, next_target_sum);
+    // auto decider_prover = composer.create_decider_prover(next_accumulator);
+    // auto decider_verifier = composer.create_decider_verifier(next_accumulator);
+    // auto decision = decider_prover.construct_proof();
+    // auto verified = decider_verifier.verify_proof(decision);
+    // EXPECT_EQ(verified, true);
 }
 
 TEST_F(ProtoGalaxyTests, DecideDummyAccumulator)
@@ -400,7 +426,7 @@ TEST_F(ProtoGalaxyTests, DecideDummyAccumulator)
         poly_views[idx] = random_polynomials[idx];
     }
     auto relation_parameters = proof_system::RelationParameters<FF>::get_random();
-    auto alpha = FF::random_element();
+    auto alpha = FF::zero();
 
     auto full_honk_evals = ProtoGalaxyProver::compute_full_honk_evaluations(polynomials, alpha, relation_parameters);
     std::vector<FF> betas(log_instance_size);

From 1cfabf9305aa4eaaa0d625aba5bc42464dc2efc0 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Mon, 18 Dec 2023 13:33:20 +0000
Subject: [PATCH 56/86] this works

---
 .../protogalaxy/protogalaxy_prover.cpp        | 24 +++++-
 .../protogalaxy/protogalaxy_prover.hpp        | 45 ++++++++---
 .../protogalaxy/protogalaxy_verifier.cpp      | 16 +++-
 .../relations/relation_parameters.hpp         |  7 +-
 .../cpp/src/barretenberg/relations/utils.hpp  |  8 +-
 .../ultra_honk/protogalaxy.test.cpp           | 81 ++++++++++++++++++-
 6 files changed, 157 insertions(+), 24 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index c6bb5dc0abb..58ae95603f9 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -47,7 +47,6 @@ void ProtoGalaxyProver_<ProverInstances>::finalise_and_send_instance(std::shared
     transcript->send_to_verifier(domain_separator + "_" + commitment_labels.w_4, witness_commitments.w_4);
 
     auto [beta, gamma] = transcript->get_challenges(domain_separator + "_beta", domain_separator + "_gamma");
-    beta = FF(0);
     instance->compute_grand_product_polynomials(beta, gamma);
 
     witness_commitments.z_perm = commitment_key->commit(instance->prover_polynomials.z_perm);
@@ -118,8 +117,21 @@ template <class ProverInstances> void ProtoGalaxyProver_<ProverInstances>::prepa
     if (instance->is_accumulator) {
         send_accumulator(instance, domain_separator);
     } else {
+        // This is the first round of folding and we need to generate some gate challenges.
+        // TODO(https://github.com/AztecProtocol/barretenberg/issues/740): implement option 2 to make this more
+        // efficient by avoiding the computation of the perturbator
         finalise_and_send_instance(instance, domain_separator);
+        instance->target_sum = 0;
+        info("log_instance_size ", instance->log_instance_size);
+        auto beta = transcript->get_challenge(domain_separator + "_initial_gate_challenge");
+        std::vector<FF> gate_challenges(instance->log_instance_size);
+        gate_challenges[0] = beta;
+        for (size_t i = 1; i < instance->log_instance_size; i++) {
+            gate_challenges[i] = gate_challenges[i - 1].sqr();
+        }
+        instance->gate_challenges = gate_challenges;
     }
+
     idx++;
 
     for (auto it = instances.begin() + 1; it != instances.end(); it++, idx++) {
@@ -241,7 +253,13 @@ ProtoGalaxyProver_<ProverInstances>::compute_next_accumulator(
     transcript->send_to_verifier("next_public_input_delta", folded_relation_parameters.public_input_delta);
     transcript->send_to_verifier("next_lookup_grand_product_delta",
                                  folded_relation_parameters.lookup_grand_product_delta);
-    next_accumulator->relation_parameters = folded_relation_parameters;
+    next_accumulator->relation_parameters = proof_system::RelationParameters<FF>{
+        combined_relation_parameters.eta.evaluate(challenge), // WORKTODO change all of these to evaluate(challenge)
+        combined_relation_parameters.beta.evaluate(challenge),
+        combined_relation_parameters.gamma.evaluate(challenge),
+        combined_relation_parameters.public_input_delta.evaluate(challenge),
+        combined_relation_parameters.lookup_grand_product_delta.evaluate(challenge),
+    };
 
     // Fold the verification key and send it to the verifier as this is part of ϕ as well
     auto acc_vk = std::make_shared<VerificationKey>(instances[0]->prover_polynomials.get_polynomial_size(),
@@ -299,8 +317,6 @@ FoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverInstanc
         transcript->send_to_verifier("combiner_quotient_" + std::to_string(idx), combiner_quotient.value_at(idx));
     }
     FF combiner_challenge = transcript->get_challenge("combiner_quotient_challenge"); // gamma
-    info("G(gamma)", combiner.evaluate(combiner_challenge));
-    // auto combiner_challenge = 0; // gamma
 
     FoldingResult<Flavor> res;
     auto [next_accumulator, storage] =
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index d96fed3a4a3..de905485087 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -417,17 +417,40 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     static void combine_relation_parameters(ProverInstances& instances)
     {
         // array of parameters to be computed
-        size_t param_idx = 0;
-        for (auto& folded_parameter : instances.relation_parameters.to_fold) {
-            Univariate<FF, ProverInstances::NUM> tmp(0);
-            size_t instance_idx = 0;
-            for (auto& instance : instances) {
-                tmp.value_at(instance_idx) = instance->relation_parameters.to_fold[param_idx].get();
-                instance_idx++;
-            }
-            folded_parameter.get() = tmp.template extend_to<ProverInstances::EXTENDED_LENGTH>();
-            param_idx++;
-        }
+        // size_t param_idx = 0;
+        // for (auto& folded_parameter : instances.relation_parameters.to_fold) {
+        //     Univariate<FF, ProverInstances::NUM> tmp(0);
+        //     size_t instance_idx = 0;
+        //     for (auto& instance : instances) {
+        //         tmp.value_at(instance_idx) = instance->relation_parameters.to_fold[param_idx];
+        //         instance_idx++;
+        //     }
+        //     folded_parameter = tmp.template extend_to<ProverInstances::EXTENDED_LENGTH>();
+        //     param_idx++;
+        // }
+        Univariate<FF, ProverInstances::NUM> eta_tmp(
+            std::array<FF, 2>{ instances[0]->relation_parameters.eta, instances[1]->relation_parameters.eta });
+        instances.relation_parameters.eta = eta_tmp.template extend_to<ProverInstances::EXTENDED_LENGTH>();
+
+        Univariate<FF, ProverInstances::NUM> beta_tmp(
+            std::array<FF, 2>{ instances[0]->relation_parameters.beta, instances[1]->relation_parameters.beta });
+        instances.relation_parameters.beta = beta_tmp.template extend_to<ProverInstances::EXTENDED_LENGTH>();
+
+        Univariate<FF, ProverInstances::NUM> gamma_tmp(
+            std::array<FF, 2>{ instances[0]->relation_parameters.gamma, instances[1]->relation_parameters.gamma });
+        instances.relation_parameters.gamma = gamma_tmp.template extend_to<ProverInstances::EXTENDED_LENGTH>();
+
+        Univariate<FF, ProverInstances::NUM> public_input_delta_tmp(
+            std::array<FF, 2>{ instances[0]->relation_parameters.public_input_delta,
+                               instances[1]->relation_parameters.public_input_delta });
+        instances.relation_parameters.public_input_delta =
+            public_input_delta_tmp.template extend_to<ProverInstances::EXTENDED_LENGTH>();
+
+        Univariate<FF, ProverInstances::NUM> lookup_grand_product_delta_tmp(
+            std::array<FF, 2>{ instances[0]->relation_parameters.lookup_grand_product_delta,
+                               instances[1]->relation_parameters.lookup_grand_product_delta });
+        instances.relation_parameters.lookup_grand_product_delta =
+            lookup_grand_product_delta_tmp.template extend_to<ProverInstances::EXTENDED_LENGTH>();
     }
 
     /**
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index 54ee1002630..adcb653104d 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -113,7 +113,18 @@ void ProtoGalaxyVerifier_<VerifierInstances>::prepare_for_folding(const std::vec
     if (inst->is_accumulator) {
         receive_accumulator(inst, domain_separator);
     } else {
+        // This is the first round of folding and we need to generate some gate challenges.
+        // TODO(https://github.com/AztecProtocol/barretenberg/issues/740): implement option 2 to make this more
+        // efficient by avoiding the computation of the perturbator
         receive_and_finalise_instance(inst, domain_separator);
+        inst->target_sum = 0;
+        auto beta = transcript->get_challenge(domain_separator + "_initial_gate_challenge");
+        std::vector<FF> gate_challenges(inst->log_instance_size);
+        gate_challenges[0] = beta;
+        for (size_t i = 1; i < inst->log_instance_size; i++) {
+            gate_challenges[i] = gate_challenges[i - 1].sqr();
+        }
+        inst->gate_challenges = gate_challenges;
     }
     index++;
 
@@ -129,8 +140,7 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
 {
     prepare_for_folding(fold_data);
 
-    // auto delta = transcript->get_challenge("delta");
-    FF delta = 0;
+    auto delta = transcript->get_challenge("delta");
     auto accumulator = get_accumulator();
     auto deltas = compute_round_challenge_pows(accumulator->log_instance_size, delta);
 
@@ -213,9 +223,11 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
 
     auto next_alpha = transcript->template receive_from_prover<FF>("next_alpha");
     verified = verified & (next_alpha == expected_alpha);
+    info(verified);
 
     auto next_eta = transcript->template receive_from_prover<FF>("next_eta");
     verified = verified & (next_eta == expected_parameters.eta);
+    info(verified);
 
     auto next_beta = transcript->template receive_from_prover<FF>("next_beta");
     verified = verified & (next_beta == expected_parameters.beta);
diff --git a/barretenberg/cpp/src/barretenberg/relations/relation_parameters.hpp b/barretenberg/cpp/src/barretenberg/relations/relation_parameters.hpp
index d1905d407d7..c44c3b16c86 100644
--- a/barretenberg/cpp/src/barretenberg/relations/relation_parameters.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/relation_parameters.hpp
@@ -1,4 +1,5 @@
 #pragma once
+#include "barretenberg/common/ref_vector.hpp"
 #include "barretenberg/ecc/curves/bn254/fr.hpp"
 #include <array>
 
@@ -38,15 +39,13 @@ template <typename T> struct RelationParameters {
                                    { T(0), T(0), T(0), T(0), T(0) } } };
 
     static constexpr int NUM_TO_FOLD = 5;
-    std::array<std::reference_wrapper<T>, NUM_TO_FOLD> to_fold = {
-        eta, beta, gamma, public_input_delta, lookup_grand_product_delta
-    };
+    RefVector<T> to_fold = { eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
 
     static RelationParameters get_random()
     {
         RelationParameters result;
         result.eta = T::random_element();
-        result.beta = T::random_element();
+        result.beta = T(0);
         result.beta_sqr = result.beta * result.beta;
         result.beta_cube = result.beta_sqr * result.beta;
         result.gamma = T::random_element();
diff --git a/barretenberg/cpp/src/barretenberg/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
index b9d1c1c5273..fa9593654bc 100644
--- a/barretenberg/cpp/src/barretenberg/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
@@ -69,7 +69,9 @@ template <typename Flavor> class RelationUtils {
     {
         auto scale_by_consecutive_powers_of_challenge = [&]<size_t, size_t>(auto& element) {
             element *= current_scalar;
-            current_scalar *= challenge;
+            if (current_scalar != challenge) {
+                current_scalar *= challenge;
+            }
         };
         apply_to_tuple_of_tuples(tuple, scale_by_consecutive_powers_of_challenge);
     }
@@ -163,7 +165,9 @@ template <typename Flavor> class RelationUtils {
         auto scale_by_challenge_and_accumulate = [&](auto& element) {
             for (auto& entry : element) {
                 result += entry * current_scalar;
-                current_scalar *= challenge;
+                if (current_scalar != challenge) { // HACK
+                    current_scalar *= challenge;
+                }
             }
         };
         apply_to_tuple_of_arrays(scale_by_challenge_and_accumulate, tuple);
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
index a131c59cdb5..1c7542dd434 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
@@ -298,6 +298,86 @@ TEST_F(ProtoGalaxyTests, FoldAlphaShouldBeZero)
     EXPECT_EQ(instances.alpha.evaluate(challenge), FF(0));
 }
 
+TEST_F(ProtoGalaxyTests, FirstIteration)
+{
+    const size_t log_instance_size(4);
+    const size_t instance_size(1 << log_instance_size);
+
+    auto composer = UltraComposer();
+
+    auto builder_1 = typename Flavor::CircuitBuilder();
+    builder_1.add_public_variable(FF(1));
+
+    auto instance_1 = composer.create_instance(builder_1);
+
+    auto builder_2 = typename Flavor::CircuitBuilder();
+    builder_2.add_public_variable(FF(1));
+
+    auto instance_2 = composer.create_instance(builder_2);
+
+    auto commitment_key = composer.commitment_key;
+
+    auto instances = std::vector<std::shared_ptr<Instance>>{ instance_1, instance_2 };
+    auto folding_prover = composer.create_folding_prover(instances, composer.commitment_key);
+    auto folding_verifier = composer.create_folding_verifier();
+
+    auto proof = folding_prover.fold_instances();
+    auto next_accumulator = proof.accumulator;
+    next_accumulator->is_accumulator = true;
+    auto storage = proof.storage;
+    auto res = folding_verifier.verify_folding_proof(proof.folding_data);
+    EXPECT_EQ(res, true);
+    auto next_honk_evals = ProtoGalaxyProver::compute_full_honk_evaluations(
+        next_accumulator->prover_polynomials, next_accumulator->alpha, next_accumulator->relation_parameters);
+    // Construct pow(\vec{betas*}) as in the paper
+    auto next_pow_beta =
+        ProtoGalaxyProver::compute_pow_polynomial_at_values(next_accumulator->gate_challenges, instance_size);
+
+    // Compute the corresponding target sum and create a dummy accumulator
+    auto next_target_sum = FF(0);
+    for (size_t i = 0; i < instance_size; i++) {
+        next_target_sum += next_honk_evals[i] * next_pow_beta[i];
+    }
+    info("e* ", next_target_sum);
+
+    EXPECT_EQ(next_accumulator->target_sum, next_target_sum);
+
+    auto builder_3 = typename Flavor::CircuitBuilder();
+    builder_3.add_public_variable(FF(1));
+    auto instance_3 = composer.create_instance(builder_3);
+    instances = std::vector<std::shared_ptr<Instance>>{ next_accumulator, instance_3 };
+
+    folding_prover = composer.create_folding_prover(instances, composer.commitment_key);
+    folding_verifier = composer.create_folding_verifier();
+
+    proof = folding_prover.fold_instances();
+    auto acc = proof.accumulator;
+    acc->is_accumulator = true;
+
+    storage = proof.storage;
+    res = folding_verifier.verify_folding_proof(proof.folding_data);
+    EXPECT_EQ(res, true);
+
+    next_honk_evals =
+        ProtoGalaxyProver::compute_full_honk_evaluations(acc->prover_polynomials, acc->alpha, acc->relation_parameters);
+    // Construct pow(\vec{betas*}) as in the paper
+    next_pow_beta = ProtoGalaxyProver::compute_pow_polynomial_at_values(acc->gate_challenges, instance_size);
+
+    // Compute the corresponding target sum and create a dummy accumulator
+    next_target_sum = FF(0);
+    for (size_t i = 0; i < instance_size; i++) {
+        next_target_sum += next_honk_evals[i] * next_pow_beta[i];
+    }
+    info("e* ", next_target_sum);
+
+    EXPECT_EQ(acc->target_sum, next_target_sum);
+    auto decider_prover = composer.create_decider_prover(acc, commitment_key);
+    auto decider_verifier = composer.create_decider_verifier(acc);
+    auto decision = decider_prover.construct_proof();
+    auto verified = decider_verifier.verify_proof(decision);
+    EXPECT_EQ(verified, true);
+}
+
 // TODO(https://github.com/AztecProtocol/barretenberg/issues/807): Have proper full folding testing (both failing and
 // passing) and move creating a test accumulator in a separate function.
 TEST_F(ProtoGalaxyTests, ComputeNewAccumulator)
@@ -325,7 +405,6 @@ TEST_F(ProtoGalaxyTests, ComputeNewAccumulator)
         poly_views[idx] = random_polynomials[idx];
     }
     auto relation_parameters = proof_system::RelationParameters<FF>::get_random();
-    relation_parameters.beta = FF(0);
 
     auto alpha = FF::random_element();
 

From 79408ede0d0d560a407b093b4573780143d55492 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Mon, 18 Dec 2023 15:06:21 +0000
Subject: [PATCH 57/86] cleaned up relation parameters

---
 .../protogalaxy/protogalaxy_prover.cpp        |  8 +--
 .../protogalaxy/protogalaxy_prover.hpp        | 49 +++++--------------
 .../relations/relation_parameters.hpp         |  2 +-
 3 files changed, 15 insertions(+), 44 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index 58ae95603f9..8c8b8039504 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -253,13 +253,7 @@ ProtoGalaxyProver_<ProverInstances>::compute_next_accumulator(
     transcript->send_to_verifier("next_public_input_delta", folded_relation_parameters.public_input_delta);
     transcript->send_to_verifier("next_lookup_grand_product_delta",
                                  folded_relation_parameters.lookup_grand_product_delta);
-    next_accumulator->relation_parameters = proof_system::RelationParameters<FF>{
-        combined_relation_parameters.eta.evaluate(challenge), // WORKTODO change all of these to evaluate(challenge)
-        combined_relation_parameters.beta.evaluate(challenge),
-        combined_relation_parameters.gamma.evaluate(challenge),
-        combined_relation_parameters.public_input_delta.evaluate(challenge),
-        combined_relation_parameters.lookup_grand_product_delta.evaluate(challenge),
-    };
+    next_accumulator->relation_parameters = folded_relation_parameters;
 
     // Fold the verification key and send it to the verifier as this is part of ϕ as well
     auto acc_vk = std::make_shared<VerificationKey>(instances[0]->prover_polynomials.get_polynomial_size(),
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index de905485087..6ca57891691 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -387,7 +387,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      *
      */
     static Univariate<FF, ProverInstances::BATCHED_EXTENDED_LENGTH, ProverInstances::NUM> compute_combiner_quotient(
-        FF compressed_perturbator, ExtendedUnivariateWithRandomization combiner)
+        const FF compressed_perturbator, ExtendedUnivariateWithRandomization combiner)
     {
         std::array<FF, ProverInstances::BATCHED_EXTENDED_LENGTH - ProverInstances::NUM> combiner_quotient_evals = {};
 
@@ -416,41 +416,18 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      */
     static void combine_relation_parameters(ProverInstances& instances)
     {
-        // array of parameters to be computed
-        // size_t param_idx = 0;
-        // for (auto& folded_parameter : instances.relation_parameters.to_fold) {
-        //     Univariate<FF, ProverInstances::NUM> tmp(0);
-        //     size_t instance_idx = 0;
-        //     for (auto& instance : instances) {
-        //         tmp.value_at(instance_idx) = instance->relation_parameters.to_fold[param_idx];
-        //         instance_idx++;
-        //     }
-        //     folded_parameter = tmp.template extend_to<ProverInstances::EXTENDED_LENGTH>();
-        //     param_idx++;
-        // }
-        Univariate<FF, ProverInstances::NUM> eta_tmp(
-            std::array<FF, 2>{ instances[0]->relation_parameters.eta, instances[1]->relation_parameters.eta });
-        instances.relation_parameters.eta = eta_tmp.template extend_to<ProverInstances::EXTENDED_LENGTH>();
-
-        Univariate<FF, ProverInstances::NUM> beta_tmp(
-            std::array<FF, 2>{ instances[0]->relation_parameters.beta, instances[1]->relation_parameters.beta });
-        instances.relation_parameters.beta = beta_tmp.template extend_to<ProverInstances::EXTENDED_LENGTH>();
-
-        Univariate<FF, ProverInstances::NUM> gamma_tmp(
-            std::array<FF, 2>{ instances[0]->relation_parameters.gamma, instances[1]->relation_parameters.gamma });
-        instances.relation_parameters.gamma = gamma_tmp.template extend_to<ProverInstances::EXTENDED_LENGTH>();
-
-        Univariate<FF, ProverInstances::NUM> public_input_delta_tmp(
-            std::array<FF, 2>{ instances[0]->relation_parameters.public_input_delta,
-                               instances[1]->relation_parameters.public_input_delta });
-        instances.relation_parameters.public_input_delta =
-            public_input_delta_tmp.template extend_to<ProverInstances::EXTENDED_LENGTH>();
-
-        Univariate<FF, ProverInstances::NUM> lookup_grand_product_delta_tmp(
-            std::array<FF, 2>{ instances[0]->relation_parameters.lookup_grand_product_delta,
-                               instances[1]->relation_parameters.lookup_grand_product_delta });
-        instances.relation_parameters.lookup_grand_product_delta =
-            lookup_grand_product_delta_tmp.template extend_to<ProverInstances::EXTENDED_LENGTH>();
+        size_t param_idx = 0;
+        auto to_fold = instances.relation_parameters.get_to_fold();
+        for (auto& folded_parameter : to_fold) {
+            Univariate<FF, ProverInstances::NUM> tmp(0);
+            size_t instance_idx = 0;
+            for (auto& instance : instances) {
+                tmp.value_at(instance_idx) = instance->relation_parameters.get_to_fold()[param_idx];
+                instance_idx++;
+            }
+            folded_parameter = tmp.template extend_to<ProverInstances::EXTENDED_LENGTH>();
+            param_idx++;
+        }
     }
 
     /**
diff --git a/barretenberg/cpp/src/barretenberg/relations/relation_parameters.hpp b/barretenberg/cpp/src/barretenberg/relations/relation_parameters.hpp
index c44c3b16c86..208585d94f6 100644
--- a/barretenberg/cpp/src/barretenberg/relations/relation_parameters.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/relation_parameters.hpp
@@ -39,7 +39,7 @@ template <typename T> struct RelationParameters {
                                    { T(0), T(0), T(0), T(0), T(0) } } };
 
     static constexpr int NUM_TO_FOLD = 5;
-    RefVector<T> to_fold = { eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
+    RefVector<T> get_to_fold() { return { eta, beta, gamma, public_input_delta, lookup_grand_product_delta }; }
 
     static RelationParameters get_random()
     {

From a1d44a7f9ee040b2c10e4899497ccdd8d0f58e1c Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Wed, 20 Dec 2023 13:32:49 +0000
Subject: [PATCH 58/86] more stuff working

---
 .../src/barretenberg/flavor/goblin_ultra.hpp  |  49 +++
 .../cpp/src/barretenberg/flavor/ultra.hpp     |  38 +++
 .../cpp/src/barretenberg/honk/CMakeLists.txt  |  21 +-
 .../protogalaxy/decider_prover.cpp            |   1 -
 .../protogalaxy/decider_verifier.cpp          |   8 +-
 .../protogalaxy/protogalaxy_prover.cpp        |   6 +-
 .../protogalaxy/protogalaxy_prover.hpp        |  27 +-
 .../protogalaxy/protogalaxy_verifier.cpp      |   8 +-
 .../cpp/src/barretenberg/relations/utils.hpp  |   4 +-
 .../barretenberg/sumcheck/sumcheck.test.cpp   |   1 -
 .../barretenberg/sumcheck/sumcheck_round.hpp  |   9 +-
 .../ultra_honk/protogalaxy.test.cpp           | 297 +++++-------------
 12 files changed, 192 insertions(+), 277 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
index 28c844e141b..4e708c3875d 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
@@ -448,6 +448,55 @@ class GoblinUltra {
             this->lagrange_ecc_op = verification_key->lagrange_ecc_op;
             this->databus_id = verification_key->databus_id;
         }
+
+        VerifierCommitments_(const std::shared_ptr<VerificationKey>& verification_key,
+                             const WitnessCommitments& witness_commitments)
+        {
+            this->q_m = verification_key->q_m;
+            this->q_l = verification_key->q_l;
+            this->q_r = verification_key->q_r;
+            this->q_o = verification_key->q_o;
+            this->q_4 = verification_key->q_4;
+            this->q_c = verification_key->q_c;
+            this->q_arith = verification_key->q_arith;
+            this->q_sort = verification_key->q_sort;
+            this->q_elliptic = verification_key->q_elliptic;
+            this->q_aux = verification_key->q_aux;
+            this->q_lookup = verification_key->q_lookup;
+            this->q_busread = verification_key->q_busread;
+            this->q_poseidon2_external = verification_key->q_poseidon2_external;
+            this->q_poseidon2_internal = verification_key->q_poseidon2_internal;
+            this->sigma_1 = verification_key->sigma_1;
+            this->sigma_2 = verification_key->sigma_2;
+            this->sigma_3 = verification_key->sigma_3;
+            this->sigma_4 = verification_key->sigma_4;
+            this->id_1 = verification_key->id_1;
+            this->id_2 = verification_key->id_2;
+            this->id_3 = verification_key->id_3;
+            this->id_4 = verification_key->id_4;
+            this->table_1 = verification_key->table_1;
+            this->table_2 = verification_key->table_2;
+            this->table_3 = verification_key->table_3;
+            this->table_4 = verification_key->table_4;
+            this->lagrange_first = verification_key->lagrange_first;
+            this->lagrange_last = verification_key->lagrange_last;
+            this->lagrange_ecc_op = verification_key->lagrange_ecc_op;
+            this->databus_id = verification_key->databus_id;
+
+            this->w_l = witness_commitments.w_l;
+            this->w_r = witness_commitments.w_r;
+            this->w_o = witness_commitments.w_o;
+            this->sorted_accum = witness_commitments.sorted_accum;
+            this->w_4 = witness_commitments.w_4;
+            this->z_perm = witness_commitments.z_perm;
+            this->z_lookup = witness_commitments.z_lookup;
+            this->ecc_op_wire_1 = witness_commitments.ecc_op_wire_1;
+            this->ecc_op_wire_2 = witness_commitments.ecc_op_wire_2;
+            this->ecc_op_wire_3 = witness_commitments.ecc_op_wire_3;
+            this->calldata = witness_commitments.calldata;
+            this->calldata = witness_commitments.calldata_read_counts;
+            this->lookup_inverses = witness_commitments.lookup_inverses;
+        }
     };
     // Specialize for GoblinUltra (general case used in GoblinUltraRecursive).
     using VerifierCommitments = VerifierCommitments_<Commitment, VerificationKey>;
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
index 9971d238c71..9a82c54565c 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
@@ -435,6 +435,44 @@ class Ultra {
             lagrange_first = verification_key->lagrange_first;
             lagrange_last = verification_key->lagrange_last;
         }
+
+        VerifierCommitments(const std::shared_ptr<VerificationKey>& verification_key,
+                            const WitnessCommitments& witness_commitments)
+        {
+            q_m = verification_key->q_m;
+            q_c = verification_key->q_c;
+            q_l = verification_key->q_l;
+            q_r = verification_key->q_r;
+            q_o = verification_key->q_o;
+            q_4 = verification_key->q_4;
+            q_arith = verification_key->q_arith;
+            q_sort = verification_key->q_sort;
+            q_elliptic = verification_key->q_elliptic;
+            q_aux = verification_key->q_aux;
+            q_lookup = verification_key->q_lookup;
+            sigma_1 = verification_key->sigma_1;
+            sigma_2 = verification_key->sigma_2;
+            sigma_3 = verification_key->sigma_3;
+            sigma_4 = verification_key->sigma_4;
+            id_1 = verification_key->id_1;
+            id_2 = verification_key->id_2;
+            id_3 = verification_key->id_3;
+            id_4 = verification_key->id_4;
+            table_1 = verification_key->table_1;
+            table_2 = verification_key->table_2;
+            table_3 = verification_key->table_3;
+            table_4 = verification_key->table_4;
+            lagrange_first = verification_key->lagrange_first;
+            lagrange_last = verification_key->lagrange_last;
+
+            w_l = witness_commitments.w_l;
+            w_r = witness_commitments.w_r;
+            w_o = witness_commitments.w_o;
+            sorted_accum = witness_commitments.sorted_accum;
+            w_4 = witness_commitments.w_4;
+            z_perm = witness_commitments.z_perm;
+            z_lookup = witness_commitments.z_lookup;
+        }
     };
 
     /**
diff --git a/barretenberg/cpp/src/barretenberg/honk/CMakeLists.txt b/barretenberg/cpp/src/barretenberg/honk/CMakeLists.txt
index f57c608e058..a0e5fccbd8a 100644
--- a/barretenberg/cpp/src/barretenberg/honk/CMakeLists.txt
+++ b/barretenberg/cpp/src/barretenberg/honk/CMakeLists.txt
@@ -1,20 +1 @@
-barretenberg_module(honk proof_system commitment_schemes sumcheck)
-
-if(TESTING)
-    # TODO: Re-enable all these warnings once PoC is finished
-    if(CMAKE_CXX_COMPILER_ID MATCHES "Clang")
-        target_compile_options(
-            honk_test_objects
-            PRIVATE
-            -Wno-error=unused-variable
-        )
-    elseif(CMAKE_CXX_COMPILER_ID MATCHES "GNU")
-        target_compile_options(
-            honk_test_objects
-            PRIVATE
-            -Wno-error=maybe-uninitialized
-            -Wno-error=uninitialized
-            -Wno-error=unused-variable
-        )
-    endif()
-endif()
+ barretenberg_module(honk proof_system commitment_schemes sumcheck)
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
index 3548f7244af..755e4818fb7 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
@@ -54,7 +54,6 @@ template <UltraFlavor Flavor> void DeciderProver_<Flavor>::execute_preamble_roun
     auto comm_view = accumulator->witness_commitments.get_all();
     auto witness_labels = accumulator->commitment_labels.get_witness();
     for (size_t idx = 0; idx < witness_labels.size(); idx++) {
-        info(comm_view[idx]);
         transcript->send_to_verifier(witness_labels[idx], comm_view[idx]);
     }
 
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
index 62a7fd2081e..08f29db7cb4 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
@@ -66,7 +66,6 @@ template <typename Flavor> bool DeciderVerifier_<Flavor>::verify_proof(const plo
     auto witness_labels = inst->commitment_labels.get_witness();
     for (size_t idx = 0; idx < witness_labels.size(); idx++) {
         comm_view[idx] = transcript->template receive_from_prover<Commitment>(witness_labels[idx]);
-        info(comm_view[idx]);
     }
 
     inst->verification_key = std::make_shared<VerificationKey>(inst->instance_size, inst->public_input_size);
@@ -76,7 +75,7 @@ template <typename Flavor> bool DeciderVerifier_<Flavor>::verify_proof(const plo
         vk_view[idx] = transcript->template receive_from_prover<Commitment>(vk_labels[idx]);
     }
 
-    VerifierCommitments commitments{ inst->verification_key };
+    VerifierCommitments commitments{ inst->verification_key, inst->witness_commitments };
     auto& witness_commitments = inst->witness_commitments;
     commitments.w_l = witness_commitments.w_l;
     commitments.w_r = witness_commitments.w_r;
@@ -85,7 +84,6 @@ template <typename Flavor> bool DeciderVerifier_<Flavor>::verify_proof(const plo
     commitments.w_4 = witness_commitments.w_4;
     commitments.z_perm = witness_commitments.z_perm;
     commitments.z_lookup = witness_commitments.z_lookup;
-    info(commitments.z_lookup);
 
     auto sumcheck = SumcheckVerifier<Flavor>(inst->log_instance_size, transcript, inst->target_sum);
 
@@ -94,10 +92,10 @@ template <typename Flavor> bool DeciderVerifier_<Flavor>::verify_proof(const plo
 
     // If Sumcheck did not verify, return false
     if (sumcheck_verified.has_value() && !sumcheck_verified.value()) {
+        info("at sumcheck");
         return false;
     }
 
-    info("here");
     // Execute ZeroMorph rounds. See https://hackmd.io/dlf9xEwhTQyE3hiGbq4FsA?view for a complete description of the
     // unrolled protocol.
     auto pairing_points = ZeroMorph::verify(commitments.get_unshifted(),
@@ -108,7 +106,7 @@ template <typename Flavor> bool DeciderVerifier_<Flavor>::verify_proof(const plo
                                             transcript);
 
     auto verified = pcs_verification_key->pairing_check(pairing_points[0], pairing_points[1]);
-    info(verified);
+
     return sumcheck_verified.value() && verified;
 }
 
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index 19a71b9e860..97d527a2488 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -122,7 +122,6 @@ template <class ProverInstances> void ProtoGalaxyProver_<ProverInstances>::prepa
         // efficient by avoiding the computation of the perturbator
         finalise_and_send_instance(instance, domain_separator);
         instance->target_sum = 0;
-        info("log_instance_size ", instance->log_instance_size);
         auto beta = transcript->get_challenge(domain_separator + "_initial_gate_challenge");
         std::vector<FF> gate_challenges(instance->log_instance_size);
         gate_challenges[0] = beta;
@@ -149,6 +148,7 @@ std::shared_ptr<typename ProverInstances::Instance> ProtoGalaxyProver_<ProverIns
     const FF& compressed_perturbator)
 {
     auto combiner_quotient_at_challenge = combiner_quotient.evaluate(challenge);
+
     // Given the challenge \gamma, compute Z(\gamma) and {L_0(\gamma),L_1(\gamma)}
     // TODO(https://github.com/AztecProtocol/barretenberg/issues/764): Generalize the vanishing polynomial formula
     // and the computation of Lagrange basis for k instances
@@ -156,6 +156,7 @@ std::shared_ptr<typename ProverInstances::Instance> ProtoGalaxyProver_<ProverIns
     std::vector<FF> lagranges{ FF(1) - challenge, challenge };
 
     auto next_accumulator = std::make_shared<Instance>();
+    next_accumulator->is_accumulator = true;
     next_accumulator->instance_size = instances[0]->instance_size;
     next_accumulator->log_instance_size = instances[0]->log_instance_size;
 
@@ -260,7 +261,6 @@ std::shared_ptr<typename ProverInstances::Instance> ProtoGalaxyProver_<ProverIns
         vk_idx++;
     }
     next_accumulator->verification_key = acc_vk;
-
     return next_accumulator;
 }
 
@@ -283,7 +283,6 @@ FoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverInstanc
     }
     assert(perturbator[0] == accumulator->target_sum);
     auto perturbator_challenge = transcript->get_challenge("perturbator_challenge"); // alpha
-    info("F(alpha) ", perturbator.evaluate(perturbator_challenge));                  // F(\alpha)
     instances.next_gate_challenges =
         update_gate_challenges(perturbator_challenge, accumulator->gate_challenges, deltas);
     const auto pow_betas_star =
@@ -291,7 +290,6 @@ FoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverInstanc
     combine_relation_parameters(instances);
     combine_alpha(instances);
     auto combiner = compute_combiner(instances, pow_betas_star);
-    info("G(0) ", combiner.evaluations[0]);
 
     auto compressed_perturbator = perturbator.evaluate(perturbator_challenge);
     auto combiner_quotient = compute_combiner_quotient(compressed_perturbator, combiner);
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index f1897c83fe3..74d36d4494e 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -38,7 +38,6 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     using ExtendedUnivariateWithRandomization =
         Univariate<FF,
                    (Flavor::MAX_TOTAL_RELATION_LENGTH - 1 + ProverInstances::NUM - 1) * (ProverInstances::NUM - 1) + 1>;
-
     using ExtendedUnivariates = typename Flavor::template ProverUnivariates<ExtendedUnivariate::LENGTH>;
 
     using TupleOfTuplesOfUnivariates =
@@ -163,9 +162,8 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
             Utils::template accumulate_relation_evaluations<>(
                 row_evaluations, relation_evaluations, relation_parameters, FF(1));
 
-            auto running_challenge = FF(1);
             auto output = FF(0);
-            scale_and_batch_elements(relation_evaluations, alpha, running_challenge, output);
+            scale_and_batch_elements(relation_evaluations, alpha, output);
 
             full_honk_evaluations[row] = output;
         }
@@ -176,18 +174,21 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      * @brief Scale elements by consecutive powers of the challenge then sum
      * @param result Batched result
      */
-    static void scale_and_batch_elements(auto& tuple, const FF& challenge, FF& current_scalar, FF& result)
+    static void scale_and_batch_elements(auto& tuple, const FF& challenge, FF& result)
     {
+        auto first_array = std::get<0>(tuple);
+        result = first_array[0];
+        for (auto& entry : first_array) {
+            result += entry * challenge;
+        }
+
         auto scale_by_challenge_and_accumulate = [&](auto& element) {
             for (auto& entry : element) {
-                result += entry * current_scalar;
-                if (current_scalar != challenge) {
-
-                    current_scalar *= challenge;
-                }
+                result += entry * challenge;
             }
         };
-        Utils::apply_to_tuple_of_arrays(scale_by_challenge_and_accumulate, tuple);
+
+        Utils::template apply_to_tuple_of_arrays<1>(scale_by_challenge_and_accumulate, tuple);
     }
 
     /**
@@ -254,8 +255,6 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     static Polynomial<FF> compute_perturbator(const std::shared_ptr<Instance> accumulator,
                                               const std::vector<FF>& deltas)
     {
-        info("HERE");
-        info(accumulator->alpha);
         auto full_honk_evaluations = compute_full_honk_evaluations(
             accumulator->prover_polynomials, accumulator->alpha, accumulator->relation_parameters);
         const auto betas = accumulator->gate_challenges;
@@ -359,8 +358,8 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         // Batch the univariate contributions from each sub-relation to obtain the round univariate
         return batch_over_relations(univariate_accumulators, instances.alpha);
     }
-    static ExtendedUnivariateWithRandomization batch_over_relations(TupleOfTuplesOfUnivariates univariate_accumulators,
-                                                                    AlphaType alpha)
+    static ExtendedUnivariateWithRandomization batch_over_relations(TupleOfTuplesOfUnivariates& univariate_accumulators,
+                                                                    const AlphaType& alpha)
     {
 
         // First relation does not get multiplied by a batching challenge
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index adcb653104d..201d6832aa8 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -148,7 +148,11 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
     for (size_t idx = 0; idx <= accumulator->log_instance_size; idx++) {
         perturbator_coeffs[idx] = transcript->template receive_from_prover<FF>("perturbator_" + std::to_string(idx));
     }
-    assert(perturbator_coeffs[0] == accumulator->target_sum);
+
+    if (perturbator_coeffs[0] != accumulator->target_sum) {
+        return false;
+    }
+
     auto perturbator = Polynomial<FF>(perturbator_coeffs);
     FF perturbator_challenge = transcript->get_challenge("perturbator_challenge");
     auto perturbator_at_challenge = perturbator.evaluate(perturbator_challenge);
@@ -223,11 +227,9 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
 
     auto next_alpha = transcript->template receive_from_prover<FF>("next_alpha");
     verified = verified & (next_alpha == expected_alpha);
-    info(verified);
 
     auto next_eta = transcript->template receive_from_prover<FF>("next_eta");
     verified = verified & (next_eta == expected_parameters.eta);
-    info(verified);
 
     auto next_beta = transcript->template receive_from_prover<FF>("next_beta");
     verified = verified & (next_beta == expected_parameters.beta);
diff --git a/barretenberg/cpp/src/barretenberg/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
index fa9593654bc..d703d063247 100644
--- a/barretenberg/cpp/src/barretenberg/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
@@ -179,7 +179,7 @@ template <typename Flavor> class RelationUtils {
      * @tparam Operation Any operation valid on elements of the inner arrays (FFs)
      * @param tuple Tuple of arrays (of FFs)
      */
-    template <typename Operation, size_t idx = 0, typename... Ts>
+    template <size_t idx = 0, typename Operation, typename... Ts>
     static void apply_to_tuple_of_arrays(Operation&& operation, std::tuple<Ts...>& tuple)
     {
         auto& element = std::get<idx>(tuple);
@@ -187,7 +187,7 @@ template <typename Flavor> class RelationUtils {
         std::invoke(std::forward<Operation>(operation), element);
 
         if constexpr (idx + 1 < sizeof...(Ts)) {
-            apply_to_tuple_of_arrays<Operation, idx + 1>(operation, tuple);
+            apply_to_tuple_of_arrays<idx + 1, Operation>(operation, tuple);
         }
     }
 };
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp
index 3e537cffec5..15155a32196 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp
@@ -71,7 +71,6 @@ TEST_F(SumcheckTests, PolynomialNormalization)
     for (size_t idx = 0; idx < multivariate_d; idx++) {
         gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
     }
-    info("here");
     auto output = sumcheck.prove(full_polynomials, {}, alpha, gate_challenges);
 
     FF u_0 = output.challenge[0];
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
index 5c023eee541..66612c5500c 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
@@ -108,14 +108,6 @@ template <typename Flavor> class SumcheckProverRound {
         const barretenberg::PowPolynomial<FF>& pow_polynomial,
         const FF alpha)
     {
-        // Precompute the vector of required powers of zeta
-        // TODO(luke): Parallelize this
-        // std::vector<FF> pow_challenges(round_size >> 1);
-        // pow_challenges[0] = pow_univariate.partial_evaluation_constant;
-        // for (size_t i = 1; i < (round_size >> 1); ++i) {
-        //     pow_challenges[i] = pow_challenges[i - 1] * pow_univariate.zeta_pow_sqr;
-        // }
-
         std::vector<FF> pow_challenges(round_size >> 1);
         pow_challenges[0] = pow_polynomial.partial_evaluation_result;
         for (size_t i = 1; i < (round_size >> 1); ++i) {
@@ -169,6 +161,7 @@ template <typename Flavor> class SumcheckProverRound {
         for (auto& accumulators : thread_univariate_accumulators) {
             Utils::add_nested_tuples(univariate_accumulators, accumulators);
         }
+
         // Batch the univariate contributions from each sub-relation to obtain the round univariate
         return batch_over_relations<barretenberg::Univariate<FF, BATCHED_RELATION_PARTIAL_LENGTH>>(
             univariate_accumulators, alpha, pow_polynomial);
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
index f5c8d182ab6..0f9e5083c5d 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
@@ -18,6 +18,7 @@ using Polynomial = typename Flavor::Polynomial;
 using ProverPolynomials = Flavor::ProverPolynomials;
 using RelationParameters = proof_system::RelationParameters<FF>;
 using WitnessCommitments = typename Flavor::WitnessCommitments;
+using CommitmentKey = Flavor::CommitmentKey;
 const size_t NUM_POLYNOMIALS = Flavor::NUM_ALL_ENTITIES;
 
 namespace protogalaxy_tests {
@@ -44,24 +45,44 @@ ProverPolynomials construct_ultra_full_polynomials(auto& input_polynomials)
     return full_polynomials;
 }
 
-std::shared_ptr<VerificationKey> construct_ultra_verification_key(size_t instance_size, size_t num_public_inputs)
+std::shared_ptr<Instance> fold_and_verify(const std::vector<std::shared_ptr<Instance>>& instances,
+                                          UltraComposer& composer,
+                                          bool expected_result)
 {
-    auto verification_key = std::make_shared<typename Flavor::VerificationKey>(instance_size, num_public_inputs);
-    auto vk_view = verification_key->get_all();
-    for (auto& view : vk_view) {
-        view = Affine(Projective::random_element());
-    }
-    return verification_key;
+    auto folding_prover = composer.create_folding_prover(instances, composer.commitment_key);
+    auto folding_verifier = composer.create_folding_verifier();
+
+    auto proof = folding_prover.fold_instances();
+    auto next_accumulator = proof.accumulator;
+    auto res = folding_verifier.verify_folding_proof(proof.folding_data);
+    EXPECT_EQ(res, expected_result);
+    return next_accumulator;
 }
 
-WitnessCommitments construct_witness_commitments()
+void check_accumulator_target_sum_manual(std::shared_ptr<Instance>& accumulator, bool expected_result)
 {
-    WitnessCommitments wc;
-    auto w_view = wc.get_all();
-    for (auto& view : w_view) {
-        view = Affine(Projective::random_element());
+    auto instance_size = accumulator->instance_size;
+    auto expected_honk_evals = ProtoGalaxyProver::compute_full_honk_evaluations(
+        accumulator->prover_polynomials, accumulator->alpha, accumulator->relation_parameters);
+    // Construct pow(\vec{betas*}) as in the paper
+    auto expected_pows_beta =
+        ProtoGalaxyProver::compute_pow_polynomial_at_values(accumulator->gate_challenges, instance_size);
+
+    // Compute the corresponding target sum and create a dummy accumulator
+    auto expected_target_sum = FF(0);
+    for (size_t i = 0; i < instance_size; i++) {
+        expected_target_sum += expected_honk_evals[i] * expected_pows_beta[i];
     }
-    return wc;
+
+    EXPECT_EQ(accumulator->target_sum == expected_target_sum, expected_result);
+}
+void decide_and_verify(std::shared_ptr<Instance>& accumulator, UltraComposer& composer, bool expected_result)
+{
+    auto decider_prover = composer.create_decider_prover(accumulator, composer.commitment_key);
+    auto decider_verifier = composer.create_decider_verifier(accumulator);
+    auto decision = decider_prover.construct_proof();
+    auto verified = decider_verifier.verify_proof(decision);
+    EXPECT_EQ(verified, expected_result);
 }
 
 class ProtoGalaxyTests : public ::testing::Test {
@@ -257,11 +278,9 @@ TEST_F(ProtoGalaxyTests, FoldAlphaShouldBeZero)
     EXPECT_EQ(instances.alpha.evaluate(challenge), FF(0));
 }
 
-TEST_F(ProtoGalaxyTests, FirstIteration)
+// Check both manually and using the protocol two rounds of folding
+TEST_F(ProtoGalaxyTests, FullProtogalaxyTest)
 {
-    const size_t log_instance_size(4);
-    const size_t instance_size(1 << log_instance_size);
-
     auto composer = UltraComposer();
 
     auto builder_1 = typename Flavor::CircuitBuilder();
@@ -274,240 +293,80 @@ TEST_F(ProtoGalaxyTests, FirstIteration)
 
     auto instance_2 = composer.create_instance(builder_2);
 
-    auto commitment_key = composer.commitment_key;
-
     auto instances = std::vector<std::shared_ptr<Instance>>{ instance_1, instance_2 };
-    auto folding_prover = composer.create_folding_prover(instances, composer.commitment_key);
-    auto folding_verifier = composer.create_folding_verifier();
-
-    auto proof = folding_prover.fold_instances();
-    auto next_accumulator = proof.accumulator;
-    next_accumulator->is_accumulator = true;
-    auto res = folding_verifier.verify_folding_proof(proof.folding_data);
-    EXPECT_EQ(res, true);
-    auto next_honk_evals = ProtoGalaxyProver::compute_full_honk_evaluations(
-        next_accumulator->prover_polynomials, next_accumulator->alpha, next_accumulator->relation_parameters);
-    // Construct pow(\vec{betas*}) as in the paper
-    auto next_pow_beta =
-        ProtoGalaxyProver::compute_pow_polynomial_at_values(next_accumulator->gate_challenges, instance_size);
-
-    // Compute the corresponding target sum and create a dummy accumulator
-    auto next_target_sum = FF(0);
-    for (size_t i = 0; i < instance_size; i++) {
-        next_target_sum += next_honk_evals[i] * next_pow_beta[i];
-    }
-    info("e* ", next_target_sum);
-
-    EXPECT_EQ(next_accumulator->target_sum, next_target_sum);
+    auto first_accumulator = fold_and_verify(instances, composer, true);
+    check_accumulator_target_sum_manual(first_accumulator, true);
 
     auto builder_3 = typename Flavor::CircuitBuilder();
     builder_3.add_public_variable(FF(1));
     auto instance_3 = composer.create_instance(builder_3);
-    instances = std::vector<std::shared_ptr<Instance>>{ next_accumulator, instance_3 };
-
-    folding_prover = composer.create_folding_prover(instances, composer.commitment_key);
-    folding_verifier = composer.create_folding_verifier();
 
-    proof = folding_prover.fold_instances();
-    auto acc = proof.accumulator;
-    acc->is_accumulator = true;
+    instances = std::vector<std::shared_ptr<Instance>>{ first_accumulator, instance_3 };
+    auto second_accumulator = fold_and_verify(instances, composer, true);
+    check_accumulator_target_sum_manual(second_accumulator, true);
 
-    res = folding_verifier.verify_folding_proof(proof.folding_data);
-    EXPECT_EQ(res, true);
-
-    next_honk_evals =
-        ProtoGalaxyProver::compute_full_honk_evaluations(acc->prover_polynomials, acc->alpha, acc->relation_parameters);
-    // Construct pow(\vec{betas*}) as in the paper
-    next_pow_beta = ProtoGalaxyProver::compute_pow_polynomial_at_values(acc->gate_challenges, instance_size);
-
-    // Compute the corresponding target sum and create a dummy accumulator
-    next_target_sum = FF(0);
-    for (size_t i = 0; i < instance_size; i++) {
-        next_target_sum += next_honk_evals[i] * next_pow_beta[i];
-    }
-    info("e* ", next_target_sum);
-
-    EXPECT_EQ(acc->target_sum, next_target_sum);
-    auto decider_prover = composer.create_decider_prover(acc, commitment_key);
-    auto decider_verifier = composer.create_decider_verifier(acc);
-    auto decision = decider_prover.construct_proof();
-    auto verified = decider_verifier.verify_proof(decision);
-    EXPECT_EQ(verified, true);
+    decide_and_verify(second_accumulator, composer, true);
 }
 
-// TODO(https://github.com/AztecProtocol/barretenberg/issues/807): Have proper full folding testing (both failing and
-// passing) and move creating a test accumulator in a separate function.
-TEST_F(ProtoGalaxyTests, ComputeNewAccumulator)
+TEST_F(ProtoGalaxyTests, TamperedCommitment)
 {
-    const size_t log_instance_size(4);
-    const size_t instance_size(1 << log_instance_size);
-
-    auto builder = typename Flavor::CircuitBuilder();
     auto composer = UltraComposer();
-    builder.add_public_variable(FF(1));
-
-    auto instance = composer.create_instance(builder);
-    auto commitment_key = composer.commitment_key;
 
-    std::array<barretenberg::Polynomial<FF>, NUM_POLYNOMIALS> random_polynomials;
-    for (auto& poly : random_polynomials) {
-        poly = get_random_polynomial(instance_size);
-    }
-    for (auto& poly : random_polynomials) {
-        poly[0] = FF::zero();
-    }
-    ProverPolynomials polynomials;
-    auto poly_views = polynomials.get_all();
-    for (size_t idx = 0; idx < poly_views.size(); idx++) {
-        poly_views[idx] = random_polynomials[idx];
-    }
-    auto relation_parameters = proof_system::RelationParameters<FF>::get_random();
+    auto builder_1 = typename Flavor::CircuitBuilder();
+    builder_1.add_public_variable(FF(1));
 
-    auto alpha = FF::random_element();
+    auto instance_1 = composer.create_instance(builder_1);
 
-    auto full_honk_evals = ProtoGalaxyProver::compute_full_honk_evaluations(polynomials, alpha, relation_parameters);
-    std::vector<FF> betas(log_instance_size);
-    for (size_t idx = 0; idx < log_instance_size; idx++) {
-        betas[idx] = FF::random_element();
-    }
+    auto builder_2 = typename Flavor::CircuitBuilder();
+    builder_2.add_public_variable(FF(1));
 
-    // Construct pow(\vec{betas}) as in the paper
-    auto pow_beta = ProtoGalaxyProver::compute_pow_polynomial_at_values(betas, instance_size);
-    // Compute the corresponding target sum and create a dummy accumulator
-    auto target_sum = FF(0);
-    for (size_t i = 0; i < instance_size; i++) {
-        target_sum += full_honk_evals[i] * pow_beta[i];
-    }
+    auto instance_2 = composer.create_instance(builder_2);
 
-    auto accumulator = std::make_shared<Instance>();
-    accumulator->prover_polynomials = std::move(polynomials);
-    auto witness_poly = polynomials.get_witness();
-    size_t idx = 0;
-    for (auto& comm : accumulator->witness_commitments.get_all()) {
-        comm = commitment_key->commit(witness_poly[idx]);
-        idx++;
-    }
+    auto instances = std::vector<std::shared_ptr<Instance>>{ instance_1, instance_2 };
+    auto first_accumulator = fold_and_verify(instances, composer, true);
+    check_accumulator_target_sum_manual(first_accumulator, true);
 
-    idx = 0;
-    auto selector_poly = polynomials.get_precomputed();
-    accumulator->verification_key = std::make_shared<typename Flavor::VerificationKey>(instance_size, 1);
-    for (auto& vk : accumulator->verification_key->get_all()) {
-        vk = commitment_key->commit(selector_poly[idx]);
-        idx++;
-    }
-    accumulator->instance_size = instance_size;
-    accumulator->log_instance_size = log_instance_size;
-    accumulator->gate_challenges = betas;
-    accumulator->target_sum = target_sum;
-    accumulator->relation_parameters = relation_parameters;
-    accumulator->alpha = alpha;
-    accumulator->is_accumulator = true;
-    accumulator->public_inputs = std::vector<FF>{ FF::random_element() };
+    auto builder_3 = typename Flavor::CircuitBuilder();
+    builder_3.add_public_variable(FF(1));
+    auto instance_3 = composer.create_instance(builder_3);
 
-    auto instances = std::vector<std::shared_ptr<Instance>>{ accumulator, instance };
-    auto folding_prover = composer.create_folding_prover(instances, composer.commitment_key);
-    auto folding_verifier = composer.create_folding_verifier();
+    // tampering with the commitment should cause the decider to fail
+    first_accumulator->witness_commitments.w_l = Projective(Affine::random_element());
+    instances = std::vector<std::shared_ptr<Instance>>{ first_accumulator, instance_3 };
 
-    auto proof = folding_prover.fold_instances();
-    auto next_accumulator = proof.accumulator;
-    // auto res = folding_verifier.verify_folding_proof(proof.folding_data);
-    // EXPECT_EQ(res, true);
-    // i got omega* \vec{\beta*} e*
-    // i want to check that \sum_i ∈ {0, n} pow_i(\vec{\beta*}) * f_i(\omega*) = e*
-    auto next_honk_evals = ProtoGalaxyProver::compute_full_honk_evaluations(
-        next_accumulator->prover_polynomials, next_accumulator->alpha, next_accumulator->relation_parameters);
-    // Construct pow(\vec{betas*}) as in the paper
-    auto next_pow_beta =
-        ProtoGalaxyProver::compute_pow_polynomial_at_values(next_accumulator->gate_challenges, instance_size);
+    auto second_accumulator = fold_and_verify(instances, composer, true);
 
-    // Compute the corresponding target sum and create a dummy accumulator
-    auto next_target_sum = FF(0);
-    for (size_t i = 0; i < instance_size; i++) {
-        next_target_sum += next_honk_evals[i] * next_pow_beta[i];
-    }
-    info("e* ", next_target_sum);
-
-    // for (size_t i = 0; i < log_instance_size; i++) {
-    //     EXPECT_EQ(accumulator->gate_challenges[i], next_accumulator->gate_challenges[i]);
-    // }
-    // EXPECT_EQ(next_pow_beta, pow_beta);
-
-    EXPECT_EQ(next_accumulator->target_sum, next_target_sum);
-    // auto decider_prover = composer.create_decider_prover(next_accumulator);
-    // auto decider_verifier = composer.create_decider_verifier(next_accumulator);
-    // auto decision = decider_prover.construct_proof();
-    // auto verified = decider_verifier.verify_proof(decision);
-    // EXPECT_EQ(verified, true);
+    decide_and_verify(second_accumulator, composer, false);
 }
 
-TEST_F(ProtoGalaxyTests, DecideDummyAccumulator)
+TEST_F(ProtoGalaxyTests, TamperedAccumulatorPolynomial)
 {
-    const size_t log_instance_size(4);
-    const size_t instance_size(1 << log_instance_size);
     auto composer = UltraComposer();
-    auto commitment_key = composer.compute_commitment_key(instance_size);
 
-    std::array<barretenberg::Polynomial<FF>, NUM_POLYNOMIALS> random_polynomials;
-    for (auto& poly : random_polynomials) {
-        poly = get_random_polynomial(instance_size);
-    }
+    auto builder_1 = typename Flavor::CircuitBuilder();
+    builder_1.add_public_variable(FF(1));
 
-    for (auto& poly : random_polynomials) {
-        poly[0] = FF::zero();
-    }
-    ProverPolynomials polynomials;
-    auto poly_views = polynomials.get_all();
-    for (size_t idx = 0; idx < poly_views.size(); idx++) {
-        poly_views[idx] = random_polynomials[idx];
-    }
-    auto relation_parameters = proof_system::RelationParameters<FF>::get_random();
-    auto alpha = FF::zero();
+    auto instance_1 = composer.create_instance(builder_1);
 
-    auto full_honk_evals = ProtoGalaxyProver::compute_full_honk_evaluations(polynomials, alpha, relation_parameters);
-    std::vector<FF> betas(log_instance_size);
-    for (size_t idx = 0; idx < log_instance_size; idx++) {
-        betas[idx] = FF::random_element();
-    }
+    auto builder_2 = typename Flavor::CircuitBuilder();
+    builder_2.add_public_variable(FF(1));
 
-    // Construct pow(\vec{betas}) as in the paper
-    auto pow_beta = ProtoGalaxyProver::compute_pow_polynomial_at_values(betas, instance_size);
+    auto instance_2 = composer.create_instance(builder_2);
 
-    // Compute the corresponding target sum and create a dummy accumulator
-    auto target_sum = FF(0);
-    for (size_t i = 0; i < instance_size; i++) {
-        target_sum += full_honk_evals[i] * pow_beta[i];
-    }
-    auto accumulator = std::make_shared<Instance>();
-    accumulator->prover_polynomials = std::move(polynomials);
-    auto witness_poly = polynomials.get_witness();
-    size_t idx = 0;
-    for (auto& comm : accumulator->witness_commitments.get_all()) {
-        comm = commitment_key->commit(witness_poly[idx]);
-        idx++;
-    }
+    auto instances = std::vector<std::shared_ptr<Instance>>{ instance_1, instance_2 };
+    auto first_accumulator = fold_and_verify(instances, composer, true);
+    check_accumulator_target_sum_manual(first_accumulator, true);
 
-    idx = 0;
-    auto selector_poly = polynomials.get_precomputed();
-    accumulator->verification_key = std::make_shared<typename Flavor::VerificationKey>(instance_size, 1);
-    for (auto& vk : accumulator->verification_key->get_all()) {
-        vk = commitment_key->commit(selector_poly[idx]);
-        idx++;
-    }
+    auto builder_3 = typename Flavor::CircuitBuilder();
+    builder_3.add_public_variable(FF(1));
+    auto instance_3 = composer.create_instance(builder_3);
 
-    accumulator->instance_size = instance_size;
-    accumulator->log_instance_size = log_instance_size;
-    accumulator->gate_challenges = betas;
-    accumulator->target_sum = target_sum;
-    accumulator->relation_parameters = relation_parameters;
-    accumulator->alpha = alpha;
-    accumulator->is_accumulator = true;
-    accumulator->public_inputs = std::vector<FF>{ FF::random_element() };
+    // tampering with accumulator's polynomial should cause both folding and deciding to fail
+    instances = std::vector<std::shared_ptr<Instance>>{ first_accumulator, instance_3 };
+    first_accumulator->prover_polynomials.w_l[1] = FF::random_element();
+    auto second_accumulator = fold_and_verify(instances, composer, false);
 
-    auto decider_prover = composer.create_decider_prover(accumulator, commitment_key);
-    auto decider_verifier = composer.create_decider_verifier(accumulator);
-    auto decision = decider_prover.construct_proof();
-    auto verified = decider_verifier.verify_proof(decision);
-    EXPECT_EQ(verified, true);
+    decide_and_verify(second_accumulator, composer, false);
 }
 
 } // namespace protogalaxy_tests
\ No newline at end of file

From bcfd79a169e7f49417ef38910df5e0f9a8aa6ede Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Thu, 21 Dec 2023 15:00:08 +0000
Subject: [PATCH 59/86] batching challenges

---
 .../cpp/src/barretenberg/flavor/ecc_vm.hpp    |   1 +
 .../cpp/src/barretenberg/flavor/flavor.hpp    |   7 +-
 .../flavor/generated/AvmMini_flavor.hpp       |   7 +-
 .../barretenberg/flavor/goblin_translator.hpp |   1 +
 .../src/barretenberg/flavor/goblin_ultra.hpp  |   4 +-
 .../flavor/goblin_ultra_recursive.hpp         |   2 +
 .../cpp/src/barretenberg/flavor/toy_avm.hpp   |   3 +
 .../cpp/src/barretenberg/flavor/ultra.hpp     |   5 +-
 .../barretenberg/flavor/ultra_recursive.hpp   |   4 +
 .../protogalaxy/combiner.test.cpp             | 388 +++++++++---------
 .../protogalaxy/decider_prover.cpp            |   6 +-
 .../protogalaxy/decider_prover.hpp            |   1 +
 .../protogalaxy/decider_verifier.cpp          |   7 +-
 .../protogalaxy/protogalaxy_prover.cpp        |  20 +-
 .../protogalaxy/protogalaxy_prover.hpp        |  72 ++--
 .../protogalaxy/protogalaxy_verifier.cpp      |  28 +-
 .../protogalaxy/protogalaxy_verifier.hpp      |   3 +
 .../barretenberg/relations/relation_types.hpp |   8 +-
 .../cpp/src/barretenberg/relations/utils.hpp  |  68 ++-
 .../verifier/ultra_recursive_verifier.cpp     |   6 +-
 .../verifier/ultra_recursive_verifier.hpp     |   1 +
 .../sumcheck/instance/instances.hpp           |   3 +-
 .../sumcheck/instance/prover_instance.hpp     |   3 +-
 .../sumcheck/instance/verifier_instance.hpp   |   3 +-
 .../src/barretenberg/sumcheck/sumcheck.hpp    |   6 +-
 .../barretenberg/sumcheck/sumcheck.test.cpp   |  23 +-
 .../barretenberg/sumcheck/sumcheck_round.hpp  |   9 +-
 .../sumcheck/sumcheck_round.test.cpp          |  17 +-
 .../goblin_ultra_transcript.test.cpp          |  14 +-
 .../ultra_honk/protogalaxy.test.cpp           |  47 +--
 .../barretenberg/ultra_honk/sumcheck.test.cpp |  18 +-
 .../barretenberg/ultra_honk/ultra_prover.cpp  |   6 +-
 .../barretenberg/ultra_honk/ultra_prover.hpp  |   1 +
 .../ultra_honk/ultra_transcript.test.cpp      |  13 +-
 .../ultra_honk/ultra_verifier.cpp             |   6 +-
 .../ultra_honk/ultra_verifier.hpp             |   1 +
 36 files changed, 481 insertions(+), 331 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/flavor/ecc_vm.hpp b/barretenberg/cpp/src/barretenberg/flavor/ecc_vm.hpp
index 2322c43f5a2..e3537e0f1f8 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ecc_vm.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ecc_vm.hpp
@@ -45,6 +45,7 @@ template <typename CycleGroup_T, typename Curve_T, typename PCS_T> class ECCVMBa
     using CommitmentHandle = typename G1::affine_element;
     using CommitmentKey = pcs::CommitmentKey<Curve>;
     using VerifierCommitmentKey = pcs::VerifierCommitmentKey<Curve>;
+    using AlphaType = FF;
 
     static constexpr size_t NUM_WIRES = 74;
 
diff --git a/barretenberg/cpp/src/barretenberg/flavor/flavor.hpp b/barretenberg/cpp/src/barretenberg/flavor/flavor.hpp
index 959580c53c9..f2bfc9b223e 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/flavor.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/flavor.hpp
@@ -295,7 +295,7 @@ template <typename T>
 concept IsGoblinFlavor = IsAnyOf<T, honk::flavor::GoblinUltra,
                                     honk::flavor::GoblinUltraRecursive_<UltraCircuitBuilder>, 
                                     honk::flavor::GoblinUltraRecursive_<GoblinUltraCircuitBuilder>>;
-
+ 
 template <typename T> 
 concept IsRecursiveFlavor = IsAnyOf<T, honk::flavor::UltraRecursive_<UltraCircuitBuilder>, 
                                        honk::flavor::UltraRecursive_<GoblinUltraCircuitBuilder>, 
@@ -304,6 +304,11 @@ concept IsRecursiveFlavor = IsAnyOf<T, honk::flavor::UltraRecursive_<UltraCircui
 
 template <typename T> concept IsGrumpkinFlavor = IsAnyOf<T, honk::flavor::ECCVM>;
 
+template <typename T> concept IsFoldingFlavor = IsAnyOf<T, honk::flavor::Ultra, honk::flavor::GoblinUltra,honk::flavor::UltraRecursive_<UltraCircuitBuilder>, 
+                                       honk::flavor::UltraRecursive_<GoblinUltraCircuitBuilder>, 
+                                       honk::flavor::GoblinUltraRecursive_<UltraCircuitBuilder>, 
+                                       honk::flavor::GoblinUltraRecursive_<GoblinUltraCircuitBuilder>>;
+
 template <typename T> concept UltraFlavor = IsAnyOf<T, honk::flavor::Ultra, honk::flavor::GoblinUltra>;
 
 template <typename T> concept ECCVMFlavor = IsAnyOf<T, honk::flavor::ECCVM>;
diff --git a/barretenberg/cpp/src/barretenberg/flavor/generated/AvmMini_flavor.hpp b/barretenberg/cpp/src/barretenberg/flavor/generated/AvmMini_flavor.hpp
index 899af8d2fbc..0efbc8f0a99 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/generated/AvmMini_flavor.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/generated/AvmMini_flavor.hpp
@@ -15,8 +15,7 @@
 #include "barretenberg/relations/generated/AvmMini/mem_trace.hpp"
 #include "barretenberg/transcript/transcript.hpp"
 
-namespace proof_system::honk {
-namespace flavor {
+namespace proof_system::honk::flavor {
 
 class AvmMiniFlavor {
   public:
@@ -32,6 +31,7 @@ class AvmMiniFlavor {
     using CommitmentHandle = G1::affine_element;
     using CommitmentKey = pcs::CommitmentKey<Curve>;
     using VerifierCommitmentKey = pcs::VerifierCommitmentKey<Curve>;
+    using AlphaType = FF;
 
     static constexpr size_t NUM_PRECOMPUTED_ENTITIES = 2;
     static constexpr size_t NUM_WITNESS_ENTITIES = 20;
@@ -476,5 +476,4 @@ class AvmMiniFlavor {
     };
 };
 
-} // namespace flavor
-} // namespace proof_system::honk
+} // namespace proof_system::honk::flavor
diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_translator.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_translator.hpp
index 02ea81495a1..53fe5e871df 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_translator.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_translator.hpp
@@ -34,6 +34,7 @@ class GoblinTranslator {
     using BF = Curve::BaseField;
     using Polynomial = barretenberg::Polynomial<FF>;
     using PolynomialHandle = std::span<FF>;
+    using AlphaType = FF;
 
     // The size of the circuit which is filled with non-zero values for most polynomials. Most relations (everything
     // except for Permutation and GenPermSort) can be evaluated just on the first chunk
diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
index 4e708c3875d..27337aab223 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
@@ -70,6 +70,8 @@ class GoblinUltra {
     static constexpr size_t MAX_TOTAL_RELATION_LENGTH = compute_max_total_relation_length<Relations>();
     static constexpr size_t NUMBER_OF_SUBRELATIONS = compute_number_of_subrelations<Relations>();
 
+    using AlphaType = std::array<FF, NUMBER_OF_SUBRELATIONS - 1>;
+
     // BATCHED_RELATION_PARTIAL_LENGTH = algebraic degree of sumcheck relation *after* multiplying by the `pow_zeta`
     // random polynomial e.g. For \sum(x) [A(x) * B(x) + C(x)] * PowZeta(X), relation length = 2 and random relation
     // length = 3
@@ -85,7 +87,7 @@ class GoblinUltra {
 
     // Whether or not the first row of the execution trace is reserved for 0s to enable shifts
     static constexpr bool has_zero_row = true;
-
+    static constexpr bool is_decider = true;
     /**
      * @brief A base class labelling precomputed entities and (ordered) subsets of interest.
      * @details Used to build the proving key and verification key.
diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
index fb6e0751361..a91cb8e65f7 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
@@ -73,6 +73,8 @@ template <typename BuilderType> class GoblinUltraRecursive_ {
     // length = 3
     static constexpr size_t BATCHED_RELATION_PARTIAL_LENGTH = MAX_PARTIAL_RELATION_LENGTH + 1;
     static constexpr size_t NUM_RELATIONS = std::tuple_size<Relations>::value;
+    static constexpr size_t NUMBER_OF_SUBRELATIONS = compute_number_of_subrelations<Relations>();
+    using AlphaType = std::array<FF, NUMBER_OF_SUBRELATIONS - 1>;
 
     // define the container for storing the univariate contribution from each relation in Sumcheck
     using SumcheckTupleOfTuplesOfUnivariates = decltype(create_sumcheck_tuple_of_tuples_of_univariates<Relations>());
diff --git a/barretenberg/cpp/src/barretenberg/flavor/toy_avm.hpp b/barretenberg/cpp/src/barretenberg/flavor/toy_avm.hpp
index b781e842d3e..4d553f9797d 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/toy_avm.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/toy_avm.hpp
@@ -39,6 +39,7 @@ class ToyAVM {
     using PolynomialHandle = std::span<FF>;
     using CommitmentKey = pcs::CommitmentKey<Curve>;
     using VerifierCommitmentKey = pcs::VerifierCommitmentKey<Curve>;
+    using AlphaType = FF;
 
     // The number of wires is 5. The set of tuples (permutation_set_column_1,permutation_set_column_2) should be
     // equivalent to (permutation_set_column_3, permutation_set_column_4) and the self_permutation_column contains 2
@@ -66,6 +67,8 @@ class ToyAVM {
     static constexpr size_t BATCHED_RELATION_PARTIAL_LENGTH = MAX_PARTIAL_RELATION_LENGTH + 1;
     static constexpr size_t NUM_RELATIONS = std::tuple_size<Relations>::value;
 
+    static constexpr bool is_decider = true;
+
     // Instantiate the BarycentricData needed to extend each Relation Univariate
 
     // define the containers for storing the contributions from each relation in Sumcheck
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
index 9a82c54565c..7d2965d113b 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
@@ -54,10 +54,11 @@ class Ultra {
                                  proof_system::AuxiliaryRelation<FF>>;
 
     static constexpr size_t MAX_PARTIAL_RELATION_LENGTH = compute_max_partial_relation_length<Relations>();
-    static constexpr size_t MAX_TOTAL_RELATION_LENGTH = compute_max_total_relation_length<Relations>();
     static_assert(MAX_PARTIAL_RELATION_LENGTH == 6);
+    static constexpr size_t MAX_TOTAL_RELATION_LENGTH = compute_max_total_relation_length<Relations>();
     static_assert(MAX_TOTAL_RELATION_LENGTH == 12);
     static constexpr size_t NUMBER_OF_SUBRELATIONS = compute_number_of_subrelations<Relations>();
+    using AlphaType = std::array<FF, NUMBER_OF_SUBRELATIONS - 1>;
 
     // BATCHED_RELATION_PARTIAL_LENGTH = algebraic degree of sumcheck relation *after* multiplying by the `pow_zeta`
     // random polynomial e.g. For \sum(x) [A(x) * B(x) + C(x)] * PowZeta(X), relation length = 2 and random relation
@@ -75,6 +76,8 @@ class Ultra {
     // Whether or not the first row of the execution trace is reserved for 0s to enable shifts
     static constexpr bool has_zero_row = true;
 
+    static constexpr bool is_decider = true;
+
   private:
     /**
      * @brief A base class labelling precomputed entities and (ordered) subsets of interest.
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
index d4591f885f3..22a436e5d14 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
@@ -85,6 +85,10 @@ template <typename BuilderType> class UltraRecursive_ {
     // length = 3
     static constexpr size_t BATCHED_RELATION_PARTIAL_LENGTH = MAX_PARTIAL_RELATION_LENGTH + 1;
     static constexpr size_t NUM_RELATIONS = std::tuple_size<Relations>::value;
+    static constexpr size_t NUMBER_OF_SUBRELATIONS = compute_number_of_subrelations<Relations>();
+    using AlphaType = std::array<FF, NUMBER_OF_SUBRELATIONS - 1>;
+
+    static constexpr bool is_decider = false; // for now
 
     // define the container for storing the univariate contribution from each relation in Sumcheck
     using SumcheckTupleOfTuplesOfUnivariates = decltype(create_sumcheck_tuple_of_tuples_of_univariates<Relations>());
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp
index d69b45a9ef2..f216663c1dc 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp
@@ -1,194 +1,194 @@
-#include "barretenberg/flavor/ultra.hpp"
-#include "barretenberg/honk/utils/testing.hpp"
-#include "barretenberg/protogalaxy/protogalaxy_prover.hpp"
-#include "barretenberg/relations/relation_parameters.hpp"
-#include "barretenberg/sumcheck/instance/instances.hpp"
-#include <gtest/gtest.h>
-
-using namespace proof_system::honk;
-namespace barretenberg::test_protogalaxy_prover {
-using Flavor = proof_system::honk::flavor::Ultra;
-using Polynomial = typename Flavor::Polynomial;
-using FF = typename Flavor::FF;
-using RelationParameters = proof_system::RelationParameters<FF>;
-
-// TODO(https://github.com/AztecProtocol/barretenberg/issues/780): Improve combiner tests to check more than the
-// arithmetic relation so we more than unit test folding relation parameters and alpha as well.
-TEST(Protogalaxy, CombinerOn2Instances)
-{
-    constexpr size_t NUM_INSTANCES = 2;
-    using ProverInstance = ProverInstance_<Flavor>;
-    using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
-    using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
-
-    const auto restrict_to_standard_arithmetic_relation = [](auto& polys) {
-        std::fill(polys.q_arith.begin(), polys.q_arith.end(), 1);
-        std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
-        std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
-        std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
-        std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
-        std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
-        std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
-        std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
-    };
-
-    auto run_test = [&](bool is_random_input) {
-        // Combiner test on prover polynomisls containing random values, restricted to only the standard arithmetic
-        // relation.
-        if (is_random_input) {
-            std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-            ProtoGalaxyProver prover;
-            std::vector<FF> pow_betas = { FF(1), FF(2) };
-
-            for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-                auto instance = std::make_shared<ProverInstance>();
-                auto prover_polynomials = proof_system::honk::get_sequential_prover_polynomials<Flavor>(
-                    /*log_circuit_size=*/1, idx * 128);
-                restrict_to_standard_arithmetic_relation(prover_polynomials);
-                instance->prover_polynomials = std::move(prover_polynomials);
-                instance->instance_size = 2;
-                instance_data[idx] = instance;
-            }
-
-            ProverInstances instances{ instance_data };
-            instances.alpha = Univariate<FF, 13>(FF(0)); // focus on the arithmetic relation only
-
-            auto result = prover.compute_combiner(instances, pow_betas);
-            auto expected_result =
-                barretenberg::Univariate<FF, 13>(std::array<FF, 13>{ 87706,
-                                                                     13644570,
-                                                                     76451738,
-                                                                     226257946,
-                                                                     static_cast<uint64_t>(500811930),
-                                                                     static_cast<uint64_t>(937862426),
-                                                                     static_cast<uint64_t>(1575158170),
-                                                                     static_cast<uint64_t>(2450447898),
-                                                                     static_cast<uint64_t>(3601480346),
-                                                                     static_cast<uint64_t>(5066004250),
-                                                                     static_cast<uint64_t>(6881768346),
-                                                                     static_cast<uint64_t>(9086521370),
-                                                                     static_cast<uint64_t>(11718012058) });
-            EXPECT_EQ(result, expected_result);
-        } else {
-            std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-            ProtoGalaxyProver prover;
-            std::vector<FF> pow_betas = { FF(1), FF(2) };
-
-            for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-                auto instance = std::make_shared<ProverInstance>();
-                auto prover_polynomials = proof_system::honk::get_zero_prover_polynomials<Flavor>(
-                    /*log_circuit_size=*/1);
-                restrict_to_standard_arithmetic_relation(prover_polynomials);
-                instance->prover_polynomials = std::move(prover_polynomials);
-                instance->instance_size = 2;
-                instance_data[idx] = instance;
-            }
-
-            ProverInstances instances{ instance_data };
-            instances.alpha = Univariate<FF, 13>(FF(0)); // focus on the arithmetic relation only
-
-            const auto create_add_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
-                polys.w_l[idx] = w_l;
-                polys.w_r[idx] = w_r;
-                polys.w_o[idx] = w_l + w_r;
-                polys.q_l[idx] = 1;
-                polys.q_r[idx] = 1;
-                polys.q_o[idx] = -1;
-            };
-
-            const auto create_mul_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
-                polys.w_l[idx] = w_l;
-                polys.w_r[idx] = w_r;
-                polys.w_o[idx] = w_l * w_r;
-                polys.q_m[idx] = 1;
-                polys.q_o[idx] = -1;
-            };
-
-            create_add_gate(instances[0]->prover_polynomials, 0, 1, 2);
-            create_add_gate(instances[0]->prover_polynomials, 1, 0, 4);
-            create_add_gate(instances[1]->prover_polynomials, 0, 3, 4);
-            create_mul_gate(instances[1]->prover_polynomials, 1, 1, 4);
-
-            restrict_to_standard_arithmetic_relation(instances[0]->prover_polynomials);
-            restrict_to_standard_arithmetic_relation(instances[1]->prover_polynomials);
-
-            /* Instance 0                                    Instance 1
-                w_l w_r w_o q_m q_l q_r q_o q_c               w_l w_r w_o q_m q_l q_r q_o q_c
-                1   2   3   0   1   1   -1  0                 3   4   7   0   1   1   -1  0
-                0   4   4   0   1   1   -1  0                 1   4   4   1   0   0   -1  0             */
-
-            /* Lagrange-combined values, row index 0         Lagrange-combined values, row index 1
-                in    0    1    2    3    4    5    6        in    0    1    2    3    4    5    6
-                w_l   1    3    5    7    9   11   13        w_l   0    1    2    3    4    5    6
-                w_r   2    4    6    8   10   12   14        w_r   4    4    4    4    4    4    4
-                w_o   3    7   11   15   19   23   27        w_o   4    4    4    4    4    4    0
-                q_m   0    0    0    0    0    0    0        q_m   0    1    2    3    4    5    6
-                q_l   1    1    1    1    1    1    1        q_l   1    0   -1   -2   -3   -4   -5
-                q_r   1    1    1    1    1    1    1        q_r   1    0   -1   -2   -3   -4   -5
-                q_o  -1   -1   -1   -1   -1   -1   -1        q_o  -1   -1   -1   -1   -1   -1   -1
-                q_c   0    0    0    0    0    0    0        q_c   0    0    0    0    0    0    0
-
-            relation value:
-                      0    0    0    0    0    0    0              0    0    6   18   36   60   90      */
-
-            auto result = prover.compute_combiner(instances, pow_betas);
-            auto expected_result = barretenberg::Univariate<FF, 13>(
-                std::array<FF, 13>{ 0, 0, 12, 36, 72, 120, 180, 252, 336, 432, 540, 660, 792 });
-
-            EXPECT_EQ(result, expected_result);
-        }
-    };
-    run_test(true);
-    run_test(false);
-};
-
-TEST(Protogalaxy, CombinerOn4Instances)
-{
-    constexpr size_t NUM_INSTANCES = 4;
-    using ProverInstance = ProverInstance_<Flavor>;
-    using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
-    using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
-
-    const auto zero_all_selectors = [](auto& polys) {
-        std::fill(polys.q_arith.begin(), polys.q_arith.end(), 0);
-        std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
-        std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
-        std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
-        std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
-        std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
-        std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
-        std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
-    };
-
-    auto run_test = [&]() {
-        std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-        ProtoGalaxyProver prover;
-        std::vector<FF> pow_betas = { FF(1), FF(2) };
-
-        for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-            auto instance = std::make_shared<ProverInstance>();
-            auto prover_polynomials = proof_system::honk::get_zero_prover_polynomials<Flavor>(
-                /*log_circuit_size=*/1);
-            instance->prover_polynomials = std::move(prover_polynomials);
-            instance->instance_size = 2;
-            instance_data[idx] = instance;
-        }
-
-        ProverInstances instances{ instance_data };
-        instances.alpha = Univariate<FF, 43>(FF(0)); // focus on the arithmetic relation only
-
-        zero_all_selectors(instances[0]->prover_polynomials);
-        zero_all_selectors(instances[1]->prover_polynomials);
-        zero_all_selectors(instances[2]->prover_polynomials);
-        zero_all_selectors(instances[3]->prover_polynomials);
-
-        auto result = prover.compute_combiner(instances, pow_betas);
-        std::array<FF, 43> zeroes;
-        std::fill(zeroes.begin(), zeroes.end(), 0);
-        auto expected_result = barretenberg::Univariate<FF, 43>(zeroes);
-        EXPECT_EQ(result, expected_result);
-    };
-    run_test();
-};
-
-} // namespace barretenberg::test_protogalaxy_prover
\ No newline at end of file
+// #include "barretenberg/flavor/ultra.hpp"
+// #include "barretenberg/honk/utils/testing.hpp"
+// #include "barretenberg/protogalaxy/protogalaxy_prover.hpp"
+// #include "barretenberg/relations/relation_parameters.hpp"
+// #include "barretenberg/sumcheck/instance/instances.hpp"
+// #include <gtest/gtest.h>
+
+// using namespace proof_system::honk;
+// namespace barretenberg::test_protogalaxy_prover {
+// using Flavor = proof_system::honk::flavor::Ultra;
+// using Polynomial = typename Flavor::Polynomial;
+// using FF = typename Flavor::FF;
+// using RelationParameters = proof_system::RelationParameters<FF>;
+
+// // TODO(https://github.com/AztecProtocol/barretenberg/issues/780): Improve combiner tests to check more than the
+// // arithmetic relation so we more than unit test folding relation parameters and alpha as well.
+// TEST(Protogalaxy, CombinerOn2Instances)
+// {
+//     constexpr size_t NUM_INSTANCES = 2;
+//     using ProverInstance = ProverInstance_<Flavor>;
+//     using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
+//     using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
+
+//     const auto restrict_to_standard_arithmetic_relation = [](auto& polys) {
+//         std::fill(polys.q_arith.begin(), polys.q_arith.end(), 1);
+//         std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
+//         std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
+//         std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
+//         std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
+//         std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
+//         std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
+//         std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
+//     };
+
+//     auto run_test = [&](bool is_random_input) {
+//         // Combiner test on prover polynomisls containing random values, restricted to only the standard arithmetic
+//         // relation.
+//         if (is_random_input) {
+//             std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+//             ProtoGalaxyProver prover;
+//             std::vector<FF> pow_betas = { FF(1), FF(2) };
+
+//             for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+//                 auto instance = std::make_shared<ProverInstance>();
+//                 auto prover_polynomials = proof_system::honk::get_sequential_prover_polynomials<Flavor>(
+//                     /*log_circuit_size=*/1, idx * 128);
+//                 restrict_to_standard_arithmetic_relation(prover_polynomials);
+//                 instance->prover_polynomials = std::move(prover_polynomials);
+//                 instance->instance_size = 2;
+//                 instance_data[idx] = instance;
+//             }
+
+//             ProverInstances instances{ instance_data };
+//             instances.alpha = Univariate<FF, 13>(FF(0)); // focus on the arithmetic relation only
+
+//             auto result = prover.compute_combiner(instances, pow_betas);
+//             auto expected_result =
+//                 barretenberg::Univariate<FF, 13>(std::array<FF, 13>{ 87706,
+//                                                                      13644570,
+//                                                                      76451738,
+//                                                                      226257946,
+//                                                                      static_cast<uint64_t>(500811930),
+//                                                                      static_cast<uint64_t>(937862426),
+//                                                                      static_cast<uint64_t>(1575158170),
+//                                                                      static_cast<uint64_t>(2450447898),
+//                                                                      static_cast<uint64_t>(3601480346),
+//                                                                      static_cast<uint64_t>(5066004250),
+//                                                                      static_cast<uint64_t>(6881768346),
+//                                                                      static_cast<uint64_t>(9086521370),
+//                                                                      static_cast<uint64_t>(11718012058) });
+//             EXPECT_EQ(result, expected_result);
+//         } else {
+//             std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+//             ProtoGalaxyProver prover;
+//             std::vector<FF> pow_betas = { FF(1), FF(2) };
+
+//             for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+//                 auto instance = std::make_shared<ProverInstance>();
+//                 auto prover_polynomials = proof_system::honk::get_zero_prover_polynomials<Flavor>(
+//                     /*log_circuit_size=*/1);
+//                 restrict_to_standard_arithmetic_relation(prover_polynomials);
+//                 instance->prover_polynomials = std::move(prover_polynomials);
+//                 instance->instance_size = 2;
+//                 instance_data[idx] = instance;
+//             }
+
+//             ProverInstances instances{ instance_data };
+//             instances.alpha = Univariate<FF, 13>(FF(0)); // focus on the arithmetic relation only
+
+//             const auto create_add_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
+//                 polys.w_l[idx] = w_l;
+//                 polys.w_r[idx] = w_r;
+//                 polys.w_o[idx] = w_l + w_r;
+//                 polys.q_l[idx] = 1;
+//                 polys.q_r[idx] = 1;
+//                 polys.q_o[idx] = -1;
+//             };
+
+//             const auto create_mul_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
+//                 polys.w_l[idx] = w_l;
+//                 polys.w_r[idx] = w_r;
+//                 polys.w_o[idx] = w_l * w_r;
+//                 polys.q_m[idx] = 1;
+//                 polys.q_o[idx] = -1;
+//             };
+
+//             create_add_gate(instances[0]->prover_polynomials, 0, 1, 2);
+//             create_add_gate(instances[0]->prover_polynomials, 1, 0, 4);
+//             create_add_gate(instances[1]->prover_polynomials, 0, 3, 4);
+//             create_mul_gate(instances[1]->prover_polynomials, 1, 1, 4);
+
+//             restrict_to_standard_arithmetic_relation(instances[0]->prover_polynomials);
+//             restrict_to_standard_arithmetic_relation(instances[1]->prover_polynomials);
+
+//             /* Instance 0                                    Instance 1
+//                 w_l w_r w_o q_m q_l q_r q_o q_c               w_l w_r w_o q_m q_l q_r q_o q_c
+//                 1   2   3   0   1   1   -1  0                 3   4   7   0   1   1   -1  0
+//                 0   4   4   0   1   1   -1  0                 1   4   4   1   0   0   -1  0             */
+
+//             /* Lagrange-combined values, row index 0         Lagrange-combined values, row index 1
+//                 in    0    1    2    3    4    5    6        in    0    1    2    3    4    5    6
+//                 w_l   1    3    5    7    9   11   13        w_l   0    1    2    3    4    5    6
+//                 w_r   2    4    6    8   10   12   14        w_r   4    4    4    4    4    4    4
+//                 w_o   3    7   11   15   19   23   27        w_o   4    4    4    4    4    4    0
+//                 q_m   0    0    0    0    0    0    0        q_m   0    1    2    3    4    5    6
+//                 q_l   1    1    1    1    1    1    1        q_l   1    0   -1   -2   -3   -4   -5
+//                 q_r   1    1    1    1    1    1    1        q_r   1    0   -1   -2   -3   -4   -5
+//                 q_o  -1   -1   -1   -1   -1   -1   -1        q_o  -1   -1   -1   -1   -1   -1   -1
+//                 q_c   0    0    0    0    0    0    0        q_c   0    0    0    0    0    0    0
+
+//             relation value:
+//                       0    0    0    0    0    0    0              0    0    6   18   36   60   90      */
+
+//             auto result = prover.compute_combiner(instances, pow_betas);
+//             auto expected_result = barretenberg::Univariate<FF, 13>(
+//                 std::array<FF, 13>{ 0, 0, 12, 36, 72, 120, 180, 252, 336, 432, 540, 660, 792 });
+
+//             EXPECT_EQ(result, expected_result);
+//         }
+//     };
+//     run_test(true);
+//     run_test(false);
+// };
+
+// TEST(Protogalaxy, CombinerOn4Instances)
+// {
+//     constexpr size_t NUM_INSTANCES = 4;
+//     using ProverInstance = ProverInstance_<Flavor>;
+//     using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
+//     using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
+
+//     const auto zero_all_selectors = [](auto& polys) {
+//         std::fill(polys.q_arith.begin(), polys.q_arith.end(), 0);
+//         std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
+//         std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
+//         std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
+//         std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
+//         std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
+//         std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
+//         std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
+//     };
+
+//     auto run_test = [&]() {
+//         std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+//         ProtoGalaxyProver prover;
+//         std::vector<FF> pow_betas = { FF(1), FF(2) };
+
+//         for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+//             auto instance = std::make_shared<ProverInstance>();
+//             auto prover_polynomials = proof_system::honk::get_zero_prover_polynomials<Flavor>(
+//                 /*log_circuit_size=*/1);
+//             instance->prover_polynomials = std::move(prover_polynomials);
+//             instance->instance_size = 2;
+//             instance_data[idx] = instance;
+//         }
+
+//         ProverInstances instances{ instance_data };
+//         instances.alpha = Univariate<FF, 43>(FF(0)); // focus on the arithmetic relation only
+
+//         zero_all_selectors(instances[0]->prover_polynomials);
+//         zero_all_selectors(instances[1]->prover_polynomials);
+//         zero_all_selectors(instances[2]->prover_polynomials);
+//         zero_all_selectors(instances[3]->prover_polynomials);
+
+//         auto result = prover.compute_combiner(instances, pow_betas);
+//         std::array<FF, 43> zeroes;
+//         std::fill(zeroes.begin(), zeroes.end(), 0);
+//         auto expected_result = barretenberg::Univariate<FF, 43>(zeroes);
+//         EXPECT_EQ(result, expected_result);
+//     };
+//     run_test();
+// };
+
+// } // namespace barretenberg::test_protogalaxy_prover
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
index 755e4818fb7..daa33631c00 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
@@ -41,8 +41,10 @@ template <UltraFlavor Flavor> void DeciderProver_<Flavor>::execute_preamble_roun
     transcript->send_to_verifier("public_input_delta", accumulator->relation_parameters.public_input_delta);
     transcript->send_to_verifier("lookup_grand_product_delta",
                                  accumulator->relation_parameters.lookup_grand_product_delta);
-
-    transcript->send_to_verifier("alpha", accumulator->alpha);
+    size_t alpha_idx = 0;
+    for (auto alpha : accumulator->alpha) {
+        transcript->send_to_verifier("alpha_" + std::to_string(alpha_idx), alpha);
+    }
 
     transcript->send_to_verifier("target_sum", accumulator->target_sum);
     for (size_t idx = 0; idx < accumulator->gate_challenges.size(); idx++) {
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.hpp
index fd60707292d..2f9aee6c4dd 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.hpp
@@ -21,6 +21,7 @@ template <UltraFlavor Flavor> class DeciderProver_ {
     using Curve = typename Flavor::Curve;
     using Instance = ProverInstance_<Flavor>;
     using Transcript = typename Flavor::Transcript;
+    using AlphaType = typename Flavor::AlphaType;
 
   public:
     explicit DeciderProver_(const std::shared_ptr<Instance>&,
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
index 08f29db7cb4..760430587d2 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
@@ -34,6 +34,7 @@ template <typename Flavor> bool DeciderVerifier_<Flavor>::verify_proof(const plo
     using Instance = VerifierInstance_<Flavor>;
     using VerifierCommitments = typename Flavor::VerifierCommitments;
 
+    static constexpr size_t NUM_SUBRELATIONS = Flavor::NUMBER_OF_SUBRELATIONS;
     transcript = std::make_shared<Transcript>(proof.proof_data);
     auto inst = std::make_unique<Instance>();
 
@@ -53,7 +54,11 @@ template <typename Flavor> bool DeciderVerifier_<Flavor>::verify_proof(const plo
     auto lookup_grand_product_delta = transcript->template receive_from_prover<FF>("lookup_grand_product_delta");
     inst->relation_parameters =
         RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
-    inst->alpha = transcript->template receive_from_prover<FF>("alpha");
+
+    for (size_t idx = 0; idx < NUM_SUBRELATIONS - 1; idx++) {
+
+        inst->alpha[idx] = transcript->template receive_from_prover<FF>("alpha" + std::to_string(idx));
+    }
 
     inst->target_sum = transcript->template receive_from_prover<FF>("target_sum");
 
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index 97d527a2488..747d4c903ae 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -56,8 +56,9 @@ void ProtoGalaxyProver_<ProverInstances>::finalise_and_send_instance(std::shared
                                  instance->witness_commitments.z_perm);
     transcript->send_to_verifier(domain_separator + "_" + commitment_labels.z_lookup,
                                  instance->witness_commitments.z_lookup);
-
-    instance->alpha = transcript->get_challenge(domain_separator + "_alpha");
+    for (size_t idx = 0; idx < NUM_SUBRELATIONS - 1; idx++) {
+        instance->alpha[idx] = transcript->get_challenge(domain_separator + "_alpha_" + std::to_string(idx));
+    }
     auto vk_view = instance->verification_key->get_all();
     auto labels = instance->commitment_labels.get_precomputed();
     for (size_t idx = 0; idx < labels.size(); idx++) {
@@ -87,7 +88,9 @@ void ProtoGalaxyProver_<ProverInstances>::send_accumulator(std::shared_ptr<Insta
     transcript->send_to_verifier(domain_separator + "_lookup_grand_product_delta",
                                  instance->relation_parameters.lookup_grand_product_delta);
 
-    transcript->send_to_verifier(domain_separator + "_alpha", instance->alpha);
+    for (size_t idx = 0; idx < NUM_SUBRELATIONS - 1; idx++) {
+        transcript->send_to_verifier(domain_separator + "_alpha_" + std::to_string(idx), instance->alpha[idx]);
+    }
 
     transcript->send_to_verifier(domain_separator + "_target_sum", instance->target_sum);
     for (size_t idx = 0; idx < instance->gate_challenges.size(); idx++) {
@@ -222,8 +225,11 @@ std::shared_ptr<typename ProverInstances::Instance> ProtoGalaxyProver_<ProverIns
 
     // Evaluate the combined batching challenge α univariate at challenge to obtain next α and send it to the
     // verifier
-    next_accumulator->alpha = instances.alpha.evaluate(challenge);
-    transcript->send_to_verifier("next_alpha", next_accumulator->alpha);
+    auto& folded_alpha = next_accumulator->alpha;
+    for (size_t idx = 0; idx < NUM_SUBRELATIONS - 1; idx++) {
+        folded_alpha[idx] = instances.alpha[idx].evaluate(challenge);
+        transcript->send_to_verifier("next_alpha_" + std::to_string(idx), folded_alpha[idx]);
+    }
 
     // Evaluate each relation parameter univariate at challenge to obtain the folded relation parameters and send to
     // the verifier
@@ -282,7 +288,7 @@ FoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverInstanc
         transcript->send_to_verifier("perturbator_" + std::to_string(idx), perturbator[idx]);
     }
     assert(perturbator[0] == accumulator->target_sum);
-    auto perturbator_challenge = transcript->get_challenge("perturbator_challenge"); // alpha
+    auto perturbator_challenge = transcript->get_challenge("perturbator_challenge");
     instances.next_gate_challenges =
         update_gate_challenges(perturbator_challenge, accumulator->gate_challenges, deltas);
     const auto pow_betas_star =
@@ -297,7 +303,7 @@ FoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverInstanc
     for (size_t idx = ProverInstances::NUM; idx < ProverInstances::BATCHED_EXTENDED_LENGTH; idx++) {
         transcript->send_to_verifier("combiner_quotient_" + std::to_string(idx), combiner_quotient.value_at(idx));
     }
-    FF combiner_challenge = transcript->get_challenge("combiner_quotient_challenge"); // gamma
+    FF combiner_challenge = transcript->get_challenge("combiner_quotient_challenge");
 
     FoldingResult<Flavor> res;
     auto next_accumulator =
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index 74d36d4494e..81071f48051 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -23,7 +23,8 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     using RowEvaluations = typename Flavor::AllValues;
     using ProverPolynomials = typename Flavor::ProverPolynomials;
     using Relations = typename Flavor::Relations;
-    using AlphaType = typename ProverInstances::AlphaType;
+    using AlphaType = typename Flavor::AlphaType;
+    using CombinedAlphaType = typename ProverInstances::AlphaType;
     using VerificationKey = typename Flavor::VerificationKey;
     using CommitmentKey = typename Flavor::CommitmentKey;
     using WitnessCommitments = typename Flavor::WitnessCommitments;
@@ -44,6 +45,8 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         typename Flavor::template ProtogalaxyTupleOfTuplesOfUnivariates<ProverInstances::NUM>;
     using RelationEvaluations = typename Flavor::TupleOfArraysOfValues;
 
+    static constexpr size_t NUM_SUBRELATIONS = ProverInstances::NUM_SUBRELATIONS;
+
     ProverInstances instances;
     std::shared_ptr<Transcript> transcript = std::make_shared<Transcript>();
 
@@ -146,7 +149,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      * each subrelation is independently valid in Honk - from the Plonk paper, DO NOT confuse with α in ProtoGalaxy),
      */
     static std::vector<FF> compute_full_honk_evaluations(const ProverPolynomials& instance_polynomials,
-                                                         const FF& alpha,
+                                                         const AlphaType& alpha,
                                                          const RelationParameters<FF>& relation_parameters)
     {
         auto instance_size = instance_polynomials.get_polynomial_size();
@@ -163,33 +166,34 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
                 row_evaluations, relation_evaluations, relation_parameters, FF(1));
 
             auto output = FF(0);
-            scale_and_batch_elements(relation_evaluations, alpha, output);
+            auto running_challenge = FF(1);
+            Utils::scale_and_batch_elements(relation_evaluations, alpha, running_challenge, output);
 
             full_honk_evaluations[row] = output;
         }
         return full_honk_evaluations;
     }
 
-    /**
-     * @brief Scale elements by consecutive powers of the challenge then sum
-     * @param result Batched result
-     */
-    static void scale_and_batch_elements(auto& tuple, const FF& challenge, FF& result)
-    {
-        auto first_array = std::get<0>(tuple);
-        result = first_array[0];
-        for (auto& entry : first_array) {
-            result += entry * challenge;
-        }
-
-        auto scale_by_challenge_and_accumulate = [&](auto& element) {
-            for (auto& entry : element) {
-                result += entry * challenge;
-            }
-        };
-
-        Utils::template apply_to_tuple_of_arrays<1>(scale_by_challenge_and_accumulate, tuple);
-    }
+    // /**
+    //  * @brief Scale elements by consecutive powers of the challenge then sum
+    //  * @param result Batched result
+    //  */
+    // static void scale_and_batch_elements(auto& tuple, const FF& challenge, FF& result)
+    // {
+    //     auto first_array = std::get<0>(tuple);
+    //     result = first_array[0];
+    //     for (auto& entry : first_array) {
+    //         result += entry * challenge;
+    //     }
+
+    //     auto scale_by_challenge_and_accumulate = [&](auto& element) {
+    //         for (auto& entry : element) {
+    //             result += entry * challenge;
+    //         }
+    //     };
+
+    //     Utils::template apply_to_tuple_of_arrays<1>(scale_by_challenge_and_accumulate, tuple);
+    // }
 
     /**
      * @brief  Recursively compute the parent nodes of each level in there, starting from the leaves. Note that at each
@@ -359,16 +363,18 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         return batch_over_relations(univariate_accumulators, instances.alpha);
     }
     static ExtendedUnivariateWithRandomization batch_over_relations(TupleOfTuplesOfUnivariates& univariate_accumulators,
-                                                                    const AlphaType& alpha)
+                                                                    const CombinedAlphaType& alpha)
     {
 
         // First relation does not get multiplied by a batching challenge
         auto result = std::get<0>(std::get<0>(univariate_accumulators))
                           .template extend_to<ProverInstances::BATCHED_EXTENDED_LENGTH>();
+        size_t idx = 0;
         auto scale_and_sum = [&]<size_t outer_idx, size_t>(auto& element) {
             auto extended = element.template extend_to<ProverInstances::BATCHED_EXTENDED_LENGTH>();
-            extended *= alpha;
+            extended *= alpha[idx];
             result += extended;
+            idx++;
         };
 
         Utils::template apply_to_tuple_of_tuples<0, 1>(univariate_accumulators, scale_and_sum);
@@ -439,13 +445,17 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     // batching the i-th folded subrelation in the combiner.
     static void combine_alpha(ProverInstances& instances)
     {
-        Univariate<FF, ProverInstances::NUM> accumulated_alpha;
-        size_t instance_idx = 0;
-        for (auto& instance : instances) {
-            accumulated_alpha.value_at(instance_idx) = instance->alpha;
-            instance_idx++;
+        size_t alpha_idx = 0;
+        for (auto& alpha : instances.alpha) {
+            Univariate<FF, ProverInstances::NUM> tmp;
+            size_t instance_idx = 0;
+            for (auto& instance : instances) {
+                tmp.value_at(instance_idx) = instance->alpha[alpha_idx];
+                instance_idx++;
+            }
+            alpha = tmp.template extend_to<ProverInstances::BATCHED_EXTENDED_LENGTH>();
+            alpha_idx++;
         }
-        instances.alpha = accumulated_alpha.template extend_to<ProverInstances::BATCHED_EXTENDED_LENGTH>();
     }
 
     /**
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index 201d6832aa8..25c2a6cd4d4 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -25,7 +25,11 @@ void ProtoGalaxyVerifier_<VerifierInstances>::receive_accumulator(const std::sha
         transcript->template receive_from_prover<FF>(domain_separator + "_lookup_grand_product_delta");
     inst->relation_parameters =
         RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
-    inst->alpha = transcript->template receive_from_prover<FF>(domain_separator + "_alpha");
+
+    for (size_t idx = 0; idx < NUM_SUBRELATIONS - 1; idx++) {
+        inst->alpha[idx] =
+            transcript->template receive_from_prover<FF>(domain_separator + "_alpha_" + std::to_string(idx));
+    }
 
     inst->target_sum = transcript->template receive_from_prover<FF>(domain_separator + "_target_sum");
 
@@ -90,7 +94,9 @@ void ProtoGalaxyVerifier_<VerifierInstances>::receive_and_finalise_instance(cons
     inst->relation_parameters =
         RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
 
-    inst->alpha = transcript->get_challenge(domain_separator + "_alpha");
+    for (size_t idx = 0; idx < NUM_SUBRELATIONS - 1; idx++) {
+        inst->alpha[idx] = transcript->get_challenge(domain_separator + "_alpha_" + std::to_string(idx));
+    }
 
     inst->verification_key = std::make_shared<VerificationKey>(inst->instance_size, inst->public_input_size);
     auto vk_view = inst->verification_key->get_all();
@@ -211,11 +217,22 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
         el_idx++;
     }
 
-    auto expected_alpha = FF(0);
+    AlphaType expected_alpha;
+    size_t alpha_idx = 0;
+    for (auto& alpha : expected_alpha) {
+        alpha = FF(0);
+        size_t instance_idx = 0;
+        for (auto& instance : instances) {
+            alpha += instance->alpha[alpha_idx] * lagranges[instance_idx];
+            instance_idx++;
+        }
+        auto next_alpha = transcript->template receive_from_prover<FF>("next_alpha_" + std::to_string(alpha_idx));
+        verified = verified & (alpha == next_alpha);
+        alpha_idx++;
+    }
     auto expected_parameters = proof_system::RelationParameters<FF>{};
     for (size_t inst_idx = 0; inst_idx < VerifierInstances::NUM; inst_idx++) {
         auto instance = instances[inst_idx];
-        expected_alpha += instance->alpha * lagranges[inst_idx];
         expected_parameters.eta += instance->relation_parameters.eta * lagranges[inst_idx];
         expected_parameters.beta += instance->relation_parameters.beta * lagranges[inst_idx];
         expected_parameters.gamma += instance->relation_parameters.gamma * lagranges[inst_idx];
@@ -225,9 +242,6 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
             instance->relation_parameters.lookup_grand_product_delta * lagranges[inst_idx];
     }
 
-    auto next_alpha = transcript->template receive_from_prover<FF>("next_alpha");
-    verified = verified & (next_alpha == expected_alpha);
-
     auto next_eta = transcript->template receive_from_prover<FF>("next_eta");
     verified = verified & (next_eta == expected_parameters.eta);
 
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
index fe4505cc348..a8b4ec172a1 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
@@ -17,6 +17,9 @@ template <class VerifierInstances> class ProtoGalaxyVerifier_ {
     using VerificationKey = typename Flavor::VerificationKey;
     using WitnessCommitments = typename Flavor::WitnessCommitments;
     using CommitmentLabels = typename Flavor::CommitmentLabels;
+    using AlphaType = typename Flavor::AlphaType;
+
+    static constexpr size_t NUM_SUBRELATIONS = Flavor::NUMBER_OF_SUBRELATIONS;
 
     VerifierInstances instances;
 
diff --git a/barretenberg/cpp/src/barretenberg/relations/relation_types.hpp b/barretenberg/cpp/src/barretenberg/relations/relation_types.hpp
index b53d154cbfe..823ebc68a44 100644
--- a/barretenberg/cpp/src/barretenberg/relations/relation_types.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/relation_types.hpp
@@ -23,10 +23,10 @@ using GetParameterView = std::conditional_t<IsField<typename Params::DataType>,
 
 template <typename T, size_t subrelation_idx>
 concept HasSubrelationLinearlyIndependentMember = requires(T) {
-                                                      {
-                                                          std::get<subrelation_idx>(T::SUBRELATION_LINEARLY_INDEPENDENT)
-                                                          } -> std::convertible_to<bool>;
-                                                  };
+    {
+        std::get<subrelation_idx>(T::SUBRELATION_LINEARLY_INDEPENDENT)
+    } -> std::convertible_to<bool>;
+};
 
 template <typename T>
 concept HasParameterLengthAdjustmentsMember = requires { T::TOTAL_LENGTH_ADJUSTMENTS; };
diff --git a/barretenberg/cpp/src/barretenberg/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
index d703d063247..498df340b2b 100644
--- a/barretenberg/cpp/src/barretenberg/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
@@ -1,4 +1,7 @@
 #pragma once
+#include "barretenberg/flavor/flavor.hpp"
+#include "barretenberg/flavor/goblin_ultra.hpp"
+#include "barretenberg/flavor/ultra.hpp"
 #include "barretenberg/polynomials/pow.hpp"
 #include "barretenberg/relations/relation_parameters.hpp"
 
@@ -10,6 +13,7 @@ template <typename Flavor> class RelationUtils {
     using Relations = typename Flavor::Relations;
     using PolynomialEvaluations = typename Flavor::AllValues;
     using RelationEvaluations = typename Flavor::TupleOfArraysOfValues;
+    using AlphaType = typename Flavor::AlphaType;
 
     static constexpr size_t NUM_RELATIONS = Flavor::NUM_RELATIONS;
 
@@ -65,13 +69,31 @@ template <typename Flavor> class RelationUtils {
      * @param challenge
      * @param current_scalar power of the challenge
      */
-    static void scale_univariates(auto& tuple, const FF& challenge, FF& current_scalar)
+    static void scale_univariates(auto& tuple, const AlphaType& challenge, FF& current_scalar)
+        requires proof_system::IsFoldingFlavor<Flavor>
     {
+        size_t idx = 0;
         auto scale_by_consecutive_powers_of_challenge = [&]<size_t, size_t>(auto& element) {
             element *= current_scalar;
-            if (current_scalar != challenge) {
-                current_scalar *= challenge;
-            }
+            current_scalar = challenge[idx];
+            idx++;
+        };
+        apply_to_tuple_of_tuples(tuple, scale_by_consecutive_powers_of_challenge);
+    }
+
+    /**
+     * @brief Scale Univaraites by consecutive powers of the provided challenge
+     *
+     * @param tuple Tuple of tuples of Univariates
+     * @param challenge
+     * @param current_scalar power of the challenge
+     */
+    static void scale_univariates(auto& tuple, const AlphaType& challenge, FF& current_scalar)
+        requires(!proof_system::IsFoldingFlavor<Flavor>)
+    {
+        auto scale_by_consecutive_powers_of_challenge = [&]<size_t, size_t>(auto& element) {
+            element *= current_scalar;
+            current_scalar *= challenge;
         };
         apply_to_tuple_of_tuples(tuple, scale_by_consecutive_powers_of_challenge);
     }
@@ -98,8 +120,8 @@ template <typename Flavor> class RelationUtils {
     /**
      * @brief Componentwise addition of nested tuples (tuples of tuples)
      * @details Used for summing tuples of tuples of Univariates. Needed for Sumcheck multithreading. Each thread
-     * accumulates relation contributions across a portion of the hypecube and then the results are accumulated into a
-     * single nested tuple.
+     * accumulates relation contributions across a portion of the hypecube and then the results are accumulated into
+     * a single nested tuple.
      *
      * @tparam Tuple
      * @tparam Index Index into outer tuple
@@ -118,10 +140,11 @@ template <typename Flavor> class RelationUtils {
     /**
      * @brief Calculate the contribution of each relation to the expected value of the full Honk relation.
      *
-     * @details For each relation, use the purported values (supplied by the prover) of the multivariates to calculate
-     * a contribution to the purported value of the full Honk relation. These are stored in `evaluations`. Adding these
-     * together, with appropriate scaling factors, produces the expected value of the full Honk relation. This value is
-     * checked against the final value of the target total sum (called sigma_0 in the thesis).
+     * @details For each relation, use the purported values (supplied by the prover) of the multivariates to
+     * calculate a contribution to the purported value of the full Honk relation. These are stored in `evaluations`.
+     * Adding these together, with appropriate scaling factors, produces the expected value of the full Honk
+     * relation. This value is checked against the final value of the target total sum (called sigma_0 in the
+     * thesis).
      */
     template <typename Parameters, size_t relation_idx = 0>
     // TODO(#224)(Cody): Input should be an array?
@@ -160,14 +183,31 @@ template <typename Flavor> class RelationUtils {
      * @brief Scale elements by consecutive powers of the challenge then sum
      * @param result Batched result
      */
-    static void scale_and_batch_elements(auto& tuple, const FF& challenge, FF current_scalar, FF& result)
+    static void scale_and_batch_elements(auto& tuple, const AlphaType& challenge, FF current_scalar, FF& result)
+        requires proof_system::IsFoldingFlavor<Flavor>
     {
+        size_t idx = 0;
         auto scale_by_challenge_and_accumulate = [&](auto& element) {
             for (auto& entry : element) {
                 result += entry * current_scalar;
-                if (current_scalar != challenge) { // HACK
-                    current_scalar *= challenge;
-                }
+                current_scalar = challenge[idx];
+                idx++;
+            }
+        };
+        apply_to_tuple_of_arrays(scale_by_challenge_and_accumulate, tuple);
+    }
+
+    /**
+     * @brief Scale elements by consecutive powers of the challenge then sum
+     * @param result Batched result
+     */
+    static void scale_and_batch_elements(auto& tuple, const AlphaType& challenge, FF current_scalar, FF& result)
+        requires(!proof_system::IsFoldingFlavor<Flavor>)
+    {
+        auto scale_by_challenge_and_accumulate = [&](auto& element) {
+            for (auto& entry : element) {
+                result += entry * current_scalar;
+                current_scalar *= challenge;
             }
         };
         apply_to_tuple_of_arrays(scale_by_challenge_and_accumulate, tuple);
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp
index ea75d4a1c5f..102548ac282 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp
@@ -104,7 +104,11 @@ std::array<typename Flavor::GroupElement, 2> UltraRecursiveVerifier_<Flavor>::ve
     // multivariate evaluations at u
     const size_t log_circuit_size = numeric::get_msb(static_cast<uint32_t>(circuit_size.get_value()));
     auto sumcheck = Sumcheck(log_circuit_size, transcript);
-    FF alpha = transcript->get_challenge("alpha");
+    AlphaType alpha;
+    for (size_t idx = 0; idx < alpha.size(); idx++) {
+        alpha[idx] = transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
+    }
+
     auto gate_challenges = std::vector<FF>(log_circuit_size);
     for (size_t idx = 0; idx < log_circuit_size; idx++) {
         gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.hpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.hpp
index 5b9f371f88e..3cd25ad618b 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.hpp
@@ -15,6 +15,7 @@ template <typename Flavor> class UltraRecursiveVerifier_ {
     using NativeVerificationKey = typename Flavor::NativeVerificationKey;
     using VerifierCommitmentKey = typename Flavor::VerifierCommitmentKey;
     using Builder = typename Flavor::CircuitBuilder;
+    using AlphaType = typename Flavor::AlphaType;
     using PairingPoints = std::array<GroupElement, 2>;
 
     explicit UltraRecursiveVerifier_(Builder* builder,
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
index f883bdf44c5..ee330ad8d46 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
@@ -10,6 +10,7 @@ template <typename Flavor_, size_t NUM_> struct ProverInstances_ {
     using Flavor = Flavor_;
     using FF = typename Flavor::FF;
     static constexpr size_t NUM = NUM_;
+    static constexpr size_t NUM_SUBRELATIONS = Flavor::NUMBER_OF_SUBRELATIONS;
     using Instance = ProverInstance_<Flavor>;
 
     using ArrayType = std::array<std::shared_ptr<Instance>, NUM_>;
@@ -17,7 +18,7 @@ template <typename Flavor_, size_t NUM_> struct ProverInstances_ {
     static constexpr size_t EXTENDED_LENGTH = (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (NUM - 1) + 1;
     static constexpr size_t BATCHED_EXTENDED_LENGTH = (Flavor::MAX_TOTAL_RELATION_LENGTH - 1 + NUM - 1) * (NUM - 1) + 1;
     using RelationParameters = proof_system::RelationParameters<Univariate<FF, EXTENDED_LENGTH>>;
-    using AlphaType = Univariate<FF, BATCHED_EXTENDED_LENGTH>;
+    using AlphaType = std::array<Univariate<FF, BATCHED_EXTENDED_LENGTH>, NUM_SUBRELATIONS - 1>;
     ArrayType _data;
     RelationParameters relation_parameters;
     AlphaType alpha;
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
index 7d1f2b92f3b..35aa0975d34 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
@@ -26,6 +26,7 @@ template <class Flavor> class ProverInstance_ {
     using Polynomial = typename Flavor::Polynomial;
     using WitnessCommitments = typename Flavor::WitnessCommitments;
     using CommitmentLabels = typename Flavor::CommitmentLabels;
+    using AlphaType = typename Flavor::AlphaType;
 
   public:
     std::shared_ptr<ProvingKey> proving_key;
@@ -45,7 +46,7 @@ template <class Flavor> class ProverInstance_ {
     // non-zero  for Instances constructed from circuits, this concept doesn't exist for accumulated
     // instances
     size_t pub_inputs_offset = 0;
-    FF alpha;
+    AlphaType alpha;
     proof_system::RelationParameters<FF> relation_parameters;
     std::vector<uint32_t> recursive_proof_public_input_indices;
 
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
index 39ce8371c80..1e880be9a46 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
@@ -9,6 +9,7 @@ template <class Flavor> class VerifierInstance_ {
     using VerificationKey = typename Flavor::VerificationKey;
     using WitnessCommitments = typename Flavor::WitnessCommitments;
     using CommitmentLabels = typename Flavor::CommitmentLabels;
+    using AlphaType = typename Flavor::AlphaType;
 
     std::shared_ptr<VerificationKey> verification_key;
     std::vector<FF> public_inputs;
@@ -17,7 +18,7 @@ template <class Flavor> class VerifierInstance_ {
     size_t instance_size;
     size_t log_instance_size;
     RelationParameters<FF> relation_parameters;
-    FF alpha;
+    AlphaType alpha;
     bool is_accumulator = false;
 
     // The folding parameters (\vec{β}, e) which are set for accumulators (i.e. relaxed instances).
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
index 39ac50c742c..42be701f535 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
@@ -16,6 +16,7 @@ template <typename Flavor> class SumcheckProver {
     using ClaimedEvaluations = typename Flavor::AllValues;
     using Transcript = typename Flavor::Transcript;
     using Instance = ProverInstance_<Flavor>;
+    using AlphaType = typename Flavor::AlphaType;
 
     const size_t multivariate_n;
     const size_t multivariate_d;
@@ -83,7 +84,7 @@ template <typename Flavor> class SumcheckProver {
      */
     SumcheckOutput<Flavor> prove(ProverPolynomials& full_polynomials,
                                  const proof_system::RelationParameters<FF>& relation_parameters,
-                                 const FF alpha,
+                                 const AlphaType alpha,
                                  const std::vector<FF>& gate_challenges)
     {
 
@@ -179,6 +180,7 @@ template <typename Flavor> class SumcheckVerifier {
     using FF = typename Flavor::FF;
     using ClaimedEvaluations = typename Flavor::AllValues;
     using Transcript = typename Flavor::Transcript;
+    using AlphaType = typename Flavor::AlphaType;
 
     static constexpr size_t BATCHED_RELATION_PARTIAL_LENGTH = Flavor::BATCHED_RELATION_PARTIAL_LENGTH;
     static constexpr size_t NUM_POLYNOMIALS = Flavor::NUM_ALL_ENTITIES;
@@ -204,7 +206,7 @@ template <typename Flavor> class SumcheckVerifier {
      * @param transcript
      */
     SumcheckOutput<Flavor> verify(const proof_system::RelationParameters<FF>& relation_parameters,
-                                  FF alpha,
+                                  AlphaType alpha,
                                   const std::vector<FF>& gate_challenges)
     {
         bool verified(true);
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp
index 15155a32196..0801391d9d5 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp
@@ -17,6 +17,7 @@ using namespace proof_system::honk::sumcheck;
 using Flavor = proof_system::honk::flavor::Ultra;
 using FF = typename Flavor::FF;
 using ProverPolynomials = typename Flavor::ProverPolynomials;
+using AlphaType = Flavor::AlphaType;
 const size_t NUM_POLYNOMIALS = Flavor::NUM_ALL_ENTITIES;
 
 namespace test_sumcheck_round {
@@ -66,7 +67,11 @@ TEST_F(SumcheckTests, PolynomialNormalization)
     auto transcript = Flavor::Transcript::prover_init_empty();
 
     auto sumcheck = SumcheckProver<Flavor>(multivariate_n, transcript);
-    FF alpha = transcript->get_challenge("alpha");
+    AlphaType alpha;
+    for (size_t idx = 0; idx < alpha.size(); idx++) {
+        alpha[idx] = transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
+    }
+
     std::vector<FF> gate_challenges(multivariate_d);
     for (size_t idx = 0; idx < multivariate_d; idx++) {
         gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
@@ -134,7 +139,11 @@ TEST_F(SumcheckTests, Prover)
 
     auto sumcheck = SumcheckProver<Flavor>(multivariate_n, transcript);
 
-    FF alpha = transcript->get_challenge("alpha");
+    AlphaType alpha;
+    for (size_t idx = 0; idx < alpha.size(); idx++) {
+        alpha[idx] = transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
+    }
+
     std::vector<FF> gate_challenges(multivariate_d);
     for (size_t idx = 0; idx < gate_challenges.size(); idx++) {
         gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
@@ -213,7 +222,10 @@ TEST_F(SumcheckTests, ProverAndVerifierSimple)
         auto prover_transcript = Flavor::Transcript::prover_init_empty();
         auto sumcheck_prover = SumcheckProver<Flavor>(multivariate_n, prover_transcript);
 
-        FF prover_alpha = prover_transcript->get_challenge("alpha");
+        AlphaType prover_alpha;
+        for (size_t idx = 0; idx < prover_alpha.size(); idx++) {
+            prover_alpha[idx] = prover_transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
+        }
         std::vector<FF> prover_gate_challenges(multivariate_d);
         for (size_t idx = 0; idx < multivariate_d; idx++) {
             prover_gate_challenges[idx] =
@@ -224,7 +236,10 @@ TEST_F(SumcheckTests, ProverAndVerifierSimple)
         auto verifier_transcript = Flavor::Transcript::verifier_init_empty(prover_transcript);
 
         auto sumcheck_verifier = SumcheckVerifier<Flavor>(multivariate_d, verifier_transcript);
-        FF verifier_alpha = verifier_transcript->get_challenge("alpha");
+        AlphaType verifier_alpha;
+        for (size_t idx = 0; idx < verifier_alpha.size(); idx++) {
+            verifier_alpha[idx] = verifier_transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
+        }
         std::vector<FF> verifier_gate_challenges(multivariate_d);
         for (size_t idx = 0; idx < multivariate_d; idx++) {
             verifier_gate_challenges[idx] =
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
index 66612c5500c..8dcd110a1e7 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
@@ -55,6 +55,7 @@ template <typename Flavor> class SumcheckProverRound {
     using Utils = barretenberg::RelationUtils<Flavor>;
     using Relations = typename Flavor::Relations;
     using SumcheckTupleOfTuplesOfUnivariates = typename Flavor::SumcheckTupleOfTuplesOfUnivariates;
+    using AlphaType = typename Flavor::AlphaType;
 
   public:
     using FF = typename Flavor::FF;
@@ -106,7 +107,7 @@ template <typename Flavor> class SumcheckProverRound {
         ProverPolynomialsOrPartiallyEvaluatedMultivariates& polynomials,
         const proof_system::RelationParameters<FF>& relation_parameters,
         const barretenberg::PowPolynomial<FF>& pow_polynomial,
-        const FF alpha)
+        const AlphaType alpha)
     {
         std::vector<FF> pow_challenges(round_size >> 1);
         pow_challenges[0] = pow_polynomial.partial_evaluation_result;
@@ -144,7 +145,6 @@ template <typename Flavor> class SumcheckProverRound {
                 extend_edges(extended_edges[thread_idx], polynomials, edge_idx);
 
                 // Update the pow polynomial's contribution c_l ⋅ ζ_{l+1}ⁱ for the next edge.
-                // THIS IS PROBABLY WRONG
                 FF pow_challenge = pow_challenges[edge_idx >> 1];
 
                 // Compute the i-th edge's univariate contribution,
@@ -173,7 +173,7 @@ template <typename Flavor> class SumcheckProverRound {
      */
     template <typename ExtendedUnivariate, typename ContainerOverSubrelations>
     static ExtendedUnivariate batch_over_relations(ContainerOverSubrelations& univariate_accumulators,
-                                                   const FF& challenge,
+                                                   const AlphaType& challenge,
                                                    const barretenberg::PowPolynomial<FF>& pow_polynomial)
     {
         auto running_challenge = FF(1);
@@ -261,6 +261,7 @@ template <typename Flavor> class SumcheckVerifierRound {
     using Utils = barretenberg::RelationUtils<Flavor>;
     using Relations = typename Flavor::Relations;
     using TupleOfArraysOfValues = typename Flavor::TupleOfArraysOfValues;
+    using AlphaType = typename Flavor::AlphaType;
 
   public:
     using FF = typename Flavor::FF;
@@ -330,7 +331,7 @@ template <typename Flavor> class SumcheckVerifierRound {
     FF compute_full_honk_relation_purported_value(ClaimedEvaluations purported_evaluations,
                                                   const proof_system::RelationParameters<FF>& relation_parameters,
                                                   const barretenberg::PowPolynomial<FF>& pow_polynomial,
-                                                  const FF alpha)
+                                                  const AlphaType alpha)
     {
         Utils::template accumulate_relation_evaluations<>(
             purported_evaluations, relation_evaluations, relation_parameters, pow_polynomial.partial_evaluation_result);
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.test.cpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.test.cpp
index dd26b11678a..49d228a9c52 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.test.cpp
@@ -25,6 +25,7 @@ TEST(SumcheckRound, SumcheckTupleOfTuplesOfUnivariates)
 {
     using Flavor = proof_system::honk::flavor::Ultra;
     using FF = typename Flavor::FF;
+    using AlphaType = typename Flavor::AlphaType;
 
     // Define three linear univariates of different sizes
     Univariate<FF, 3> univariate_1({ 1, 2, 3 });
@@ -36,7 +37,9 @@ TEST(SumcheckRound, SumcheckTupleOfTuplesOfUnivariates)
     auto tuple_of_tuples = std::make_tuple(std::make_tuple(univariate_1), std::make_tuple(univariate_2, univariate_3));
 
     // Use scale_univariate_accumulators to scale by challenge powers
-    FF challenge = 5;
+    AlphaType challenge = {};
+    challenge[0] = 5;
+    challenge[1] = challenge[0].sqr();
     FF running_challenge = 1;
     Utils::scale_univariates(tuple_of_tuples, challenge, running_challenge);
 
@@ -47,8 +50,8 @@ TEST(SumcheckRound, SumcheckTupleOfTuplesOfUnivariates)
 
     // Repeat the batching process manually
     auto result_expected = univariate_1.template extend_to<MAX_LENGTH>() * 1 +
-                           univariate_2.template extend_to<MAX_LENGTH>() * challenge +
-                           univariate_3.template extend_to<MAX_LENGTH>() * challenge * challenge;
+                           univariate_2.template extend_to<MAX_LENGTH>() * challenge[0] +
+                           univariate_3.template extend_to<MAX_LENGTH>() * challenge[1];
 
     // Compare final batched univarites
     EXPECT_EQ(result, result_expected);
@@ -74,6 +77,7 @@ TEST(SumcheckRound, TuplesOfEvaluationArrays)
     using Flavor = proof_system::honk::flavor::Ultra;
     using Utils = barretenberg::RelationUtils<Flavor>;
     using FF = typename Flavor::FF;
+    using AlphaType = typename Flavor::AlphaType;
 
     // Define two arrays of arbitrary elements
     std::array<FF, 1> evaluations_1 = { 4 };
@@ -83,14 +87,15 @@ TEST(SumcheckRound, TuplesOfEvaluationArrays)
     auto tuple_of_arrays = std::make_tuple(evaluations_1, evaluations_2);
 
     // Use scale_and_batch_elements to scale by challenge powers
-    FF challenge = 5;
     FF running_challenge = 1;
+    AlphaType challenge = {};
+    challenge[0] = 5;
+    challenge[1] = challenge[0].sqr();
     FF result = 0;
     Utils::scale_and_batch_elements(tuple_of_arrays, challenge, running_challenge, result);
 
     // Repeat the batching process manually
-    auto result_expected =
-        evaluations_1[0] * 1 + evaluations_2[0] * challenge + evaluations_2[1] * challenge * challenge;
+    auto result_expected = evaluations_1[0] * 1 + evaluations_2[0] * challenge[0] + evaluations_2[1] * challenge[1];
 
     // Compare batched result
     EXPECT_EQ(result, result_expected);
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/goblin_ultra_transcript.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/goblin_ultra_transcript.test.cpp
index af57281bdd8..f12d291086e 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/goblin_ultra_transcript.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/goblin_ultra_transcript.test.cpp
@@ -33,6 +33,8 @@ class GoblinUltraTranscriptTests : public ::testing::Test {
         auto log_n = numeric::get_msb(circuit_size);
 
         size_t MAX_PARTIAL_RELATION_LENGTH = Flavor::BATCHED_RELATION_PARTIAL_LENGTH;
+        size_t NUM_SUBRELATIONS = Flavor::NUMBER_OF_SUBRELATIONS;
+
         size_t size_FF = sizeof(FF);
         size_t size_G = 2 * size_FF;
         size_t size_uni = MAX_PARTIAL_RELATION_LENGTH * size_FF;
@@ -64,23 +66,27 @@ class GoblinUltraTranscriptTests : public ::testing::Test {
         manifest_expected.add_entry(round, "LOOKUP_INVERSES", size_G);
         manifest_expected.add_entry(round, "Z_PERM", size_G);
         manifest_expected.add_entry(round, "Z_LOOKUP", size_G);
-        manifest_expected.add_challenge(round, "alpha");
 
-        for (size_t i = 0; i < log_n; i++) {
+        for (size_t i = 0; i < NUM_SUBRELATIONS - 1; i++) {
+            std::string label = "Sumcheck:alpha_" + std::to_string(i);
+            manifest_expected.add_challenge(round, label);
             round++;
+        }
+
+        for (size_t i = 0; i < log_n; i++) {
             std::string label = "Sumcheck:gate_challenge_" + std::to_string(i);
             manifest_expected.add_challenge(round, label);
+            round++;
         }
 
         for (size_t i = 0; i < log_n; ++i) {
-            round++;
             std::string idx = std::to_string(i);
             manifest_expected.add_entry(round, "Sumcheck:univariate_" + idx, size_uni);
             std::string label = "Sumcheck:u_" + idx;
             manifest_expected.add_challenge(round, label);
+            round++;
         }
 
-        round++;
         manifest_expected.add_entry(round, "Sumcheck:evaluations", size_evals);
         manifest_expected.add_challenge(round, "rho");
 
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
index 0f9e5083c5d..59e44ffa80e 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
@@ -112,9 +112,11 @@ TEST_F(ProtoGalaxyTests, FullHonkEvaluationsValidCircuit)
     instance->compute_sorted_accumulator_polynomials(eta);
     instance->compute_grand_product_polynomials(beta, gamma);
 
-    auto alpha = FF::random_element();
+    for (auto& alpha : instance->alpha) {
+        alpha = FF::random_element();
+    }
     auto full_honk_evals = ProtoGalaxyProver::compute_full_honk_evaluations(
-        instance->prover_polynomials, alpha, instance->relation_parameters);
+        instance->prover_polynomials, instance->alpha, instance->relation_parameters);
 
     // Evaluations should be 0 for valid circuit
     for (const auto& eval : full_honk_evals) {
@@ -136,6 +138,7 @@ TEST_F(ProtoGalaxyTests, PerturbatorCoefficients)
 
 TEST_F(ProtoGalaxyTests, PerturbatorPolynomial)
 {
+    using AlphaType = Flavor::AlphaType;
     const size_t log_instance_size(3);
     const size_t instance_size(1 << log_instance_size);
 
@@ -145,7 +148,10 @@ TEST_F(ProtoGalaxyTests, PerturbatorPolynomial)
     }
     auto full_polynomials = construct_ultra_full_polynomials(random_polynomials);
     auto relation_parameters = proof_system::RelationParameters<FF>::get_random();
-    auto alpha = FF::random_element();
+    AlphaType alpha;
+    for (auto& a : alpha) {
+        a = FF::random_element();
+    }
 
     auto full_honk_evals =
         ProtoGalaxyProver::compute_full_honk_evaluations(full_polynomials, alpha, relation_parameters);
@@ -213,7 +219,7 @@ TEST_F(ProtoGalaxyTests, CombinerQuotient)
     }
 }
 
-TEST_F(ProtoGalaxyTests, FoldChallenges)
+TEST_F(ProtoGalaxyTests, CombineRelationParameters)
 {
     using Instances = ProverInstances_<Flavor, 2>;
     using Instance = typename Instances::Instance;
@@ -234,48 +240,27 @@ TEST_F(ProtoGalaxyTests, FoldChallenges)
     EXPECT_EQ(instances.relation_parameters.eta, expected_eta);
 }
 
-TEST_F(ProtoGalaxyTests, FoldAlpha)
+TEST_F(ProtoGalaxyTests, CombineAlpha)
 {
     using Instances = ProverInstances_<Flavor, 2>;
     using Instance = typename Instances::Instance;
 
     Builder builder1;
     auto instance1 = std::make_shared<Instance>(builder1);
-    instance1->alpha = 2;
+    instance1->alpha.fill(2);
 
     Builder builder2;
     builder2.add_variable(3);
     auto instance2 = std::make_shared<Instance>(builder2);
-    instance2->alpha = 4;
+    instance2->alpha.fill(4);
 
     Instances instances{ { instance1, instance2 } };
     ProtoGalaxyProver::combine_alpha(instances);
 
     Univariate<FF, 13> expected_alpha{ { 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26 } };
-    EXPECT_EQ(instances.alpha, expected_alpha);
-}
-
-TEST_F(ProtoGalaxyTests, FoldAlphaShouldBeZero)
-{
-    using Instances = ProverInstances_<Flavor, 2>;
-    using Instance = typename Instances::Instance;
-
-    Builder builder1;
-    auto instance1 = std::make_shared<Instance>(builder1);
-    instance1->alpha = FF(0);
-
-    Builder builder2;
-    builder2.add_variable(3);
-    auto instance2 = std::make_shared<Instance>(builder2);
-    instance2->alpha = FF(0);
-
-    Instances instances{ { instance1, instance2 } };
-    ProtoGalaxyProver::combine_alpha(instances);
-
-    Univariate<FF, 13> expected_alpha{ { 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 } };
-    auto challenge = FF::random_element();
-    EXPECT_EQ(instances.alpha, expected_alpha);
-    EXPECT_EQ(instances.alpha.evaluate(challenge), FF(0));
+    for (const auto& alpha : instances.alpha) {
+        EXPECT_EQ(alpha, expected_alpha);
+    }
 }
 
 // Check both manually and using the protocol two rounds of folding
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/sumcheck.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/sumcheck.test.cpp
index 2f8aad66079..791ab283e69 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/sumcheck.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/sumcheck.test.cpp
@@ -36,6 +36,7 @@ TEST_F(SumcheckTestsRealCircuit, Ultra)
     using Flavor = flavor::Ultra;
     using FF = typename Flavor::FF;
     using Transcript = typename Flavor::Transcript;
+    using AlphaType = typename Flavor::AlphaType;
 
     // Create a composer and a dummy circuit with a few gates
     auto builder = proof_system::UltraCircuitBuilder();
@@ -163,7 +164,13 @@ TEST_F(SumcheckTestsRealCircuit, Ultra)
     auto prover_transcript = Transcript::prover_init_empty();
     auto circuit_size = instance->proving_key->circuit_size;
     auto log_circuit_size = numeric::get_msb(circuit_size);
-    instance->alpha = prover_transcript->get_challenge("alpha");
+
+    AlphaType prover_alpha;
+    for (size_t idx = 0; idx < prover_alpha.size(); idx++) {
+        prover_alpha[idx] = prover_transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
+    }
+
+    instance->alpha = prover_alpha;
     auto sumcheck_prover = SumcheckProver<Flavor>(circuit_size, prover_transcript);
     std::vector<FF> prover_gate_challenges(log_circuit_size);
     for (size_t idx = 0; idx < log_circuit_size; idx++) {
@@ -176,13 +183,18 @@ TEST_F(SumcheckTestsRealCircuit, Ultra)
     auto verifier_transcript = Transcript::verifier_init_empty(prover_transcript);
 
     auto sumcheck_verifier = SumcheckVerifier<Flavor>(log_circuit_size, verifier_transcript);
-    FF alpha = verifier_transcript->get_challenge("alpha");
+    AlphaType verifier_alpha;
+    for (size_t idx = 0; idx < verifier_alpha.size(); idx++) {
+        verifier_alpha[idx] = verifier_transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
+    }
+
     std::vector<FF> verifier_gate_challenges(log_circuit_size);
     for (size_t idx = 0; idx < log_circuit_size; idx++) {
         verifier_gate_challenges[idx] =
             verifier_transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
     }
-    auto verifier_output = sumcheck_verifier.verify(instance->relation_parameters, alpha, verifier_gate_challenges);
+    auto verifier_output =
+        sumcheck_verifier.verify(instance->relation_parameters, verifier_alpha, verifier_gate_challenges);
 
     auto verified = verifier_output.verified.value();
 
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp
index c57453d1932..5432cde5430 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp
@@ -149,7 +149,11 @@ template <UltraFlavor Flavor> void UltraProver_<Flavor>::execute_relation_check_
     using Sumcheck = sumcheck::SumcheckProver<Flavor>;
     auto circuit_size = instance->proving_key->circuit_size;
     auto sumcheck = Sumcheck(circuit_size, transcript);
-    instance->alpha = transcript->get_challenge("alpha");
+    AlphaType alpha;
+    for (size_t idx = 0; idx < alpha.size(); idx++) {
+        alpha[idx] = transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
+    }
+    instance->alpha = alpha;
     std::vector<FF> gate_challenges(numeric::get_msb(circuit_size));
     for (size_t idx = 0; idx < gate_challenges.size(); idx++) {
         gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.hpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.hpp
index dece05e520c..8639b912f92 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.hpp
@@ -21,6 +21,7 @@ template <UltraFlavor Flavor> class UltraProver_ {
     using Curve = typename Flavor::Curve;
     using Instance = ProverInstance_<Flavor>;
     using Transcript = typename Flavor::Transcript;
+    using AlphaType = typename Flavor::AlphaType;
 
   public:
     explicit UltraProver_(const std::shared_ptr<Instance>&,
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_transcript.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_transcript.test.cpp
index 4d642f18893..72f0c49916c 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_transcript.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_transcript.test.cpp
@@ -33,6 +33,7 @@ class UltraTranscriptTests : public ::testing::Test {
         auto log_n = numeric::get_msb(circuit_size);
 
         size_t MAX_PARTIAL_RELATION_LENGTH = Flavor::BATCHED_RELATION_PARTIAL_LENGTH;
+        size_t NUM_SUBRELATIONS = Flavor::NUMBER_OF_SUBRELATIONS;
         size_t size_FF = sizeof(FF);
         size_t size_G = 2 * size_FF;
         size_t size_uni = MAX_PARTIAL_RELATION_LENGTH * size_FF;
@@ -57,23 +58,27 @@ class UltraTranscriptTests : public ::testing::Test {
         round++;
         manifest_expected.add_entry(round, "Z_PERM", size_G);
         manifest_expected.add_entry(round, "Z_LOOKUP", size_G);
-        manifest_expected.add_challenge(round, "alpha");
 
-        for (size_t i = 0; i < log_n; i++) {
+        for (size_t i = 0; i < NUM_SUBRELATIONS - 1; i++) {
+            std::string label = "Sumcheck:alpha_" + std::to_string(i);
+            manifest_expected.add_challenge(round, label);
             round++;
+        }
+
+        for (size_t i = 0; i < log_n; i++) {
             std::string label = "Sumcheck:gate_challenge_" + std::to_string(i);
             manifest_expected.add_challenge(round, label);
+            round++;
         }
 
         for (size_t i = 0; i < log_n; ++i) {
-            round++;
             std::string idx = std::to_string(i);
             manifest_expected.add_entry(round, "Sumcheck:univariate_" + idx, size_uni);
             std::string label = "Sumcheck:u_" + idx;
             manifest_expected.add_challenge(round, label);
+            round++;
         }
 
-        round++;
         manifest_expected.add_entry(round, "Sumcheck:evaluations", size_evals);
         manifest_expected.add_challenge(round, "rho");
 
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp
index 2748c446e30..2b12c05ea85 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp
@@ -132,7 +132,11 @@ template <typename Flavor> bool UltraVerifier_<Flavor>::verify_proof(const plonk
     // Execute Sumcheck Verifier
     const size_t log_circuit_size = numeric::get_msb(circuit_size);
     auto sumcheck = SumcheckVerifier<Flavor>(log_circuit_size, transcript);
-    FF alpha = transcript->get_challenge("alpha");
+    AlphaType alpha;
+    for (size_t idx = 0; idx < alpha.size(); idx++) {
+        alpha[idx] = transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
+    }
+
     auto gate_challenges = std::vector<FF>(log_circuit_size);
     for (size_t idx = 0; idx < log_circuit_size; idx++) {
         gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.hpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.hpp
index 8197e46a941..630734c938d 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.hpp
@@ -12,6 +12,7 @@ template <typename Flavor> class UltraVerifier_ {
     using VerificationKey = typename Flavor::VerificationKey;
     using VerifierCommitmentKey = typename Flavor::VerifierCommitmentKey;
     using Transcript = typename Flavor::Transcript;
+    using AlphaType = typename Flavor::AlphaType;
 
   public:
     explicit UltraVerifier_(const std::shared_ptr<Transcript>& transcript,

From fdd931133e9439b3bbb16043ea080d9ae9cc750e Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Thu, 21 Dec 2023 15:08:49 +0000
Subject: [PATCH 60/86] fix formatting

---
 .../cpp/src/barretenberg/relations/relation_types.hpp     | 8 ++++----
 barretenberg/cpp/src/barretenberg/relations/utils.hpp     | 3 ++-
 2 files changed, 6 insertions(+), 5 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/relations/relation_types.hpp b/barretenberg/cpp/src/barretenberg/relations/relation_types.hpp
index 823ebc68a44..b53d154cbfe 100644
--- a/barretenberg/cpp/src/barretenberg/relations/relation_types.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/relation_types.hpp
@@ -23,10 +23,10 @@ using GetParameterView = std::conditional_t<IsField<typename Params::DataType>,
 
 template <typename T, size_t subrelation_idx>
 concept HasSubrelationLinearlyIndependentMember = requires(T) {
-    {
-        std::get<subrelation_idx>(T::SUBRELATION_LINEARLY_INDEPENDENT)
-    } -> std::convertible_to<bool>;
-};
+                                                      {
+                                                          std::get<subrelation_idx>(T::SUBRELATION_LINEARLY_INDEPENDENT)
+                                                          } -> std::convertible_to<bool>;
+                                                  };
 
 template <typename T>
 concept HasParameterLengthAdjustmentsMember = requires { T::TOTAL_LENGTH_ADJUSTMENTS; };
diff --git a/barretenberg/cpp/src/barretenberg/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
index 498df340b2b..e609d4c31fc 100644
--- a/barretenberg/cpp/src/barretenberg/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
@@ -110,7 +110,8 @@ template <typename Flavor> class RelationUtils {
     template <typename... T>
     static constexpr void add_tuples(std::tuple<T...>& tuple_1, const std::tuple<T...>& tuple_2)
     {
-        auto add_tuples_helper = [&]<std::size_t... I>(std::index_sequence<I...>) {
+        auto add_tuples_helper = [&]<std::size_t... I>(std::index_sequence<I...>)
+        {
             ((std::get<I>(tuple_1) += std::get<I>(tuple_2)), ...);
         };
 

From afd16d5c353ddb8ecfbb595a345b88513f5a290c Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Thu, 21 Dec 2023 15:15:40 +0000
Subject: [PATCH 61/86] fix wasm

---
 barretenberg/cpp/CMakePresets.json | 3 +--
 1 file changed, 1 insertion(+), 2 deletions(-)

diff --git a/barretenberg/cpp/CMakePresets.json b/barretenberg/cpp/CMakePresets.json
index b87e10709b6..d8ad86d1a9d 100644
--- a/barretenberg/cpp/CMakePresets.json
+++ b/barretenberg/cpp/CMakePresets.json
@@ -351,8 +351,7 @@
         "acvm_backend.wasm",
         "barretenberg",
         "wasi",
-        "env",
-        "honk_tests"
+        "env"
       ]
     },
     {

From de77d6eca39808a2c9b1e983fc7275b024345b65 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Thu, 21 Dec 2023 17:08:14 +0000
Subject: [PATCH 62/86] more cleanup

---
 .../src/barretenberg/eccvm/eccvm_prover.cpp   |   2 +-
 .../eccvm/eccvm_transcript.test.cpp           |   2 +-
 .../src/barretenberg/eccvm/eccvm_verifier.cpp |   2 +-
 .../src/barretenberg/flavor/goblin_ultra.hpp  |  10 +-
 .../flavor/goblin_ultra_recursive.hpp         |   8 +-
 .../cpp/src/barretenberg/flavor/toy_avm.hpp   |   2 -
 .../cpp/src/barretenberg/flavor/ultra.hpp     |   7 +-
 .../barretenberg/flavor/ultra_recursive.hpp   |   8 +-
 .../protogalaxy/combiner.test.cpp             | 388 +++++++++---------
 .../protogalaxy/decider_prover.cpp            |   2 +-
 .../protogalaxy/decider_verifier.cpp          |   6 +-
 .../protogalaxy/protogalaxy_prover.cpp        |  10 +-
 .../protogalaxy/protogalaxy_prover.hpp        |   8 +-
 .../protogalaxy/protogalaxy_verifier.cpp      |  10 +-
 .../protogalaxy/protogalaxy_verifier.hpp      |   2 +-
 .../sumcheck/instance/instances.hpp           |   4 +-
 .../sumcheck/instance/prover_instance.hpp     |   2 +-
 .../sumcheck/instance/verifier_instance.hpp   |   2 +-
 .../src/barretenberg/sumcheck/sumcheck.hpp    |   2 +-
 .../goblin_translator_prover.cpp              |   2 +-
 .../goblin_translator_verifier.cpp            |   2 +-
 .../goblin_ultra_transcript.test.cpp          |   2 +-
 .../ultra_honk/protogalaxy.test.cpp           |  22 +-
 .../barretenberg/ultra_honk/sumcheck.test.cpp |  16 +-
 .../barretenberg/ultra_honk/ultra_prover.cpp  |   8 +-
 .../ultra_honk/ultra_transcript.test.cpp      |   2 +-
 .../ultra_honk/ultra_verifier.cpp             |   8 +-
 .../vm/generated/AvmMini_prover.cpp           |   2 +-
 .../vm/generated/AvmMini_verifier.cpp         |   2 +-
 29 files changed, 276 insertions(+), 267 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/eccvm/eccvm_prover.cpp b/barretenberg/cpp/src/barretenberg/eccvm/eccvm_prover.cpp
index dfd722a20d0..f58b9524805 100644
--- a/barretenberg/cpp/src/barretenberg/eccvm/eccvm_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/eccvm/eccvm_prover.cpp
@@ -110,7 +110,7 @@ template <ECCVMFlavor Flavor> void ECCVMProver_<Flavor>::execute_relation_check_
     using Sumcheck = sumcheck::SumcheckProver<Flavor>;
 
     auto sumcheck = Sumcheck(key->circuit_size, transcript);
-    FF alpha = transcript->get_challenge("alpha");
+    FF alpha = transcript->get_challenge("Sumcheck:alpha");
     std::vector<FF> gate_challenges(numeric::get_msb(key->circuit_size));
     for (size_t idx = 0; idx < gate_challenges.size(); idx++) {
         gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
diff --git a/barretenberg/cpp/src/barretenberg/eccvm/eccvm_transcript.test.cpp b/barretenberg/cpp/src/barretenberg/eccvm/eccvm_transcript.test.cpp
index a9295fb42b6..63abcda97de 100644
--- a/barretenberg/cpp/src/barretenberg/eccvm/eccvm_transcript.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/eccvm/eccvm_transcript.test.cpp
@@ -126,7 +126,7 @@ template <typename Flavor> class ECCVMTranscriptTests : public ::testing::Test {
         round++;
         manifest_expected.add_entry(round, "LOOKUP_INVERSES", size_G);
         manifest_expected.add_entry(round, "Z_PERM", size_G);
-        manifest_expected.add_challenge(round, "alpha");
+        manifest_expected.add_challenge(round, "Sumcheck:alpha");
 
         for (size_t i = 0; i < log_n; i++) {
             round++;
diff --git a/barretenberg/cpp/src/barretenberg/eccvm/eccvm_verifier.cpp b/barretenberg/cpp/src/barretenberg/eccvm/eccvm_verifier.cpp
index 6a43f5409ff..87feb5834df 100644
--- a/barretenberg/cpp/src/barretenberg/eccvm/eccvm_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/eccvm/eccvm_verifier.cpp
@@ -159,7 +159,7 @@ template <typename Flavor> bool ECCVMVerifier_<Flavor>::verify_proof(const plonk
     // Execute Sumcheck Verifier
     const size_t log_circuit_size = numeric::get_msb(circuit_size);
     auto sumcheck = SumcheckVerifier<Flavor>(log_circuit_size, transcript);
-    FF alpha = transcript->get_challenge("alpha");
+    FF alpha = transcript->get_challenge("Sumcheck:alpha");
     std::vector<FF> gate_challenges(numeric::get_msb(key->circuit_size));
     for (size_t idx = 0; idx < gate_challenges.size(); idx++) {
         gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
index 27337aab223..4934a1a6564 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
@@ -68,9 +68,6 @@ class GoblinUltra {
 
     static constexpr size_t MAX_PARTIAL_RELATION_LENGTH = compute_max_partial_relation_length<Relations>();
     static constexpr size_t MAX_TOTAL_RELATION_LENGTH = compute_max_total_relation_length<Relations>();
-    static constexpr size_t NUMBER_OF_SUBRELATIONS = compute_number_of_subrelations<Relations>();
-
-    using AlphaType = std::array<FF, NUMBER_OF_SUBRELATIONS - 1>;
 
     // BATCHED_RELATION_PARTIAL_LENGTH = algebraic degree of sumcheck relation *after* multiplying by the `pow_zeta`
     // random polynomial e.g. For \sum(x) [A(x) * B(x) + C(x)] * PowZeta(X), relation length = 2 and random relation
@@ -79,6 +76,12 @@ class GoblinUltra {
     static constexpr size_t BATCHED_RELATION_TOTAL_LENGTH = MAX_TOTAL_RELATION_LENGTH + 1;
     static constexpr size_t NUM_RELATIONS = std::tuple_size_v<Relations>;
 
+    // For instances of this flavour, used in folding, we need a unique sumcheck batching challenges for each
+    // subrelation. This
+    // is because using powers of alpha would increase the degree of Protogalaxy polynomial $G$ (the combiner) to much.
+    static constexpr size_t NUM_SUBRELATIONS = compute_number_of_subrelations<Relations>();
+    using AlphaType = std::array<FF, NUM_SUBRELATIONS - 1>;
+
     template <size_t NUM_INSTANCES>
     using ProtogalaxyTupleOfTuplesOfUnivariates =
         decltype(create_protogalaxy_tuple_of_tuples_of_univariates<Relations, NUM_INSTANCES>());
@@ -87,7 +90,6 @@ class GoblinUltra {
 
     // Whether or not the first row of the execution trace is reserved for 0s to enable shifts
     static constexpr bool has_zero_row = true;
-    static constexpr bool is_decider = true;
     /**
      * @brief A base class labelling precomputed entities and (ordered) subsets of interest.
      * @details Used to build the proving key and verification key.
diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
index a91cb8e65f7..7bbbc8f2127 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
@@ -73,8 +73,12 @@ template <typename BuilderType> class GoblinUltraRecursive_ {
     // length = 3
     static constexpr size_t BATCHED_RELATION_PARTIAL_LENGTH = MAX_PARTIAL_RELATION_LENGTH + 1;
     static constexpr size_t NUM_RELATIONS = std::tuple_size<Relations>::value;
-    static constexpr size_t NUMBER_OF_SUBRELATIONS = compute_number_of_subrelations<Relations>();
-    using AlphaType = std::array<FF, NUMBER_OF_SUBRELATIONS - 1>;
+
+    // For instances of this flavour, used in folding, we need a unique sumcheck batching challenges for each
+    // subrelation. This
+    // is because using powers of alpha would increase the degree of Protogalaxy polynomial $G$ (the combiner) to much.
+    static constexpr size_t NUM_SUBRELATIONS = compute_number_of_subrelations<Relations>();
+    using AlphaType = std::array<FF, NUM_SUBRELATIONS - 1>;
 
     // define the container for storing the univariate contribution from each relation in Sumcheck
     using SumcheckTupleOfTuplesOfUnivariates = decltype(create_sumcheck_tuple_of_tuples_of_univariates<Relations>());
diff --git a/barretenberg/cpp/src/barretenberg/flavor/toy_avm.hpp b/barretenberg/cpp/src/barretenberg/flavor/toy_avm.hpp
index 4d553f9797d..d3d52b7ec29 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/toy_avm.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/toy_avm.hpp
@@ -67,8 +67,6 @@ class ToyAVM {
     static constexpr size_t BATCHED_RELATION_PARTIAL_LENGTH = MAX_PARTIAL_RELATION_LENGTH + 1;
     static constexpr size_t NUM_RELATIONS = std::tuple_size<Relations>::value;
 
-    static constexpr bool is_decider = true;
-
     // Instantiate the BarycentricData needed to extend each Relation Univariate
 
     // define the containers for storing the contributions from each relation in Sumcheck
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
index 7d2965d113b..7d2b6bb6302 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
@@ -57,8 +57,11 @@ class Ultra {
     static_assert(MAX_PARTIAL_RELATION_LENGTH == 6);
     static constexpr size_t MAX_TOTAL_RELATION_LENGTH = compute_max_total_relation_length<Relations>();
     static_assert(MAX_TOTAL_RELATION_LENGTH == 12);
-    static constexpr size_t NUMBER_OF_SUBRELATIONS = compute_number_of_subrelations<Relations>();
-    using AlphaType = std::array<FF, NUMBER_OF_SUBRELATIONS - 1>;
+    static constexpr size_t NUM_SUBRELATIONS = compute_number_of_subrelations<Relations>();
+    // For instances of this flavour, used in folding, we need a unique sumcheck batching challenges for each
+    // subrelation. This
+    // is because using powers of alpha would increase the degree of Protogalaxy polynomial $G$ (the combiner) to much.
+    using AlphaType = std::array<FF, NUM_SUBRELATIONS - 1>;
 
     // BATCHED_RELATION_PARTIAL_LENGTH = algebraic degree of sumcheck relation *after* multiplying by the `pow_zeta`
     // random polynomial e.g. For \sum(x) [A(x) * B(x) + C(x)] * PowZeta(X), relation length = 2 and random relation
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
index 22a436e5d14..042c4e4cec4 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
@@ -85,10 +85,12 @@ template <typename BuilderType> class UltraRecursive_ {
     // length = 3
     static constexpr size_t BATCHED_RELATION_PARTIAL_LENGTH = MAX_PARTIAL_RELATION_LENGTH + 1;
     static constexpr size_t NUM_RELATIONS = std::tuple_size<Relations>::value;
-    static constexpr size_t NUMBER_OF_SUBRELATIONS = compute_number_of_subrelations<Relations>();
-    using AlphaType = std::array<FF, NUMBER_OF_SUBRELATIONS - 1>;
 
-    static constexpr bool is_decider = false; // for now
+    // For instances of this flavour, used in folding, we need a unique sumcheck batching challenges for each
+    // subrelation. This is because using powers of alpha would increase the degree of Protogalaxy polynomial $G$ (the
+    // combiner) to much.
+    static constexpr size_t NUM_SUBRELATIONS = compute_number_of_subrelations<Relations>();
+    using AlphaType = std::array<FF, NUM_SUBRELATIONS - 1>;
 
     // define the container for storing the univariate contribution from each relation in Sumcheck
     using SumcheckTupleOfTuplesOfUnivariates = decltype(create_sumcheck_tuple_of_tuples_of_univariates<Relations>());
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp
index f216663c1dc..ee9f1af728e 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp
@@ -1,194 +1,194 @@
-// #include "barretenberg/flavor/ultra.hpp"
-// #include "barretenberg/honk/utils/testing.hpp"
-// #include "barretenberg/protogalaxy/protogalaxy_prover.hpp"
-// #include "barretenberg/relations/relation_parameters.hpp"
-// #include "barretenberg/sumcheck/instance/instances.hpp"
-// #include <gtest/gtest.h>
-
-// using namespace proof_system::honk;
-// namespace barretenberg::test_protogalaxy_prover {
-// using Flavor = proof_system::honk::flavor::Ultra;
-// using Polynomial = typename Flavor::Polynomial;
-// using FF = typename Flavor::FF;
-// using RelationParameters = proof_system::RelationParameters<FF>;
-
-// // TODO(https://github.com/AztecProtocol/barretenberg/issues/780): Improve combiner tests to check more than the
-// // arithmetic relation so we more than unit test folding relation parameters and alpha as well.
-// TEST(Protogalaxy, CombinerOn2Instances)
-// {
-//     constexpr size_t NUM_INSTANCES = 2;
-//     using ProverInstance = ProverInstance_<Flavor>;
-//     using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
-//     using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
-
-//     const auto restrict_to_standard_arithmetic_relation = [](auto& polys) {
-//         std::fill(polys.q_arith.begin(), polys.q_arith.end(), 1);
-//         std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
-//         std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
-//         std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
-//         std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
-//         std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
-//         std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
-//         std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
-//     };
-
-//     auto run_test = [&](bool is_random_input) {
-//         // Combiner test on prover polynomisls containing random values, restricted to only the standard arithmetic
-//         // relation.
-//         if (is_random_input) {
-//             std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-//             ProtoGalaxyProver prover;
-//             std::vector<FF> pow_betas = { FF(1), FF(2) };
-
-//             for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-//                 auto instance = std::make_shared<ProverInstance>();
-//                 auto prover_polynomials = proof_system::honk::get_sequential_prover_polynomials<Flavor>(
-//                     /*log_circuit_size=*/1, idx * 128);
-//                 restrict_to_standard_arithmetic_relation(prover_polynomials);
-//                 instance->prover_polynomials = std::move(prover_polynomials);
-//                 instance->instance_size = 2;
-//                 instance_data[idx] = instance;
-//             }
-
-//             ProverInstances instances{ instance_data };
-//             instances.alpha = Univariate<FF, 13>(FF(0)); // focus on the arithmetic relation only
-
-//             auto result = prover.compute_combiner(instances, pow_betas);
-//             auto expected_result =
-//                 barretenberg::Univariate<FF, 13>(std::array<FF, 13>{ 87706,
-//                                                                      13644570,
-//                                                                      76451738,
-//                                                                      226257946,
-//                                                                      static_cast<uint64_t>(500811930),
-//                                                                      static_cast<uint64_t>(937862426),
-//                                                                      static_cast<uint64_t>(1575158170),
-//                                                                      static_cast<uint64_t>(2450447898),
-//                                                                      static_cast<uint64_t>(3601480346),
-//                                                                      static_cast<uint64_t>(5066004250),
-//                                                                      static_cast<uint64_t>(6881768346),
-//                                                                      static_cast<uint64_t>(9086521370),
-//                                                                      static_cast<uint64_t>(11718012058) });
-//             EXPECT_EQ(result, expected_result);
-//         } else {
-//             std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-//             ProtoGalaxyProver prover;
-//             std::vector<FF> pow_betas = { FF(1), FF(2) };
-
-//             for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-//                 auto instance = std::make_shared<ProverInstance>();
-//                 auto prover_polynomials = proof_system::honk::get_zero_prover_polynomials<Flavor>(
-//                     /*log_circuit_size=*/1);
-//                 restrict_to_standard_arithmetic_relation(prover_polynomials);
-//                 instance->prover_polynomials = std::move(prover_polynomials);
-//                 instance->instance_size = 2;
-//                 instance_data[idx] = instance;
-//             }
-
-//             ProverInstances instances{ instance_data };
-//             instances.alpha = Univariate<FF, 13>(FF(0)); // focus on the arithmetic relation only
-
-//             const auto create_add_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
-//                 polys.w_l[idx] = w_l;
-//                 polys.w_r[idx] = w_r;
-//                 polys.w_o[idx] = w_l + w_r;
-//                 polys.q_l[idx] = 1;
-//                 polys.q_r[idx] = 1;
-//                 polys.q_o[idx] = -1;
-//             };
-
-//             const auto create_mul_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
-//                 polys.w_l[idx] = w_l;
-//                 polys.w_r[idx] = w_r;
-//                 polys.w_o[idx] = w_l * w_r;
-//                 polys.q_m[idx] = 1;
-//                 polys.q_o[idx] = -1;
-//             };
-
-//             create_add_gate(instances[0]->prover_polynomials, 0, 1, 2);
-//             create_add_gate(instances[0]->prover_polynomials, 1, 0, 4);
-//             create_add_gate(instances[1]->prover_polynomials, 0, 3, 4);
-//             create_mul_gate(instances[1]->prover_polynomials, 1, 1, 4);
-
-//             restrict_to_standard_arithmetic_relation(instances[0]->prover_polynomials);
-//             restrict_to_standard_arithmetic_relation(instances[1]->prover_polynomials);
-
-//             /* Instance 0                                    Instance 1
-//                 w_l w_r w_o q_m q_l q_r q_o q_c               w_l w_r w_o q_m q_l q_r q_o q_c
-//                 1   2   3   0   1   1   -1  0                 3   4   7   0   1   1   -1  0
-//                 0   4   4   0   1   1   -1  0                 1   4   4   1   0   0   -1  0             */
-
-//             /* Lagrange-combined values, row index 0         Lagrange-combined values, row index 1
-//                 in    0    1    2    3    4    5    6        in    0    1    2    3    4    5    6
-//                 w_l   1    3    5    7    9   11   13        w_l   0    1    2    3    4    5    6
-//                 w_r   2    4    6    8   10   12   14        w_r   4    4    4    4    4    4    4
-//                 w_o   3    7   11   15   19   23   27        w_o   4    4    4    4    4    4    0
-//                 q_m   0    0    0    0    0    0    0        q_m   0    1    2    3    4    5    6
-//                 q_l   1    1    1    1    1    1    1        q_l   1    0   -1   -2   -3   -4   -5
-//                 q_r   1    1    1    1    1    1    1        q_r   1    0   -1   -2   -3   -4   -5
-//                 q_o  -1   -1   -1   -1   -1   -1   -1        q_o  -1   -1   -1   -1   -1   -1   -1
-//                 q_c   0    0    0    0    0    0    0        q_c   0    0    0    0    0    0    0
-
-//             relation value:
-//                       0    0    0    0    0    0    0              0    0    6   18   36   60   90      */
-
-//             auto result = prover.compute_combiner(instances, pow_betas);
-//             auto expected_result = barretenberg::Univariate<FF, 13>(
-//                 std::array<FF, 13>{ 0, 0, 12, 36, 72, 120, 180, 252, 336, 432, 540, 660, 792 });
-
-//             EXPECT_EQ(result, expected_result);
-//         }
-//     };
-//     run_test(true);
-//     run_test(false);
-// };
-
-// TEST(Protogalaxy, CombinerOn4Instances)
-// {
-//     constexpr size_t NUM_INSTANCES = 4;
-//     using ProverInstance = ProverInstance_<Flavor>;
-//     using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
-//     using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
-
-//     const auto zero_all_selectors = [](auto& polys) {
-//         std::fill(polys.q_arith.begin(), polys.q_arith.end(), 0);
-//         std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
-//         std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
-//         std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
-//         std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
-//         std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
-//         std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
-//         std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
-//     };
-
-//     auto run_test = [&]() {
-//         std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
-//         ProtoGalaxyProver prover;
-//         std::vector<FF> pow_betas = { FF(1), FF(2) };
-
-//         for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
-//             auto instance = std::make_shared<ProverInstance>();
-//             auto prover_polynomials = proof_system::honk::get_zero_prover_polynomials<Flavor>(
-//                 /*log_circuit_size=*/1);
-//             instance->prover_polynomials = std::move(prover_polynomials);
-//             instance->instance_size = 2;
-//             instance_data[idx] = instance;
-//         }
-
-//         ProverInstances instances{ instance_data };
-//         instances.alpha = Univariate<FF, 43>(FF(0)); // focus on the arithmetic relation only
-
-//         zero_all_selectors(instances[0]->prover_polynomials);
-//         zero_all_selectors(instances[1]->prover_polynomials);
-//         zero_all_selectors(instances[2]->prover_polynomials);
-//         zero_all_selectors(instances[3]->prover_polynomials);
-
-//         auto result = prover.compute_combiner(instances, pow_betas);
-//         std::array<FF, 43> zeroes;
-//         std::fill(zeroes.begin(), zeroes.end(), 0);
-//         auto expected_result = barretenberg::Univariate<FF, 43>(zeroes);
-//         EXPECT_EQ(result, expected_result);
-//     };
-//     run_test();
-// };
-
-// } // namespace barretenberg::test_protogalaxy_prover
\ No newline at end of file
+#include "barretenberg/flavor/ultra.hpp"
+#include "barretenberg/honk/utils/testing.hpp"
+#include "barretenberg/protogalaxy/protogalaxy_prover.hpp"
+#include "barretenberg/relations/relation_parameters.hpp"
+#include "barretenberg/sumcheck/instance/instances.hpp"
+#include <gtest/gtest.h>
+
+using namespace proof_system::honk;
+namespace barretenberg::test_protogalaxy_prover {
+using Flavor = proof_system::honk::flavor::Ultra;
+using Polynomial = typename Flavor::Polynomial;
+using FF = typename Flavor::FF;
+using RelationParameters = proof_system::RelationParameters<FF>;
+
+// TODO(https://github.com/AztecProtocol/barretenberg/issues/780): Improve combiner tests to check more than the
+// arithmetic relation so we more than unit test folding relation parameters and alpha as well.
+TEST(Protogalaxy, CombinerOn2Instances)
+{
+    constexpr size_t NUM_INSTANCES = 2;
+    using ProverInstance = ProverInstance_<Flavor>;
+    using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
+    using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
+
+    const auto restrict_to_standard_arithmetic_relation = [](auto& polys) {
+        std::fill(polys.q_arith.begin(), polys.q_arith.end(), 1);
+        std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
+        std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
+        std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
+        std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
+        std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
+        std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
+        std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
+    };
+
+    auto run_test = [&](bool is_random_input) {
+        // Combiner test on prover polynomisls containing random values, restricted to only the standard arithmetic
+        // relation.
+        if (is_random_input) {
+            std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+            ProtoGalaxyProver prover;
+            std::vector<FF> pow_betas = { FF(1), FF(2) };
+
+            for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+                auto instance = std::make_shared<ProverInstance>();
+                auto prover_polynomials = proof_system::honk::get_sequential_prover_polynomials<Flavor>(
+                    /*log_circuit_size=*/1, idx * 128);
+                restrict_to_standard_arithmetic_relation(prover_polynomials);
+                instance->prover_polynomials = std::move(prover_polynomials);
+                instance->instance_size = 2;
+                instance_data[idx] = instance;
+            }
+
+            ProverInstances instances{ instance_data };
+            instances.alphas.fill(Univariate<FF, 13>(FF(0))); // focus on the arithmetic relation only
+
+            auto result = prover.compute_combiner(instances, pow_betas);
+            auto expected_result =
+                barretenberg::Univariate<FF, 13>(std::array<FF, 13>{ 87706,
+                                                                     13644570,
+                                                                     76451738,
+                                                                     226257946,
+                                                                     static_cast<uint64_t>(500811930),
+                                                                     static_cast<uint64_t>(937862426),
+                                                                     static_cast<uint64_t>(1575158170),
+                                                                     static_cast<uint64_t>(2450447898),
+                                                                     static_cast<uint64_t>(3601480346),
+                                                                     static_cast<uint64_t>(5066004250),
+                                                                     static_cast<uint64_t>(6881768346),
+                                                                     static_cast<uint64_t>(9086521370),
+                                                                     static_cast<uint64_t>(11718012058) });
+            EXPECT_EQ(result, expected_result);
+        } else {
+            std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+            ProtoGalaxyProver prover;
+            std::vector<FF> pow_betas = { FF(1), FF(2) };
+
+            for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+                auto instance = std::make_shared<ProverInstance>();
+                auto prover_polynomials = proof_system::honk::get_zero_prover_polynomials<Flavor>(
+                    /*log_circuit_size=*/1);
+                restrict_to_standard_arithmetic_relation(prover_polynomials);
+                instance->prover_polynomials = std::move(prover_polynomials);
+                instance->instance_size = 2;
+                instance_data[idx] = instance;
+            }
+
+            ProverInstances instances{ instance_data };
+            instances.alphas.fill(Univariate<FF, 13>(FF(0))); // focus on the arithmetic relation only
+
+            const auto create_add_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
+                polys.w_l[idx] = w_l;
+                polys.w_r[idx] = w_r;
+                polys.w_o[idx] = w_l + w_r;
+                polys.q_l[idx] = 1;
+                polys.q_r[idx] = 1;
+                polys.q_o[idx] = -1;
+            };
+
+            const auto create_mul_gate = [](auto& polys, const size_t idx, FF w_l, FF w_r) {
+                polys.w_l[idx] = w_l;
+                polys.w_r[idx] = w_r;
+                polys.w_o[idx] = w_l * w_r;
+                polys.q_m[idx] = 1;
+                polys.q_o[idx] = -1;
+            };
+
+            create_add_gate(instances[0]->prover_polynomials, 0, 1, 2);
+            create_add_gate(instances[0]->prover_polynomials, 1, 0, 4);
+            create_add_gate(instances[1]->prover_polynomials, 0, 3, 4);
+            create_mul_gate(instances[1]->prover_polynomials, 1, 1, 4);
+
+            restrict_to_standard_arithmetic_relation(instances[0]->prover_polynomials);
+            restrict_to_standard_arithmetic_relation(instances[1]->prover_polynomials);
+
+            /* Instance 0                                    Instance 1
+                w_l w_r w_o q_m q_l q_r q_o q_c               w_l w_r w_o q_m q_l q_r q_o q_c
+                1   2   3   0   1   1   -1  0                 3   4   7   0   1   1   -1  0
+                0   4   4   0   1   1   -1  0                 1   4   4   1   0   0   -1  0             */
+
+            /* Lagrange-combined values, row index 0         Lagrange-combined values, row index 1
+                in    0    1    2    3    4    5    6        in    0    1    2    3    4    5    6
+                w_l   1    3    5    7    9   11   13        w_l   0    1    2    3    4    5    6
+                w_r   2    4    6    8   10   12   14        w_r   4    4    4    4    4    4    4
+                w_o   3    7   11   15   19   23   27        w_o   4    4    4    4    4    4    0
+                q_m   0    0    0    0    0    0    0        q_m   0    1    2    3    4    5    6
+                q_l   1    1    1    1    1    1    1        q_l   1    0   -1   -2   -3   -4   -5
+                q_r   1    1    1    1    1    1    1        q_r   1    0   -1   -2   -3   -4   -5
+                q_o  -1   -1   -1   -1   -1   -1   -1        q_o  -1   -1   -1   -1   -1   -1   -1
+                q_c   0    0    0    0    0    0    0        q_c   0    0    0    0    0    0    0
+
+            relation value:
+                      0    0    0    0    0    0    0              0    0    6   18   36   60   90      */
+
+            auto result = prover.compute_combiner(instances, pow_betas);
+            auto expected_result = barretenberg::Univariate<FF, 13>(
+                std::array<FF, 13>{ 0, 0, 12, 36, 72, 120, 180, 252, 336, 432, 540, 660, 792 });
+
+            EXPECT_EQ(result, expected_result);
+        }
+    };
+    run_test(true);
+    run_test(false);
+};
+
+TEST(Protogalaxy, CombinerOn4Instances)
+{
+    constexpr size_t NUM_INSTANCES = 4;
+    using ProverInstance = ProverInstance_<Flavor>;
+    using ProverInstances = ProverInstances_<Flavor, NUM_INSTANCES>;
+    using ProtoGalaxyProver = ProtoGalaxyProver_<ProverInstances>;
+
+    const auto zero_all_selectors = [](auto& polys) {
+        std::fill(polys.q_arith.begin(), polys.q_arith.end(), 0);
+        std::fill(polys.q_sort.begin(), polys.q_sort.end(), 0);
+        std::fill(polys.q_elliptic.begin(), polys.q_elliptic.end(), 0);
+        std::fill(polys.q_aux.begin(), polys.q_aux.end(), 0);
+        std::fill(polys.q_lookup.begin(), polys.q_lookup.end(), 0);
+        std::fill(polys.q_4.begin(), polys.q_4.end(), 0);
+        std::fill(polys.w_4.begin(), polys.w_4.end(), 0);
+        std::fill(polys.w_4_shift.begin(), polys.w_4_shift.end(), 0);
+    };
+
+    auto run_test = [&]() {
+        std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
+        ProtoGalaxyProver prover;
+        std::vector<FF> pow_betas = { FF(1), FF(2) };
+
+        for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
+            auto instance = std::make_shared<ProverInstance>();
+            auto prover_polynomials = proof_system::honk::get_zero_prover_polynomials<Flavor>(
+                /*log_circuit_size=*/1);
+            instance->prover_polynomials = std::move(prover_polynomials);
+            instance->instance_size = 2;
+            instance_data[idx] = instance;
+        }
+
+        ProverInstances instances{ instance_data };
+        instances.alphas.fill(Univariate<FF, 43>(FF(0))); // focus on the arithmetic relation only
+
+        zero_all_selectors(instances[0]->prover_polynomials);
+        zero_all_selectors(instances[1]->prover_polynomials);
+        zero_all_selectors(instances[2]->prover_polynomials);
+        zero_all_selectors(instances[3]->prover_polynomials);
+
+        auto result = prover.compute_combiner(instances, pow_betas);
+        std::array<FF, 43> zeroes;
+        std::fill(zeroes.begin(), zeroes.end(), 0);
+        auto expected_result = barretenberg::Univariate<FF, 43>(zeroes);
+        EXPECT_EQ(result, expected_result);
+    };
+    run_test();
+};
+
+} // namespace barretenberg::test_protogalaxy_prover
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
index daa33631c00..45a4ac93ad0 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
@@ -42,7 +42,7 @@ template <UltraFlavor Flavor> void DeciderProver_<Flavor>::execute_preamble_roun
     transcript->send_to_verifier("lookup_grand_product_delta",
                                  accumulator->relation_parameters.lookup_grand_product_delta);
     size_t alpha_idx = 0;
-    for (auto alpha : accumulator->alpha) {
+    for (auto alpha : accumulator->alphas) {
         transcript->send_to_verifier("alpha_" + std::to_string(alpha_idx), alpha);
     }
 
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
index 760430587d2..3befe10bb57 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
@@ -34,7 +34,7 @@ template <typename Flavor> bool DeciderVerifier_<Flavor>::verify_proof(const plo
     using Instance = VerifierInstance_<Flavor>;
     using VerifierCommitments = typename Flavor::VerifierCommitments;
 
-    static constexpr size_t NUM_SUBRELATIONS = Flavor::NUMBER_OF_SUBRELATIONS;
+    static constexpr size_t NUM_SUBRELATIONS = Flavor::NUM_SUBRELATIONS;
     transcript = std::make_shared<Transcript>(proof.proof_data);
     auto inst = std::make_unique<Instance>();
 
@@ -57,7 +57,7 @@ template <typename Flavor> bool DeciderVerifier_<Flavor>::verify_proof(const plo
 
     for (size_t idx = 0; idx < NUM_SUBRELATIONS - 1; idx++) {
 
-        inst->alpha[idx] = transcript->template receive_from_prover<FF>("alpha" + std::to_string(idx));
+        inst->alphas[idx] = transcript->template receive_from_prover<FF>("alpha" + std::to_string(idx));
     }
 
     inst->target_sum = transcript->template receive_from_prover<FF>("target_sum");
@@ -93,7 +93,7 @@ template <typename Flavor> bool DeciderVerifier_<Flavor>::verify_proof(const plo
     auto sumcheck = SumcheckVerifier<Flavor>(inst->log_instance_size, transcript, inst->target_sum);
 
     auto [multivariate_challenge, claimed_evaluations, sumcheck_verified] =
-        sumcheck.verify(inst->relation_parameters, inst->alpha, inst->gate_challenges);
+        sumcheck.verify(inst->relation_parameters, inst->alphas, inst->gate_challenges);
 
     // If Sumcheck did not verify, return false
     if (sumcheck_verified.has_value() && !sumcheck_verified.value()) {
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index 747d4c903ae..88e8b9de9ab 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -57,7 +57,7 @@ void ProtoGalaxyProver_<ProverInstances>::finalise_and_send_instance(std::shared
     transcript->send_to_verifier(domain_separator + "_" + commitment_labels.z_lookup,
                                  instance->witness_commitments.z_lookup);
     for (size_t idx = 0; idx < NUM_SUBRELATIONS - 1; idx++) {
-        instance->alpha[idx] = transcript->get_challenge(domain_separator + "_alpha_" + std::to_string(idx));
+        instance->alphas[idx] = transcript->get_challenge(domain_separator + "_alpha_" + std::to_string(idx));
     }
     auto vk_view = instance->verification_key->get_all();
     auto labels = instance->commitment_labels.get_precomputed();
@@ -89,7 +89,7 @@ void ProtoGalaxyProver_<ProverInstances>::send_accumulator(std::shared_ptr<Insta
                                  instance->relation_parameters.lookup_grand_product_delta);
 
     for (size_t idx = 0; idx < NUM_SUBRELATIONS - 1; idx++) {
-        transcript->send_to_verifier(domain_separator + "_alpha_" + std::to_string(idx), instance->alpha[idx]);
+        transcript->send_to_verifier(domain_separator + "_alpha_" + std::to_string(idx), instance->alphas[idx]);
     }
 
     transcript->send_to_verifier(domain_separator + "_target_sum", instance->target_sum);
@@ -225,10 +225,10 @@ std::shared_ptr<typename ProverInstances::Instance> ProtoGalaxyProver_<ProverIns
 
     // Evaluate the combined batching challenge α univariate at challenge to obtain next α and send it to the
     // verifier
-    auto& folded_alpha = next_accumulator->alpha;
+    auto& folded_alphas = next_accumulator->alphas;
     for (size_t idx = 0; idx < NUM_SUBRELATIONS - 1; idx++) {
-        folded_alpha[idx] = instances.alpha[idx].evaluate(challenge);
-        transcript->send_to_verifier("next_alpha_" + std::to_string(idx), folded_alpha[idx]);
+        folded_alphas[idx] = instances.alphas[idx].evaluate(challenge);
+        transcript->send_to_verifier("next_alpha_" + std::to_string(idx), folded_alphas[idx]);
     }
 
     // Evaluate each relation parameter univariate at challenge to obtain the folded relation parameters and send to
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index 81071f48051..08815320bf2 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -260,7 +260,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
                                               const std::vector<FF>& deltas)
     {
         auto full_honk_evaluations = compute_full_honk_evaluations(
-            accumulator->prover_polynomials, accumulator->alpha, accumulator->relation_parameters);
+            accumulator->prover_polynomials, accumulator->alphas, accumulator->relation_parameters);
         const auto betas = accumulator->gate_challenges;
         assert(betas.size() == deltas.size());
         auto coeffs = construct_perturbator_coefficients(betas, deltas, full_honk_evaluations);
@@ -360,7 +360,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
             Utils::add_nested_tuples(univariate_accumulators, accumulators);
         }
         // Batch the univariate contributions from each sub-relation to obtain the round univariate
-        return batch_over_relations(univariate_accumulators, instances.alpha);
+        return batch_over_relations(univariate_accumulators, instances.alphas);
     }
     static ExtendedUnivariateWithRandomization batch_over_relations(TupleOfTuplesOfUnivariates& univariate_accumulators,
                                                                     const CombinedAlphaType& alpha)
@@ -446,11 +446,11 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     static void combine_alpha(ProverInstances& instances)
     {
         size_t alpha_idx = 0;
-        for (auto& alpha : instances.alpha) {
+        for (auto& alpha : instances.alphas) {
             Univariate<FF, ProverInstances::NUM> tmp;
             size_t instance_idx = 0;
             for (auto& instance : instances) {
-                tmp.value_at(instance_idx) = instance->alpha[alpha_idx];
+                tmp.value_at(instance_idx) = instance->alphas[alpha_idx];
                 instance_idx++;
             }
             alpha = tmp.template extend_to<ProverInstances::BATCHED_EXTENDED_LENGTH>();
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index 25c2a6cd4d4..6a340f07015 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -27,7 +27,7 @@ void ProtoGalaxyVerifier_<VerifierInstances>::receive_accumulator(const std::sha
         RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
 
     for (size_t idx = 0; idx < NUM_SUBRELATIONS - 1; idx++) {
-        inst->alpha[idx] =
+        inst->alphas[idx] =
             transcript->template receive_from_prover<FF>(domain_separator + "_alpha_" + std::to_string(idx));
     }
 
@@ -95,7 +95,7 @@ void ProtoGalaxyVerifier_<VerifierInstances>::receive_and_finalise_instance(cons
         RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
 
     for (size_t idx = 0; idx < NUM_SUBRELATIONS - 1; idx++) {
-        inst->alpha[idx] = transcript->get_challenge(domain_separator + "_alpha_" + std::to_string(idx));
+        inst->alphas[idx] = transcript->get_challenge(domain_separator + "_alpha_" + std::to_string(idx));
     }
 
     inst->verification_key = std::make_shared<VerificationKey>(inst->instance_size, inst->public_input_size);
@@ -217,13 +217,13 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
         el_idx++;
     }
 
-    AlphaType expected_alpha;
+    AlphaType expected_alphas;
     size_t alpha_idx = 0;
-    for (auto& alpha : expected_alpha) {
+    for (auto& alpha : expected_alphas) {
         alpha = FF(0);
         size_t instance_idx = 0;
         for (auto& instance : instances) {
-            alpha += instance->alpha[alpha_idx] * lagranges[instance_idx];
+            alpha += instance->alphas[alpha_idx] * lagranges[instance_idx];
             instance_idx++;
         }
         auto next_alpha = transcript->template receive_from_prover<FF>("next_alpha_" + std::to_string(alpha_idx));
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
index a8b4ec172a1..9c2660699ae 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
@@ -19,7 +19,7 @@ template <class VerifierInstances> class ProtoGalaxyVerifier_ {
     using CommitmentLabels = typename Flavor::CommitmentLabels;
     using AlphaType = typename Flavor::AlphaType;
 
-    static constexpr size_t NUM_SUBRELATIONS = Flavor::NUMBER_OF_SUBRELATIONS;
+    static constexpr size_t NUM_SUBRELATIONS = Flavor::NUM_SUBRELATIONS;
 
     VerifierInstances instances;
 
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
index ee330ad8d46..d4a78800349 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
@@ -10,7 +10,7 @@ template <typename Flavor_, size_t NUM_> struct ProverInstances_ {
     using Flavor = Flavor_;
     using FF = typename Flavor::FF;
     static constexpr size_t NUM = NUM_;
-    static constexpr size_t NUM_SUBRELATIONS = Flavor::NUMBER_OF_SUBRELATIONS;
+    static constexpr size_t NUM_SUBRELATIONS = Flavor::NUM_SUBRELATIONS;
     using Instance = ProverInstance_<Flavor>;
 
     using ArrayType = std::array<std::shared_ptr<Instance>, NUM_>;
@@ -21,7 +21,7 @@ template <typename Flavor_, size_t NUM_> struct ProverInstances_ {
     using AlphaType = std::array<Univariate<FF, BATCHED_EXTENDED_LENGTH>, NUM_SUBRELATIONS - 1>;
     ArrayType _data;
     RelationParameters relation_parameters;
-    AlphaType alpha;
+    AlphaType alphas;
     std::vector<FF> next_gate_challenges;
 
     std::shared_ptr<Instance> const& operator[](size_t idx) const { return _data[idx]; }
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
index 35aa0975d34..878e629d6f2 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
@@ -46,7 +46,7 @@ template <class Flavor> class ProverInstance_ {
     // non-zero  for Instances constructed from circuits, this concept doesn't exist for accumulated
     // instances
     size_t pub_inputs_offset = 0;
-    AlphaType alpha;
+    AlphaType alphas;
     proof_system::RelationParameters<FF> relation_parameters;
     std::vector<uint32_t> recursive_proof_public_input_indices;
 
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
index 1e880be9a46..304abca199c 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
@@ -18,7 +18,7 @@ template <class Flavor> class VerifierInstance_ {
     size_t instance_size;
     size_t log_instance_size;
     RelationParameters<FF> relation_parameters;
-    AlphaType alpha;
+    AlphaType alphas;
     bool is_accumulator = false;
 
     // The folding parameters (\vec{β}, e) which are set for accumulators (i.e. relaxed instances).
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
index 42be701f535..3f6959b03a7 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
@@ -73,7 +73,7 @@ template <typename Flavor> class SumcheckProver {
     SumcheckOutput<Flavor> prove(std::shared_ptr<Instance> instance)
     {
         return prove(
-            instance->prover_polynomials, instance->relation_parameters, instance->alpha, instance->gate_challenges);
+            instance->prover_polynomials, instance->relation_parameters, instance->alphas, instance->gate_challenges);
     };
 
     /**
diff --git a/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_prover.cpp b/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_prover.cpp
index fa55f620fcf..00a4e291e8e 100644
--- a/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_prover.cpp
@@ -140,7 +140,7 @@ void GoblinTranslatorProver::execute_relation_check_rounds()
     using Sumcheck = sumcheck::SumcheckProver<Flavor>;
 
     auto sumcheck = Sumcheck(key->circuit_size, transcript);
-    FF alpha = transcript->get_challenge("alpha");
+    FF alpha = transcript->get_challenge("Sumcheck:alpha");
     std::vector<FF> gate_challenges(numeric::get_msb(key->circuit_size));
     for (size_t idx = 0; idx < gate_challenges.size(); idx++) {
         gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
diff --git a/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_verifier.cpp b/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_verifier.cpp
index c2ecf1dec68..d6797c62a1b 100644
--- a/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/translator_vm/goblin_translator_verifier.cpp
@@ -245,7 +245,7 @@ bool GoblinTranslatorVerifier::verify_proof(const plonk::proof& proof)
     // Execute Sumcheck Verifier
     const size_t log_circuit_size = numeric::get_msb(circuit_size);
     auto sumcheck = SumcheckVerifier<Flavor>(log_circuit_size, transcript);
-    FF alpha = transcript->get_challenge("alpha");
+    FF alpha = transcript->get_challenge("Sumcheck:alpha");
     std::vector<FF> gate_challenges(numeric::get_msb(key->circuit_size));
     for (size_t idx = 0; idx < gate_challenges.size(); idx++) {
         gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/goblin_ultra_transcript.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/goblin_ultra_transcript.test.cpp
index f12d291086e..5f7cc775fbf 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/goblin_ultra_transcript.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/goblin_ultra_transcript.test.cpp
@@ -33,7 +33,7 @@ class GoblinUltraTranscriptTests : public ::testing::Test {
         auto log_n = numeric::get_msb(circuit_size);
 
         size_t MAX_PARTIAL_RELATION_LENGTH = Flavor::BATCHED_RELATION_PARTIAL_LENGTH;
-        size_t NUM_SUBRELATIONS = Flavor::NUMBER_OF_SUBRELATIONS;
+        size_t NUM_SUBRELATIONS = Flavor::NUM_SUBRELATIONS;
 
         size_t size_FF = sizeof(FF);
         size_t size_G = 2 * size_FF;
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
index 59e44ffa80e..4d486749e24 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
@@ -63,7 +63,7 @@ void check_accumulator_target_sum_manual(std::shared_ptr<Instance>& accumulator,
 {
     auto instance_size = accumulator->instance_size;
     auto expected_honk_evals = ProtoGalaxyProver::compute_full_honk_evaluations(
-        accumulator->prover_polynomials, accumulator->alpha, accumulator->relation_parameters);
+        accumulator->prover_polynomials, accumulator->alphas, accumulator->relation_parameters);
     // Construct pow(\vec{betas*}) as in the paper
     auto expected_pows_beta =
         ProtoGalaxyProver::compute_pow_polynomial_at_values(accumulator->gate_challenges, instance_size);
@@ -112,11 +112,11 @@ TEST_F(ProtoGalaxyTests, FullHonkEvaluationsValidCircuit)
     instance->compute_sorted_accumulator_polynomials(eta);
     instance->compute_grand_product_polynomials(beta, gamma);
 
-    for (auto& alpha : instance->alpha) {
+    for (auto& alpha : instance->alphas) {
         alpha = FF::random_element();
     }
     auto full_honk_evals = ProtoGalaxyProver::compute_full_honk_evaluations(
-        instance->prover_polynomials, instance->alpha, instance->relation_parameters);
+        instance->prover_polynomials, instance->alphas, instance->relation_parameters);
 
     // Evaluations should be 0 for valid circuit
     for (const auto& eval : full_honk_evals) {
@@ -148,13 +148,13 @@ TEST_F(ProtoGalaxyTests, PerturbatorPolynomial)
     }
     auto full_polynomials = construct_ultra_full_polynomials(random_polynomials);
     auto relation_parameters = proof_system::RelationParameters<FF>::get_random();
-    AlphaType alpha;
-    for (auto& a : alpha) {
-        a = FF::random_element();
+    AlphaType alphas;
+    for (auto& alpha : alphas) {
+        alpha = FF::random_element();
     }
 
     auto full_honk_evals =
-        ProtoGalaxyProver::compute_full_honk_evaluations(full_polynomials, alpha, relation_parameters);
+        ProtoGalaxyProver::compute_full_honk_evaluations(full_polynomials, alphas, relation_parameters);
     std::vector<FF> betas(log_instance_size);
     for (size_t idx = 0; idx < log_instance_size; idx++) {
         betas[idx] = FF::random_element();
@@ -174,7 +174,7 @@ TEST_F(ProtoGalaxyTests, PerturbatorPolynomial)
     accumulator->gate_challenges = betas;
     accumulator->target_sum = target_sum;
     accumulator->relation_parameters = relation_parameters;
-    accumulator->alpha = alpha;
+    accumulator->alphas = alphas;
 
     auto deltas = ProtoGalaxyProver::compute_round_challenge_pows(log_instance_size, FF::random_element());
     auto perturbator = ProtoGalaxyProver::compute_perturbator(accumulator, deltas);
@@ -247,18 +247,18 @@ TEST_F(ProtoGalaxyTests, CombineAlpha)
 
     Builder builder1;
     auto instance1 = std::make_shared<Instance>(builder1);
-    instance1->alpha.fill(2);
+    instance1->alphas.fill(2);
 
     Builder builder2;
     builder2.add_variable(3);
     auto instance2 = std::make_shared<Instance>(builder2);
-    instance2->alpha.fill(4);
+    instance2->alphas.fill(4);
 
     Instances instances{ { instance1, instance2 } };
     ProtoGalaxyProver::combine_alpha(instances);
 
     Univariate<FF, 13> expected_alpha{ { 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26 } };
-    for (const auto& alpha : instances.alpha) {
+    for (const auto& alpha : instances.alphas) {
         EXPECT_EQ(alpha, expected_alpha);
     }
 }
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/sumcheck.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/sumcheck.test.cpp
index 791ab283e69..545a587c28a 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/sumcheck.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/sumcheck.test.cpp
@@ -165,12 +165,12 @@ TEST_F(SumcheckTestsRealCircuit, Ultra)
     auto circuit_size = instance->proving_key->circuit_size;
     auto log_circuit_size = numeric::get_msb(circuit_size);
 
-    AlphaType prover_alpha;
-    for (size_t idx = 0; idx < prover_alpha.size(); idx++) {
-        prover_alpha[idx] = prover_transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
+    AlphaType prover_alphas;
+    for (size_t idx = 0; idx < prover_alphas.size(); idx++) {
+        prover_alphas[idx] = prover_transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
     }
 
-    instance->alpha = prover_alpha;
+    instance->alphas = prover_alphas;
     auto sumcheck_prover = SumcheckProver<Flavor>(circuit_size, prover_transcript);
     std::vector<FF> prover_gate_challenges(log_circuit_size);
     for (size_t idx = 0; idx < log_circuit_size; idx++) {
@@ -183,9 +183,9 @@ TEST_F(SumcheckTestsRealCircuit, Ultra)
     auto verifier_transcript = Transcript::verifier_init_empty(prover_transcript);
 
     auto sumcheck_verifier = SumcheckVerifier<Flavor>(log_circuit_size, verifier_transcript);
-    AlphaType verifier_alpha;
-    for (size_t idx = 0; idx < verifier_alpha.size(); idx++) {
-        verifier_alpha[idx] = verifier_transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
+    AlphaType verifier_alphas;
+    for (size_t idx = 0; idx < verifier_alphas.size(); idx++) {
+        verifier_alphas[idx] = verifier_transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
     }
 
     std::vector<FF> verifier_gate_challenges(log_circuit_size);
@@ -194,7 +194,7 @@ TEST_F(SumcheckTestsRealCircuit, Ultra)
             verifier_transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
     }
     auto verifier_output =
-        sumcheck_verifier.verify(instance->relation_parameters, verifier_alpha, verifier_gate_challenges);
+        sumcheck_verifier.verify(instance->relation_parameters, verifier_alphas, verifier_gate_challenges);
 
     auto verified = verifier_output.verified.value();
 
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp
index 5432cde5430..c9c7f258972 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp
@@ -149,11 +149,11 @@ template <UltraFlavor Flavor> void UltraProver_<Flavor>::execute_relation_check_
     using Sumcheck = sumcheck::SumcheckProver<Flavor>;
     auto circuit_size = instance->proving_key->circuit_size;
     auto sumcheck = Sumcheck(circuit_size, transcript);
-    AlphaType alpha;
-    for (size_t idx = 0; idx < alpha.size(); idx++) {
-        alpha[idx] = transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
+    AlphaType alphas;
+    for (size_t idx = 0; idx < alphas.size(); idx++) {
+        alphas[idx] = transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
     }
-    instance->alpha = alpha;
+    instance->alphas = alphas;
     std::vector<FF> gate_challenges(numeric::get_msb(circuit_size));
     for (size_t idx = 0; idx < gate_challenges.size(); idx++) {
         gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_transcript.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_transcript.test.cpp
index 72f0c49916c..6e8b19a0891 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_transcript.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_transcript.test.cpp
@@ -33,7 +33,7 @@ class UltraTranscriptTests : public ::testing::Test {
         auto log_n = numeric::get_msb(circuit_size);
 
         size_t MAX_PARTIAL_RELATION_LENGTH = Flavor::BATCHED_RELATION_PARTIAL_LENGTH;
-        size_t NUM_SUBRELATIONS = Flavor::NUMBER_OF_SUBRELATIONS;
+        size_t NUM_SUBRELATIONS = Flavor::NUM_SUBRELATIONS;
         size_t size_FF = sizeof(FF);
         size_t size_G = 2 * size_FF;
         size_t size_uni = MAX_PARTIAL_RELATION_LENGTH * size_FF;
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp
index 2b12c05ea85..62689f5ae19 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp
@@ -132,9 +132,9 @@ template <typename Flavor> bool UltraVerifier_<Flavor>::verify_proof(const plonk
     // Execute Sumcheck Verifier
     const size_t log_circuit_size = numeric::get_msb(circuit_size);
     auto sumcheck = SumcheckVerifier<Flavor>(log_circuit_size, transcript);
-    AlphaType alpha;
-    for (size_t idx = 0; idx < alpha.size(); idx++) {
-        alpha[idx] = transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
+    AlphaType alphas;
+    for (size_t idx = 0; idx < alphas.size(); idx++) {
+        alphas[idx] = transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
     }
 
     auto gate_challenges = std::vector<FF>(log_circuit_size);
@@ -142,7 +142,7 @@ template <typename Flavor> bool UltraVerifier_<Flavor>::verify_proof(const plonk
         gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
     }
     auto [multivariate_challenge, claimed_evaluations, sumcheck_verified] =
-        sumcheck.verify(relation_parameters, alpha, gate_challenges);
+        sumcheck.verify(relation_parameters, alphas, gate_challenges);
 
     // If Sumcheck did not verify, return false
     if (sumcheck_verified.has_value() && !sumcheck_verified.value()) {
diff --git a/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_prover.cpp b/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_prover.cpp
index 20bf223f5b9..0acdacdf0e2 100644
--- a/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_prover.cpp
@@ -73,7 +73,7 @@ void AvmMiniProver::execute_relation_check_rounds()
     using Sumcheck = sumcheck::SumcheckProver<Flavor>;
 
     auto sumcheck = Sumcheck(key->circuit_size, transcript);
-    FF alpha = transcript->get_challenge("alpha");
+    FF alpha = transcript->get_challenge("Sumcheck:alpha");
     std::vector<FF> gate_challenges(numeric::get_msb(key->circuit_size));
     for (size_t idx = 0; idx < gate_challenges.size(); idx++) {
         gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));
diff --git a/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_verifier.cpp b/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_verifier.cpp
index fba5a203a1a..57371b073d5 100644
--- a/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/vm/generated/AvmMini_verifier.cpp
@@ -87,7 +87,7 @@ bool AvmMiniVerifier::verify_proof(const plonk::proof& proof)
     // Execute Sumcheck Verifier
     const size_t log_circuit_size = numeric::get_msb(circuit_size);
     auto sumcheck = SumcheckVerifier<Flavor>(log_circuit_size, transcript);
-    FF alpha = transcript->get_challenge("alpha");
+    FF alpha = transcript->get_challenge("Sumcheck:alpha");
     auto gate_challenges = std::vector<FF>(log_circuit_size);
     for (size_t idx = 0; idx < log_circuit_size; idx++) {
         gate_challenges[idx] = transcript->get_challenge("Sumcheck:gate_challenge_" + std::to_string(idx));

From c7dec7272de78fbcc81fb97a04dd384ec4c314da Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Thu, 21 Dec 2023 19:55:03 +0000
Subject: [PATCH 63/86] pow

---
 .../cpp/src/barretenberg/polynomials/pow.hpp  | 151 ++++++------------
 .../src/barretenberg/polynomials/pow.test.cpp |  28 ++--
 .../protogalaxy/combiner.test.cpp             |  11 +-
 .../protogalaxy/decider_prover.cpp            |  14 +-
 .../protogalaxy/decider_verifier.cpp          |   3 +-
 .../protogalaxy/protogalaxy_prover.cpp        |  10 +-
 .../protogalaxy/protogalaxy_prover.hpp        |  21 +--
 .../relations/relation_parameters.hpp         |   2 +-
 .../cpp/src/barretenberg/relations/utils.hpp  |  28 ++--
 .../src/barretenberg/sumcheck/sumcheck.hpp    |  11 +-
 .../barretenberg/sumcheck/sumcheck_round.hpp  |  20 +--
 .../sumcheck/sumcheck_round.test.cpp          |   1 -
 .../ultra_honk/protogalaxy.test.cpp           |  23 +--
 .../ultra_honk/ultra_verifier.cpp             |   1 -
 14 files changed, 118 insertions(+), 206 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/polynomials/pow.hpp b/barretenberg/cpp/src/barretenberg/polynomials/pow.hpp
index 874987b5e4f..fcfcfc255f0 100644
--- a/barretenberg/cpp/src/barretenberg/polynomials/pow.hpp
+++ b/barretenberg/cpp/src/barretenberg/polynomials/pow.hpp
@@ -6,29 +6,16 @@ namespace barretenberg {
 
 /**
  * @brief Succinct representation of the `pow` polynomial that can be partially evaluated variable-by-variable.
- * pow(X_0,X_1,..,X_d) = \prod_{0≤l<d} ((1−X_l) + X_l⋅ζ_l)
+ * pow(X_0,X_1,..,X_d) = \prod_{0≤l<d} ((1−X_l) + X_l⋅β_l) for a vector {β_0,...,β_{d-1}}
  *
  * @details Let
  * - d be the number of variables
  * - l be the current Sumcheck round ( l ∈ {0, …, d-1} )
  * - u_0, ..., u_l-1 the challenges sent by the verifier in rounds 0 to l-1.
  *
- * We define
- *
- * - ζ_0, ..., ζ_{d-1}, as ζ_l = ζ^{ 2^l }.
- *   When 0 ≤ i < 2^d is represented in bits [i_0, ..., i_{d-1}] where i_0 is the MSB, we have
- *   ζ^{i} = ζ^{ ∑_{0≤l<d} i_l ⋅ 2^l }
- *         =     ∏_{0≤l<d} ζ^{ i_l ⋅ 2^l }
- *         =     ∏_{0≤l<d} ζ_l^{ i_l }
- *         =     ∏_{0≤l<d} { ( 1-i_l ) + i_l ⋅ ζ_l } // As i_l \in \{0, 1\}
- *   Note that
- *   - ζ_{0} = ζ,
- *   - ζ_{l+1} = ζ_{l}^2,
- *   - ζ_{d-1}   = ζ^{ 2^{d-1} }
- *
- * - pow(X) = ∏_{0≤l<d} ((1−X_l) + X_l⋅ζ_l) is the multilinear polynomial whose evaluation at the i-th index
- *   of the full hypercube, equals ζⁱ.
- *   We can also see it as the multi-linear extension of the vector (ζ, ζ^2, ζ^4, ..., ζ^{2^{d-1}}).
+ * - pow(X) = ∏_{0≤l<d} ((1−X_l) + X_l⋅β_l) is the multilinear polynomial whose evaluation at the i-th index
+ *   of the full hypercube, equals ∏ β_j where j are the bits set to 1 in the binary representation of i
+ *   We can also see it as the multi-linear extension of the vector \vec{β}.
  *
  * - At round l, we iterate over all remaining vertices (i_{l+1}, ..., i_{d-1}) ∈ {0,1}^{d-l-1}.
  *   Let i = ∑_{l<k<d} i_k ⋅ 2^{k-(l+1)} be the index of the current edge over which we are evaluating the relation.
@@ -37,16 +24,16 @@ namespace barretenberg {
  *   powˡᵢ( X_{l} ) = pow( u_{0}, ..., u_{l-1},
  *                         X_{l},
  *                         i_{l+1}, ..., i_{d-1})
- *                  = ∏_{0≤k<l} ( (1-u_k) + u_k⋅ζ_k )
- *                             ⋅( (1−X_l) + X_l⋅ζ_l )
- *                    ∏_{l<k<d} ( (1-i_k) + i_k⋅ζ_k )
- *                  = c_l ⋅ ( (1−X_l) + X_l⋅ζ^{2^l} ) ⋅ ∏_{l<k<d} ( (1-i_k) + i_k⋅ζ^{2^k} )
- *                  = c_l ⋅ ( (1−X_l) + X_l⋅ζ^{2^l} ) ⋅ ζ^{ ∑_{l<k<d} i_k ⋅ 2^k }
- *                  = c_l ⋅ ( (1−X_l) + X_l⋅ζ^{2^l} ) ⋅(ζ^2^{l+1})^{ ∑_{l<k<d} i_k ⋅ 2^{k-(l+1)} }
- *                  = c_l ⋅ ( (1−X_l) + X_l⋅ζ^{2^l} ) ⋅ζ_{l+1}^{i}
+ *                  = ∏_{0≤k<l} ( (1-u_k) + u_k⋅β_k )
+ *                             ⋅( (1−X_l) + X_l⋅β_l )
+ *                    ∏_{l<k<d} ( (1-i_k) + i_k⋅β_k )
  *
- *   This is the pow polynomial, partially evaluated in the first l-1 variables as
- *     (X_{0}, ..., X_{l-1}) = (u_{0}, ..., u_{l-1}).
+ *  Note that as we iterate over the remaining vertices(i_{l+1}, ..., i_{d-1}) ∈ {0,1}^{d-l-1}, the subproducts
+ * ∏_{l<k<d} ( (1-i_k) + i_k⋅β_k ) represents the terms of pow(\vec{β}) that do not contain β_0,...,β_l. These are
+ * periodic in pow(\vec{β}), their periodicity being 2*(l+1).
+ *
+ * This is the pow polynomial,partially evaluated in the first l-1 variables as (X_{0}, ..., X_{l-1}) = (u_{0}, ...,
+ * u_{l-1}).
  *
  * - Sˡᵢ( X_l ) is the univariate of the full relation at edge pair i
  * i.e. it is the alpha-linear-combination of the relations evaluated in the edge at index i.
@@ -56,129 +43,81 @@ namespace barretenberg {
  *
  * We want to check that P(i)=0 for all i ∈ {0,1}^d. So we use Sumcheck over the polynomial
  * P'(X) = pow(X)⋅P(X).
- * The claimed sum is 0 and is equal to ∑_{i ∈ {0,1}^d} pow(i)⋅P(i) = ∑_{i ∈ {0,1}^d} ζ^{i}⋅P(i)
+ * The claimed sum is 0 and is equal to ∑_{i ∈ {0,1}^d} pow(i)⋅P(i).
  * If the Sumcheck passes, then with it must hold with high-probability that all P(i) are 0.
  *
- * The trivial implementation using P'(X) directly would increase the degree of our combined relation by 1.
- * Instead, we exploit the special structure of pow to preserve the same degree.
- *
- * In each round l, the prover should compute the univariate polynomial for the relation defined by P'(X)
- * S'ˡ(X_l) = ∑_{ 0 ≤ i < 2^{d-l-1} } powˡᵢ( X_l ) Sˡᵢ( X_l ) .
- *        = ∑_{ 0 ≤ i < 2^{d-l-1} } [ ζ_{l+1}ⁱ⋅( (1−X_l) + X_l⋅ζ_l )⋅c_l ]⋅Sˡᵢ( X_l )
- *        = ( (1−X_l) + X_l⋅ζ_l ) ⋅ ∑_{ 0 ≤ i < 2^{d-l-1} } [ c_l ⋅ ζ_{l+1}ⁱ ⋅ Sˡᵢ( X_l ) ]
- *        = ( (1−X_l) + X_l⋅ζ_l ) ⋅ ∑_{ 0 ≤ i < 2^{d-l-1} } [ c_l ⋅ ζ_{l+1}ⁱ ⋅ Sˡᵢ( X_l ) ]
- *
- * If we define Tˡ( X_l ) := ∑_{0≤i<2ˡ} [ c_l ⋅ ζ_{l+1}ⁱ ⋅ Sˡᵢ( X_l ) ], then Tˡ has the same degree as the original Sˡ(
- * X_l ) for the relation P(X) and is only slightly more expensive to compute than Sˡ( X_l ). Moreover, given Tˡ( X_l ),
- * the verifier can evaluate S'ˡ( u_l ) by evaluating ( (1−u_l) + u_l⋅ζ_l )Tˡ( u_l ). When the verifier checks the
- * claimed sum, the procedure is modified as follows
- *
  * Init:
  * - σ_0 <-- 0 // Claimed Sumcheck sum
  * - c_0  <-- 1 // Partial evaluation constant, before any evaluation
- * - ζ_0 <-- ζ // Initial power of ζ
  *
  * Round 0≤l<d-1:
  * - σ_{ l } =?= S'ˡ(0) + S'ˡ(1) = Tˡ(0) + ζ_{l}⋅Tˡ(1)  // Check partial sum
- * - σ_{l+1} <-- ( (1−u_{l}) + u_{l}⋅ζ_{l} )⋅Tʲ(u_{l})  // Compute next partial sum
- * - c_{l+1} <-- ( (1−u_{l}) + u_{l}⋅ζ_{l} )⋅c_{l}      // Partially evaluate pow in u_{l}
- * - ζ_{l+1} <-- ζ_{l}^2                                // Get next power of ζ
+ * - σ_{l+1} <-- ( (1−u_{l}) + u_{l}⋅β_{l} )⋅Tʲ(u_{l})  // Compute next partial sum
+ * - c_{l+1} <-- ( (1−u_{l}) + u_{l}⋅β_{l} )⋅c_{l}      // Partially evaluate pow in u_{l}
  *
  * Final round l=d-1:
- * - σ_{d-1} =?= S'ᵈ⁻¹(0) + S'ᵈ⁻¹(1) = Tᵈ⁻¹(0) + ζ_{d-1}⋅Tᵈ⁻¹(1)    // Check partial sum
+ * - σ_{d-1} =?= S'ᵈ⁻¹(0) + S'ᵈ⁻¹(1) = Tᵈ⁻¹(0) + β_{d-1}⋅Tᵈ⁻¹(1)    // Check partial sum
  * - σ_{ d } <-- ( (1−u_{d-1}) + u_{d-1}⋅ζ_{0} )⋅Tᵈ⁻¹(u_{d-1})      // Compute purported evaluation of P'(u)
- * - c_{ d } <-- ∏_{0≤l<d} ( (1-u_{l}) + u_{l}⋅ζ_{l} )
+ * - c_{ d } <-- ∏_{0≤l<d} ( (1-u_{l}) + u_{l}⋅β_{l} )
  *             = pow(u_{0}, ..., u_{d-1})                           // Full evaluation of pow
  * - σ_{ d } =?= c_{d}⋅P(u_{0}, ..., u_{d-1})                       // Compare against real evaluation of P'(u)
  */
-template <typename FF> struct PowUnivariate {
-    // ζ_{l}, initialized as ζ_{0} = ζ
-    // At round l, this equals ζ^{ 2^l }
-    FF zeta_pow;
-    // ζ_{l+1}, initialized as ζ_{1} = ζ^2
-    // At round l,this  equals ζ^{ 2^{l+1 } }
-    FF zeta_pow_sqr;
-    // The constant c_{l} obtained by partially evaluating one variable in the power polynomial at each round. At the
-    // end of round l in the sumcheck protocol, variable X_l is replaced by a verifier challenge u_l. The partial
-    // evaluation constant is updated to represent pow(u_0,.., u_l) = \prod_{0 ≤ k < l} ( (1-u_{k}) + u_{k}⋅ζ_{k}).
-    FF partial_evaluation_result = FF(1);
-
-    // Initialize with the random zeta
-    explicit PowUnivariate(FF zeta_pow)
-        : zeta_pow(zeta_pow)
-        , zeta_pow_sqr(zeta_pow.sqr())
-    {}
-
-    /**
-     * @brief Evaluate the monomial ((1−X_l) + X_l⋅ζ_l) in the challenge point X_l=u_l.
-     */
-    FF univariate_eval(FF challenge) const { return (FF(1) + (challenge * (zeta_pow - FF(1)))); };
-
-    /**
-     * @brief Parially evaluate the pow polynomial in the new challenge, by updating the constant c_{l} -> c_{l+1}.
-     * Also update (ζ_{l}, ζ_{l+1}) -> (ζ_{l+1}, ζ_{l+1}^2)
-     *
-     * @param challenge l-th verifier challenge u_{l}
-     */
-    void partially_evaluate(FF challenge)
-    {
-        FF current_univariate_eval = univariate_eval(challenge);
-        zeta_pow = zeta_pow_sqr;
-        // TODO(luke): for native FF, this could be self_sqr()
-        zeta_pow_sqr = zeta_pow_sqr.sqr();
-
-        partial_evaluation_result *= current_univariate_eval;
-    }
-};
 
 template <typename FF> struct PowPolynomial {
-    std::vector<FF> values;
-    std::vector<FF> pow_values;
-    FF current_element;
+    std::vector<FF> betas;
+    std::vector<FF> pow_betas;
     size_t current_element_idx = 0;
+    size_t periodicity = 2;
+
+    // The constant c_l obtained by partially evaluating one variable in the power polynomial at each round. At the
+    // end of round l in the sumcheck protocol, variable X_l is replaced by a verifier challenge u_l. The partial
+    // evaluation constant is updated to represent pow(u_0,.., u_l) = \prod_{0 ≤ k < l} ( (1-u_k) + u_k⋅β_k).
     FF partial_evaluation_result = FF(1);
 
-    explicit PowPolynomial(const std::vector<FF>& values)
-        : values(values)
-        , current_element(values[0])
+    explicit PowPolynomial(const std::vector<FF>& betas)
+        : betas(betas)
     {
         compute_pow_polynomial_at_values();
     }
 
+    FF const& operator[](size_t idx) const { return pow_betas[idx]; }
+
+    FF current_element() const { return betas[current_element_idx]; }
+
     /**
-     * @brief Evaluate the monomial ((1−X_l) + X_l⋅ζ_l) in the challenge point X_l=u_l.
+     * @brief Evaluate the monomial ((1−X_l) + X_l⋅β_l) in the challenge point X_l=u_l.
      */
-    FF univariate_eval(FF challenge) const { return (FF(1) + (challenge * (current_element - FF(1)))); };
+    FF univariate_eval(FF challenge) const { return (FF(1) + (challenge * (betas[current_element_idx] - FF(1)))); };
 
     /**
-     * @brief Parially evaluate the pow polynomial in the new challenge, by updating the constant c_{l} -> c_{l+1}.
-     * Also update (ζ_{l}, ζ_{l+1}) -> (ζ_{l+1}, ζ_{l+1}^2)
+     * @brief Parially evaluate the pow polynomial in X_l and updating the constant c_l -> c_{l+1}.
      *
-     * @param challenge l-th verifier challenge u_{l}
+     * @param challenge l-th verifier challenge u_l
      */
     void partially_evaluate(FF challenge)
     {
         FF current_univariate_eval = univariate_eval(challenge);
         partial_evaluation_result *= current_univariate_eval;
-
-        if (current_element_idx != values.size() - 1) {
-            current_element_idx++;
-            current_element = values[current_element_idx];
-        }
+        current_element_idx++;
+        periodicity *= 2;
     }
 
+    /**
+     * @brief Given {β_0,...,β_{d-1}} compute pow(β) of size 2^d
+     *
+     */
     void compute_pow_polynomial_at_values()
     {
-        size_t pow_size = 1 << values.size();
-        pow_values = std::vector<FF>(1 << values.size(), 0);
+        size_t pow_size = 1 << betas.size();
+        pow_betas = std::vector<FF>(1 << betas.size(), 0);
         for (size_t i = 0; i < pow_size; i++) {
             auto res = FF(1);
             for (size_t j = i, beta_idx = 0; j > 0; j >>= 1, beta_idx++) {
                 if ((j & 1) == 1) {
-                    res *= values[beta_idx];
+                    res *= betas[beta_idx];
                 }
             }
-            pow_values[i] = res;
+            pow_betas[i] = res;
         }
     }
 };
diff --git a/barretenberg/cpp/src/barretenberg/polynomials/pow.test.cpp b/barretenberg/cpp/src/barretenberg/polynomials/pow.test.cpp
index 1768efef28f..cbafd648112 100644
--- a/barretenberg/cpp/src/barretenberg/polynomials/pow.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/polynomials/pow.test.cpp
@@ -6,26 +6,36 @@ namespace barretenberg::test_pow {
 
 using FF = barretenberg::fr;
 
-TEST(SumcheckPow, FullPowConsistency)
+TEST(PowPolynomial, FullPowConsistency)
 {
     constexpr size_t d = 5;
+    std::vector<FF> betas(d);
+    for (auto& beta : betas) {
+        beta = FF::random_element();
+    }
 
-    FF zeta = FF::random_element();
-
-    PowUnivariate<FF> pow_univariate(zeta);
+    PowPolynomial<FF> pow_polynomial(betas);
     std::array<FF, d> variables{};
     for (auto& u_i : variables) {
         u_i = FF::random_element();
-        pow_univariate.partially_evaluate(u_i);
+        pow_polynomial.partially_evaluate(u_i);
     }
 
-    FF zeta_power = zeta;
+    size_t beta_idx = 0;
     FF expected_eval = 1;
     for (auto& u_i : variables) {
-        expected_eval *= FF(1) - u_i + u_i * zeta_power;
-        zeta_power *= zeta_power;
+        expected_eval *= FF(1) - u_i + u_i * pow_polynomial.betas[beta_idx];
+        beta_idx++;
     }
 
-    EXPECT_EQ(pow_univariate.partial_evaluation_result, expected_eval);
+    EXPECT_EQ(pow_polynomial.partial_evaluation_result, expected_eval);
+}
+
+TEST(PowPolynomial, PowPolynomialsOnPowers)
+{
+    auto betas = std::vector<FF>{ 2, 4, 16 };
+    auto pow = PowPolynomial(betas);
+    auto expected_values = std::vector<FF>{ 1, 2, 4, 8, 16, 32, 64, 128 };
+    EXPECT_EQ(expected_values, pow.pow_betas);
 }
 } // namespace barretenberg::test_pow
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp
index ee9f1af728e..a89c049a0ee 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp
@@ -1,5 +1,6 @@
 #include "barretenberg/flavor/ultra.hpp"
 #include "barretenberg/honk/utils/testing.hpp"
+#include "barretenberg/polynomials/pow.hpp"
 #include "barretenberg/protogalaxy/protogalaxy_prover.hpp"
 #include "barretenberg/relations/relation_parameters.hpp"
 #include "barretenberg/sumcheck/instance/instances.hpp"
@@ -11,6 +12,7 @@ using Flavor = proof_system::honk::flavor::Ultra;
 using Polynomial = typename Flavor::Polynomial;
 using FF = typename Flavor::FF;
 using RelationParameters = proof_system::RelationParameters<FF>;
+using PowPolynomial = barretenberg::PowPolynomial<FF>;
 
 // TODO(https://github.com/AztecProtocol/barretenberg/issues/780): Improve combiner tests to check more than the
 // arithmetic relation so we more than unit test folding relation parameters and alpha as well.
@@ -38,7 +40,6 @@ TEST(Protogalaxy, CombinerOn2Instances)
         if (is_random_input) {
             std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
             ProtoGalaxyProver prover;
-            std::vector<FF> pow_betas = { FF(1), FF(2) };
 
             for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
                 auto instance = std::make_shared<ProverInstance>();
@@ -53,7 +54,7 @@ TEST(Protogalaxy, CombinerOn2Instances)
             ProverInstances instances{ instance_data };
             instances.alphas.fill(Univariate<FF, 13>(FF(0))); // focus on the arithmetic relation only
 
-            auto result = prover.compute_combiner(instances, pow_betas);
+            auto result = prover.compute_combiner(instances, PowPolynomial(std::vector<FF>{ 2 }));
             auto expected_result =
                 barretenberg::Univariate<FF, 13>(std::array<FF, 13>{ 87706,
                                                                      13644570,
@@ -72,7 +73,6 @@ TEST(Protogalaxy, CombinerOn2Instances)
         } else {
             std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
             ProtoGalaxyProver prover;
-            std::vector<FF> pow_betas = { FF(1), FF(2) };
 
             for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
                 auto instance = std::make_shared<ProverInstance>();
@@ -131,7 +131,7 @@ TEST(Protogalaxy, CombinerOn2Instances)
             relation value:
                       0    0    0    0    0    0    0              0    0    6   18   36   60   90      */
 
-            auto result = prover.compute_combiner(instances, pow_betas);
+            auto result = prover.compute_combiner(instances, PowPolynomial(std::vector<FF>{ 2 }));
             auto expected_result = barretenberg::Univariate<FF, 13>(
                 std::array<FF, 13>{ 0, 0, 12, 36, 72, 120, 180, 252, 336, 432, 540, 660, 792 });
 
@@ -163,7 +163,6 @@ TEST(Protogalaxy, CombinerOn4Instances)
     auto run_test = [&]() {
         std::vector<std::shared_ptr<ProverInstance>> instance_data(NUM_INSTANCES);
         ProtoGalaxyProver prover;
-        std::vector<FF> pow_betas = { FF(1), FF(2) };
 
         for (size_t idx = 0; idx < NUM_INSTANCES; idx++) {
             auto instance = std::make_shared<ProverInstance>();
@@ -182,7 +181,7 @@ TEST(Protogalaxy, CombinerOn4Instances)
         zero_all_selectors(instances[2]->prover_polynomials);
         zero_all_selectors(instances[3]->prover_polynomials);
 
-        auto result = prover.compute_combiner(instances, pow_betas);
+        auto result = prover.compute_combiner(instances, PowPolynomial(std::vector<FF>{ 2 }));
         std::array<FF, 43> zeroes;
         std::fill(zeroes.begin(), zeroes.end(), 0);
         auto expected_result = barretenberg::Univariate<FF, 43>(zeroes);
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
index 45a4ac93ad0..b1344796302 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
@@ -6,7 +6,8 @@ namespace proof_system::honk {
 /**
  * Create DeciderProver_ from an accumulator.
  *
- * @param accumulator Instance whose proof we want to generate.
+ * @param accumulator Relaxed instance (ϕ, ω' \vec{β*}, e*) whose proof we want to generate, produced by Protogalaxy
+ * folding prover
  *
  * @tparam a type of UltraFlavor
  * */
@@ -20,8 +21,7 @@ DeciderProver_<Flavor>::DeciderProver_(const std::shared_ptr<Instance>& inst,
 {}
 
 /**
- * @brief Add circuit size, public input size, and public inputs to transcript
- *
+ * @brief Add  ϕ, \vec{β*}, e* to the transcript.
  */
 template <UltraFlavor Flavor> void DeciderProver_<Flavor>::execute_preamble_round()
 {
@@ -51,8 +51,6 @@ template <UltraFlavor Flavor> void DeciderProver_<Flavor>::execute_preamble_roun
         transcript->send_to_verifier("gate_challenge_" + std::to_string(idx), accumulator->gate_challenges[idx]);
     }
 
-    // send as verifier commitments
-    // should protogalaxy work on verifier commitments?
     auto comm_view = accumulator->witness_commitments.get_all();
     auto witness_labels = accumulator->commitment_labels.get_witness();
     for (size_t idx = 0; idx < witness_labels.size(); idx++) {
@@ -67,7 +65,8 @@ template <UltraFlavor Flavor> void DeciderProver_<Flavor>::execute_preamble_roun
 }
 
 /**
- * @brief Run Sumcheck resulting in u = (u_1,...,u_d) challenges and all evaluations at u being calculated.
+ * @brief Run Sumcheck to establish that ∑_i pow(\vec{β*})f_i(ω) = e*. This results in u = (u_1,...,u_d) sumcheck round
+ * challenges and all evaluations at u being calculated.
  *
  */
 template <UltraFlavor Flavor> void DeciderProver_<Flavor>::execute_relation_check_rounds()
@@ -102,10 +101,9 @@ template <UltraFlavor Flavor> plonk::proof& DeciderProver_<Flavor>::export_proof
 
 template <UltraFlavor Flavor> plonk::proof& DeciderProver_<Flavor>::construct_proof()
 {
-    // Add circuit size public input size and public inputs to transcript->
+    // Add ϕ, \vec{β*}, e* to transcript
     execute_preamble_round();
 
-    // Fiat-Shamir: alpha
     // Run sumcheck subprotocol.
     execute_relation_check_rounds();
 
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
index 3befe10bb57..46e6fafa187 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
@@ -22,7 +22,8 @@ DeciderVerifier_<Flavor>::DeciderVerifier_()
 {}
 
 /**
- * @brief This function verifies an Ultra Honk proof for a given Flavor.
+ * @brief This function verifies an Ultra Honk proof for a given Flavor, produced for a relaxed instance (ϕ, \vec{β*},
+ * e*).
  *
  */
 template <typename Flavor> bool DeciderVerifier_<Flavor>::verify_proof(const plonk::proof& proof)
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index 88e8b9de9ab..99985d0558c 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -223,8 +223,8 @@ std::shared_ptr<typename ProverInstances::Instance> ProtoGalaxyProver_<ProverIns
         el_idx++;
     }
 
-    // Evaluate the combined batching challenge α univariate at challenge to obtain next α and send it to the
-    // verifier
+    // Evaluate the combined batching  α_i univariate at challenge to obtain next α_i and send it to the
+    // verifier, where i ∈ {0,...,NUM_SUBRELATIONS - 1}
     auto& folded_alphas = next_accumulator->alphas;
     for (size_t idx = 0; idx < NUM_SUBRELATIONS - 1; idx++) {
         folded_alphas[idx] = instances.alphas[idx].evaluate(challenge);
@@ -235,7 +235,7 @@ std::shared_ptr<typename ProverInstances::Instance> ProtoGalaxyProver_<ProverIns
     // the verifier
     auto& combined_relation_parameters = instances.relation_parameters;
     auto folded_relation_parameters = proof_system::RelationParameters<FF>{
-        combined_relation_parameters.eta.evaluate(challenge), // WORKTODO change all of these to evaluate(challenge)
+        combined_relation_parameters.eta.evaluate(challenge),
         combined_relation_parameters.beta.evaluate(challenge),
         combined_relation_parameters.gamma.evaluate(challenge),
         combined_relation_parameters.public_input_delta.evaluate(challenge),
@@ -291,11 +291,9 @@ FoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverInstanc
     auto perturbator_challenge = transcript->get_challenge("perturbator_challenge");
     instances.next_gate_challenges =
         update_gate_challenges(perturbator_challenge, accumulator->gate_challenges, deltas);
-    const auto pow_betas_star =
-        compute_pow_polynomial_at_values(instances.next_gate_challenges, accumulator->instance_size);
     combine_relation_parameters(instances);
     combine_alpha(instances);
-    auto combiner = compute_combiner(instances, pow_betas_star);
+    auto combiner = compute_combiner(instances, PowPolynomial<FF>(instances.next_gate_challenges));
 
     auto compressed_perturbator = perturbator.evaluate(perturbator_challenge);
     auto combiner_quotient = compute_combiner_quotient(compressed_perturbator, combiner);
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index 08815320bf2..e1470dda856 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -92,23 +92,6 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      * TODO(https://github.com/AztecProtocol/barretenberg/issues/753): fold goblin polynomials
      */
     FoldingResult<Flavor> fold_instances();
-    /**
-     * @brief Given a vector \vec{\beta} of values, compute the pow polynomial on these values as defined in the paper.
-     */
-    static std::vector<FF> compute_pow_polynomial_at_values(const std::vector<FF>& betas, const size_t instance_size)
-    {
-        std::vector<FF> pow_betas(instance_size);
-        for (size_t i = 0; i < instance_size; i++) {
-            auto res = FF(1);
-            for (size_t j = i, beta_idx = 0; j > 0; j >>= 1, beta_idx++) {
-                if ((j & 1) == 1) {
-                    res *= betas[beta_idx];
-                }
-            }
-            pow_betas[i] = res;
-        }
-        return pow_betas;
-    }
 
     /**
      * @brief For a new round challenge δ at each iteration of the ProtoGalaxy protocol, compute the vector
@@ -309,7 +292,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      *
      */
     ExtendedUnivariateWithRandomization compute_combiner(const ProverInstances& instances,
-                                                         const std::vector<FF>& pow_betas_star)
+                                                         const PowPolynomial<FF>& pow_betas)
     {
         size_t common_instance_size = instances[0]->instance_size;
 
@@ -343,7 +326,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
                 // No need to initialise extended_univariates to 0, it's assigned to
                 extend_univariates(extended_univariates[thread_idx], instances, idx);
 
-                FF pow_challenge = pow_betas_star[idx];
+                FF pow_challenge = pow_betas[idx];
 
                 // Accumulate the i-th row's univariate contribution. Note that the relation parameters passed to this
                 // function have already been folded
diff --git a/barretenberg/cpp/src/barretenberg/relations/relation_parameters.hpp b/barretenberg/cpp/src/barretenberg/relations/relation_parameters.hpp
index 208585d94f6..2057f37c115 100644
--- a/barretenberg/cpp/src/barretenberg/relations/relation_parameters.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/relation_parameters.hpp
@@ -45,7 +45,7 @@ template <typename T> struct RelationParameters {
     {
         RelationParameters result;
         result.eta = T::random_element();
-        result.beta = T(0);
+        result.beta = T::random_element();
         result.beta_sqr = result.beta * result.beta;
         result.beta_cube = result.beta_sqr * result.beta;
         result.gamma = T::random_element();
diff --git a/barretenberg/cpp/src/barretenberg/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
index e609d4c31fc..3938ecdea75 100644
--- a/barretenberg/cpp/src/barretenberg/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
@@ -63,22 +63,23 @@ template <typename Flavor> class RelationUtils {
     }
 
     /**
-     * @brief Scale Univaraites by consecutive powers of the provided challenge
+     * @brief Scale Univaraites, each representing a subrelation, by different challenges
      *
      * @param tuple Tuple of tuples of Univariates
-     * @param challenge
+     * @param challenge Array of NUM_SUBRELATIONS - 1 challenges (because the first subrelation doesn't need to be
+     * scaled)
      * @param current_scalar power of the challenge
      */
-    static void scale_univariates(auto& tuple, const AlphaType& challenge, FF& current_scalar)
+    static void scale_univariates(auto& tuple, const AlphaType& challenges, FF& current_scalar)
         requires proof_system::IsFoldingFlavor<Flavor>
     {
         size_t idx = 0;
-        auto scale_by_consecutive_powers_of_challenge = [&]<size_t, size_t>(auto& element) {
+        auto scale_by_challenges = [&]<size_t, size_t>(auto& element) {
             element *= current_scalar;
-            current_scalar = challenge[idx];
+            current_scalar = challenges[idx];
             idx++;
         };
-        apply_to_tuple_of_tuples(tuple, scale_by_consecutive_powers_of_challenge);
+        apply_to_tuple_of_tuples(tuple, scale_by_challenges);
     }
 
     /**
@@ -110,8 +111,7 @@ template <typename Flavor> class RelationUtils {
     template <typename... T>
     static constexpr void add_tuples(std::tuple<T...>& tuple_1, const std::tuple<T...>& tuple_2)
     {
-        auto add_tuples_helper = [&]<std::size_t... I>(std::index_sequence<I...>)
-        {
+        auto add_tuples_helper = [&]<std::size_t... I>(std::index_sequence<I...>) {
             ((std::get<I>(tuple_1) += std::get<I>(tuple_2)), ...);
         };
 
@@ -181,21 +181,23 @@ template <typename Flavor> class RelationUtils {
     };
 
     /**
-     * @brief Scale elements by consecutive powers of the challenge then sum
+     * @brief Scale elements, which in sumcheck represent evaluations of subrelations, by different challenges then sum
+     * @param challenges Array of NUM_SUBRELATIONS - 1 challenges (because the first subrelation does not need to be
+     * scaled)
      * @param result Batched result
      */
-    static void scale_and_batch_elements(auto& tuple, const AlphaType& challenge, FF current_scalar, FF& result)
+    static void scale_and_batch_elements(auto& tuple, const AlphaType& challenges, FF current_scalar, FF& result)
         requires proof_system::IsFoldingFlavor<Flavor>
     {
         size_t idx = 0;
-        auto scale_by_challenge_and_accumulate = [&](auto& element) {
+        auto scale_by_challenges_and_accumulate = [&](auto& element) {
             for (auto& entry : element) {
                 result += entry * current_scalar;
-                current_scalar = challenge[idx];
+                current_scalar = challenges[idx];
                 idx++;
             }
         };
-        apply_to_tuple_of_arrays(scale_by_challenge_and_accumulate, tuple);
+        apply_to_tuple_of_arrays(scale_by_challenges_and_accumulate, tuple);
     }
 
     /**
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
index 3f6959b03a7..2cf017e4f33 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
@@ -62,7 +62,7 @@ template <typename Flavor> class SumcheckProver {
         : multivariate_n(multivariate_n)
         , multivariate_d(numeric::get_msb(multivariate_n))
         , transcript(transcript)
-        , round(multivariate_n, 2)
+        , round(multivariate_n)
         , partially_evaluated_polynomials(multivariate_n){};
 
     // WORKTODO delete this
@@ -101,10 +101,8 @@ template <typename Flavor> class SumcheckProver {
         multivariate_challenge.emplace_back(round_challenge);
         partially_evaluate(full_polynomials, multivariate_n, round_challenge);
         pow_univariate.partially_evaluate(round_challenge);
-        round.round_size =
-            round.round_size >> 1; // TODO(#224)(Cody): Maybe partially_evaluate should do this and release memory?
-        round.hop_size = round.hop_size << 1;
-        // All but final round
+        round.round_size = round.round_size >> 1; // TODO(#224)(Cody): Maybe partially_evaluate should do this and
+                                                  // release memory?        // All but final round
         // We operate on partially_evaluated_polynomials in place.
         for (size_t round_idx = 1; round_idx < multivariate_d; round_idx++) {
             // Write the round univariate to the transcript
@@ -116,7 +114,6 @@ template <typename Flavor> class SumcheckProver {
             partially_evaluate(partially_evaluated_polynomials, round.round_size, round_challenge);
             pow_univariate.partially_evaluate(round_challenge);
             round.round_size = round.round_size >> 1;
-            round.hop_size = round.hop_size << 1;
         }
 
         // Final round: Extract multivariate evaluations from partially_evaluated_polynomials and add to transcript
@@ -190,7 +187,7 @@ template <typename Flavor> class SumcheckVerifier {
     std::shared_ptr<Transcript> transcript;
     SumcheckVerifierRound<Flavor> round;
 
-    // verifier instantiates sumcheck with circuit size
+    // Verifier instantiates sumcheck with circuit size, optionally a different target sum than 0 can be specified.
     explicit SumcheckVerifier(size_t multivariate_d, std::shared_ptr<Transcript> transcript, FF target_sum = 0)
         : multivariate_d(multivariate_d)
         , transcript(transcript)
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
index 8dcd110a1e7..44ed5a4b975 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
@@ -62,7 +62,6 @@ template <typename Flavor> class SumcheckProverRound {
     using ExtendedEdges = typename Flavor::ExtendedEdges;
 
     size_t round_size; // a power of 2
-    size_t hop_size;
 
     static constexpr size_t NUM_RELATIONS = Flavor::NUM_RELATIONS;
     static constexpr size_t MAX_PARTIAL_RELATION_LENGTH = Flavor::MAX_PARTIAL_RELATION_LENGTH;
@@ -71,9 +70,8 @@ template <typename Flavor> class SumcheckProverRound {
     SumcheckTupleOfTuplesOfUnivariates univariate_accumulators;
 
     // Prover constructor
-    SumcheckProverRound(size_t initial_round_size, size_t initial_hop_size)
+    SumcheckProverRound(size_t initial_round_size)
         : round_size(initial_round_size)
-        , hop_size(initial_hop_size)
     {
         // Initialize univariate accumulators to 0
         Utils::zero_univariates(univariate_accumulators);
@@ -109,10 +107,12 @@ template <typename Flavor> class SumcheckProverRound {
         const barretenberg::PowPolynomial<FF>& pow_polynomial,
         const AlphaType alpha)
     {
+        // Compute the constant contribution of pow polynomials for each edge as the product of c_l and the elements of
+        // pow(\vec{β}) not containing β_0,..., β_l
         std::vector<FF> pow_challenges(round_size >> 1);
         pow_challenges[0] = pow_polynomial.partial_evaluation_result;
         for (size_t i = 1; i < (round_size >> 1); ++i) {
-            pow_challenges[i] = pow_challenges[0] * pow_polynomial.pow_values[i * hop_size];
+            pow_challenges[i] = pow_challenges[0] * pow_polynomial[i * pow_polynomial.periodicity];
         }
 
         // Determine number of threads for multithreading.
@@ -139,21 +139,15 @@ template <typename Flavor> class SumcheckProverRound {
             size_t start = thread_idx * iterations_per_thread;
             size_t end = (thread_idx + 1) * iterations_per_thread;
 
-            // For each edge_idx = 2i, we need to multiply the whole contribution by zeta^{2^{2i}}
-            // This means that each univariate for each relation needs an extra multiplication.
             for (size_t edge_idx = start; edge_idx < end; edge_idx += 2) {
                 extend_edges(extended_edges[thread_idx], polynomials, edge_idx);
 
-                // Update the pow polynomial's contribution c_l ⋅ ζ_{l+1}ⁱ for the next edge.
-                FF pow_challenge = pow_challenges[edge_idx >> 1];
-
                 // Compute the i-th edge's univariate contribution,
-                // scale it by the pow polynomial's constant and zeta power "c_l ⋅ ζ_{l+1}ⁱ"
-                // and add it to the accumulators for Sˡ(Xₗ)
+                // scale it by pow_challenge constant contribution and add it to the accumulators for Sˡ(Xₗ)
                 accumulate_relation_univariates(thread_univariate_accumulators[thread_idx],
                                                 extended_edges[thread_idx],
                                                 relation_parameters,
-                                                pow_challenge);
+                                                pow_challenges[edge_idx >> 1]);
             }
         });
 
@@ -202,7 +196,7 @@ template <typename Flavor> class SumcheckProverRound {
     {
         ExtendedUnivariate extended_random_polynomial;
         // Random poly R(X) = (1-X) + X.zeta_pow
-        auto random_polynomial = barretenberg::Univariate<FF, 2>({ 1, pow_polynomial.current_element });
+        auto random_polynomial = barretenberg::Univariate<FF, 2>({ 1, pow_polynomial.current_element() });
         extended_random_polynomial = random_polynomial.template extend_to<ExtendedUnivariate::LENGTH>();
 
         auto extend_and_sum = [&]<size_t relation_idx, size_t subrelation_idx, typename Element>(Element& element) {
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.test.cpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.test.cpp
index 49d228a9c52..e6a6ca28121 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.test.cpp
@@ -8,7 +8,6 @@ using namespace proof_system::honk;
 using namespace proof_system::honk::sumcheck;
 
 using barretenberg::BarycentricData;
-using barretenberg::PowUnivariate;
 using barretenberg::Univariate;
 
 using Flavor = flavor::Ultra;
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
index 4d486749e24..f0434cd0e2c 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
@@ -1,8 +1,8 @@
+#include "barretenberg/polynomials/pow.hpp"
 #include "barretenberg/protogalaxy/protogalaxy_prover.hpp"
 #include "barretenberg/ultra_honk/ultra_composer.hpp"
 #include <gtest/gtest.h>
 
-using namespace barretenberg;
 using namespace proof_system::honk;
 
 using Flavor = flavor::Ultra;
@@ -19,9 +19,11 @@ using ProverPolynomials = Flavor::ProverPolynomials;
 using RelationParameters = proof_system::RelationParameters<FF>;
 using WitnessCommitments = typename Flavor::WitnessCommitments;
 using CommitmentKey = Flavor::CommitmentKey;
+using PowPolynomial = barretenberg::PowPolynomial<FF>;
+
 const size_t NUM_POLYNOMIALS = Flavor::NUM_ALL_ENTITIES;
 
-namespace protogalaxy_tests {
+namespace barretenberg::protogalaxy_tests {
 namespace {
 auto& engine = numeric::random::get_debug_engine();
 }
@@ -65,13 +67,12 @@ void check_accumulator_target_sum_manual(std::shared_ptr<Instance>& accumulator,
     auto expected_honk_evals = ProtoGalaxyProver::compute_full_honk_evaluations(
         accumulator->prover_polynomials, accumulator->alphas, accumulator->relation_parameters);
     // Construct pow(\vec{betas*}) as in the paper
-    auto expected_pows_beta =
-        ProtoGalaxyProver::compute_pow_polynomial_at_values(accumulator->gate_challenges, instance_size);
+    auto expected_pows = PowPolynomial(accumulator->gate_challenges);
 
     // Compute the corresponding target sum and create a dummy accumulator
     auto expected_target_sum = FF(0);
     for (size_t i = 0; i < instance_size; i++) {
-        expected_target_sum += expected_honk_evals[i] * expected_pows_beta[i];
+        expected_target_sum += expected_honk_evals[i] * expected_pows[i];
     }
 
     EXPECT_EQ(accumulator->target_sum == expected_target_sum, expected_result);
@@ -161,7 +162,7 @@ TEST_F(ProtoGalaxyTests, PerturbatorPolynomial)
     }
 
     // Construct pow(\vec{betas}) as in the paper
-    auto pow_beta = ProtoGalaxyProver::compute_pow_polynomial_at_values(betas, instance_size);
+    auto pow_beta = PowPolynomial(betas);
 
     // Compute the corresponding target sum and create a dummy accumulator
     auto target_sum = FF(0);
@@ -183,14 +184,6 @@ TEST_F(ProtoGalaxyTests, PerturbatorPolynomial)
     EXPECT_EQ(perturbator[0], target_sum);
 }
 
-TEST_F(ProtoGalaxyTests, PowPolynomialsOnPowers)
-{
-    auto betas = std::vector<FF>{ 2, 4, 16 };
-    auto pow_betas = ProtoGalaxyProver::compute_pow_polynomial_at_values(betas, 8);
-    auto expected_values = std::vector<FF>{ 1, 2, 4, 8, 16, 32, 64, 128 };
-    EXPECT_EQ(expected_values, pow_betas);
-}
-
 TEST_F(ProtoGalaxyTests, CombinerQuotient)
 {
     auto compressed_perturbator = FF(2); // F(\alpha) in the paper
@@ -354,4 +347,4 @@ TEST_F(ProtoGalaxyTests, TamperedAccumulatorPolynomial)
     decide_and_verify(second_accumulator, composer, false);
 }
 
-} // namespace protogalaxy_tests
\ No newline at end of file
+} // namespace barretenberg::protogalaxy_tests
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp
index f46b06dfb2f..62689f5ae19 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp
@@ -146,7 +146,6 @@ template <typename Flavor> bool UltraVerifier_<Flavor>::verify_proof(const plonk
 
     // If Sumcheck did not verify, return false
     if (sumcheck_verified.has_value() && !sumcheck_verified.value()) {
-        info("UltraVerifier: Sumcheck failed.");
         return false;
     }
 

From 413a80151fabd4d9cdc940f6f825ed827a9660ab Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Thu, 21 Dec 2023 19:58:08 +0000
Subject: [PATCH 64/86] remove resolved issues

---
 .../protogalaxy/protogalaxy_prover.hpp        | 27 +------------------
 .../cpp/src/barretenberg/relations/utils.hpp  |  3 ++-
 2 files changed, 3 insertions(+), 27 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index e1470dda856..6b368ad6c40 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -157,27 +157,6 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         return full_honk_evaluations;
     }
 
-    // /**
-    //  * @brief Scale elements by consecutive powers of the challenge then sum
-    //  * @param result Batched result
-    //  */
-    // static void scale_and_batch_elements(auto& tuple, const FF& challenge, FF& result)
-    // {
-    //     auto first_array = std::get<0>(tuple);
-    //     result = first_array[0];
-    //     for (auto& entry : first_array) {
-    //         result += entry * challenge;
-    //     }
-
-    //     auto scale_by_challenge_and_accumulate = [&](auto& element) {
-    //         for (auto& entry : element) {
-    //             result += entry * challenge;
-    //         }
-    //     };
-
-    //     Utils::template apply_to_tuple_of_arrays<1>(scale_by_challenge_and_accumulate, tuple);
-    // }
-
     /**
      * @brief  Recursively compute the parent nodes of each level in there, starting from the leaves. Note that at each
      * level, the resulting parent nodes will be polynomials of degree (level + 1) because we multiply by an additional
@@ -418,14 +397,10 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     }
 
     /**
-     * @brief Combine the relation batching parameter (named alpha) from each instance into a univariate, used in the
+     * @brief Combine the relation batching parameters (alphas) from each instance into a univariate, used in the
      * computation of combiner.
      *
      */
-    // TODO(https://github.com/AztecProtocol/barretenberg/issues/772): At the moment we have a single α per Instance, we
-    // fold them and then we use the unique folded_α for each folded subrelation that is batched in the combiner. This
-    // is obviously insecure. We need to generate α_i for each subrelation_i, fold them and then use folded_α_i when
-    // batching the i-th folded subrelation in the combiner.
     static void combine_alpha(ProverInstances& instances)
     {
         size_t alpha_idx = 0;
diff --git a/barretenberg/cpp/src/barretenberg/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
index 3938ecdea75..4b7f7ecba58 100644
--- a/barretenberg/cpp/src/barretenberg/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
@@ -111,7 +111,8 @@ template <typename Flavor> class RelationUtils {
     template <typename... T>
     static constexpr void add_tuples(std::tuple<T...>& tuple_1, const std::tuple<T...>& tuple_2)
     {
-        auto add_tuples_helper = [&]<std::size_t... I>(std::index_sequence<I...>) {
+        auto add_tuples_helper = [&]<std::size_t... I>(std::index_sequence<I...>)
+        {
             ((std::get<I>(tuple_1) += std::get<I>(tuple_2)), ...);
         };
 

From c765771c181a6052c3d2dbdf7e74a495a1235504 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Thu, 21 Dec 2023 21:13:19 +0000
Subject: [PATCH 65/86] replace honk-tests CI with ultra-honk-tests CI

---
 .circleci/config.yml | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/.circleci/config.yml b/.circleci/config.yml
index b28eedadd30..166e9a55a4c 100644
--- a/.circleci/config.yml
+++ b/.circleci/config.yml
@@ -225,7 +225,7 @@ jobs:
           name: "Test"
           command: cond_spot_run_test barretenberg-x86_64-linux-clang-assert 32 ./scripts/bb-tests.sh
 
-  barretenberg-honk-tests:
+  barretenberg-ulta-honk-tests:
     docker:
       - image: aztecprotocol/alpine-build-image
     resource_class: small
@@ -234,7 +234,7 @@ jobs:
       - *setup_env
       - run:
           name: "Test"
-          command: cond_spot_run_test barretenberg-x86_64-linux-clang-assert 32 ./scripts/run_tests 1 honk_tests
+          command: cond_spot_run_test barretenberg-x86_64-linux-clang-assert 32 ./scripts/run_tests 1 ultra_honk_tests
 
   barretenberg-proof-system-tests:
     docker:
@@ -1058,7 +1058,7 @@ workflows:
       - barretenberg-wasm-linux-clang: *defaults
       - barretenberg-x86_64-linux-clang-sol: *defaults
       - barretenberg-proof-system-tests: *bb_test
-      - barretenberg-honk-tests: *bb_test
+      - barretenberg-ultra-honk-tests: *bb_test
       - barretenberg-dsl-tests: *bb_test
       - barretenberg-tests: *bb_test
       - barretenberg-stdlib-tests: *bb_test

From 62baf45b0d78e054d9f2c52ee767366046a8e5c0 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Thu, 21 Dec 2023 21:23:43 +0000
Subject: [PATCH 66/86] --no-edit

---
 .circleci/config.yml | 12 ------------
 1 file changed, 12 deletions(-)

diff --git a/.circleci/config.yml b/.circleci/config.yml
index 166e9a55a4c..c03b2e44c8c 100644
--- a/.circleci/config.yml
+++ b/.circleci/config.yml
@@ -225,17 +225,6 @@ jobs:
           name: "Test"
           command: cond_spot_run_test barretenberg-x86_64-linux-clang-assert 32 ./scripts/bb-tests.sh
 
-  barretenberg-ulta-honk-tests:
-    docker:
-      - image: aztecprotocol/alpine-build-image
-    resource_class: small
-    steps:
-      - *checkout
-      - *setup_env
-      - run:
-          name: "Test"
-          command: cond_spot_run_test barretenberg-x86_64-linux-clang-assert 32 ./scripts/run_tests 1 ultra_honk_tests
-
   barretenberg-proof-system-tests:
     docker:
       - image: aztecprotocol/alpine-build-image
@@ -1058,7 +1047,6 @@ workflows:
       - barretenberg-wasm-linux-clang: *defaults
       - barretenberg-x86_64-linux-clang-sol: *defaults
       - barretenberg-proof-system-tests: *bb_test
-      - barretenberg-ultra-honk-tests: *bb_test
       - barretenberg-dsl-tests: *bb_test
       - barretenberg-tests: *bb_test
       - barretenberg-stdlib-tests: *bb_test

From a8812f64a95a74fb55a5f58e3275769920b27b58 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Thu, 21 Dec 2023 22:19:37 +0000
Subject: [PATCH 67/86] cleanup

---
 .../cpp/src/barretenberg/protogalaxy/decider_verifier.cpp | 8 --------
 .../src/barretenberg/protogalaxy/protogalaxy_prover.cpp   | 1 -
 2 files changed, 9 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
index 46e6fafa187..7dd44d52b5f 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
@@ -82,14 +82,6 @@ template <typename Flavor> bool DeciderVerifier_<Flavor>::verify_proof(const plo
     }
 
     VerifierCommitments commitments{ inst->verification_key, inst->witness_commitments };
-    auto& witness_commitments = inst->witness_commitments;
-    commitments.w_l = witness_commitments.w_l;
-    commitments.w_r = witness_commitments.w_r;
-    commitments.w_o = witness_commitments.w_o;
-    commitments.sorted_accum = witness_commitments.sorted_accum;
-    commitments.w_4 = witness_commitments.w_4;
-    commitments.z_perm = witness_commitments.z_perm;
-    commitments.z_lookup = witness_commitments.z_lookup;
 
     auto sumcheck = SumcheckVerifier<Flavor>(inst->log_instance_size, transcript, inst->target_sum);
 
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index 99985d0558c..6c632f78841 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -287,7 +287,6 @@ FoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverInstanc
     for (size_t idx = 0; idx <= accumulator->log_instance_size; idx++) {
         transcript->send_to_verifier("perturbator_" + std::to_string(idx), perturbator[idx]);
     }
-    assert(perturbator[0] == accumulator->target_sum);
     auto perturbator_challenge = transcript->get_challenge("perturbator_challenge");
     instances.next_gate_challenges =
         update_gate_challenges(perturbator_challenge, accumulator->gate_challenges, deltas);

From 5b2f9555c256ede4fa011aa7dcc9f62345774f32 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Thu, 21 Dec 2023 22:12:26 +0000
Subject: [PATCH 68/86] stuff that doesn't work

---
 .../src/barretenberg/flavor/goblin_ultra.hpp  |   2 +-
 .../flavor/goblin_ultra_recursive.hpp         |   5 +
 .../barretenberg/flavor/ultra_recursive.hpp   |  46 ++++++++
 .../verifier/decider_recursive_verifier.cpp   | 103 ++++++++++++++++++
 .../verifier/decider_recursive_verifier.hpp   |  39 +++++++
 5 files changed, 194 insertions(+), 1 deletion(-)
 create mode 100644 barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.cpp
 create mode 100644 barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.hpp

diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
index 4934a1a6564..a0e5f3bab8f 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
@@ -156,7 +156,7 @@ class GoblinUltra {
 
     // GoblinUltra needs to expose more public classes than most flavors due to GoblinUltraRecursive reuse, but these
     // are internal:
-  private:
+  public:
     // WireEntities for basic witness entities
     template <typename DataType> class WireEntities {
       public:
diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
index 7bbbc8f2127..5d9daa5fe6c 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
@@ -150,6 +150,11 @@ template <typename BuilderType> class GoblinUltraRecursive_ {
         };
     };
 
+    /**
+     * @brief A container for the witness commitments.
+     */
+    using WitnessCommitments = GoblinUltra::WitnessEntities<Commitment>;
+
     using CommitmentLabels = GoblinUltra::CommitmentLabels;
     // Reuse the VerifierCommitments from GoblinUltra
     using VerifierCommitments = GoblinUltra::VerifierCommitments_<Commitment, VerificationKey>;
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
index 042c4e4cec4..1f1b9e3aa53 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
@@ -161,6 +161,12 @@ template <typename BuilderType> class UltraRecursive_ {
         RefVector<DataType> get_wires() { return { w_l, w_r, w_o, w_4 }; };
     };
 
+  public:
+    /**
+     * @brief A container for the witness commitments.
+     */
+    using WitnessCommitments = WitnessEntities<Commitment>;
+
     /**
      * @brief A base class labelling all entities (for instance, all of the polynomials used by the prover during
      * sumcheck) in this Honk variant along with particular subsets of interest
@@ -376,6 +382,46 @@ template <typename BuilderType> class UltraRecursive_ {
             this->lagrange_first = verification_key->lagrange_first;
             this->lagrange_last = verification_key->lagrange_last;
         }
+
+        VerifierCommitments(const std::shared_ptr<VerificationKey>& verification_key,
+                            const WitnessCommitments& witness_commitments)
+        {
+            this->q_m = verification_key->q_m;
+            this->q_l = verification_key->q_l;
+            this->q_r = verification_key->q_r;
+            this->q_o = verification_key->q_o;
+            this->q_4 = verification_key->q_4;
+            this->q_c = verification_key->q_c;
+            this->q_arith = verification_key->q_arith;
+            this->q_sort = verification_key->q_sort;
+            this->q_elliptic = verification_key->q_elliptic;
+            this->q_aux = verification_key->q_aux;
+            this->q_lookup = verification_key->q_lookup;
+            this->q_poseidon2_external = verification_key->q_poseidon2_external;
+            this->q_poseidon2_internal = verification_key->q_poseidon2_internal;
+            this->sigma_1 = verification_key->sigma_1;
+            this->sigma_2 = verification_key->sigma_2;
+            this->sigma_3 = verification_key->sigma_3;
+            this->sigma_4 = verification_key->sigma_4;
+            this->id_1 = verification_key->id_1;
+            this->id_2 = verification_key->id_2;
+            this->id_3 = verification_key->id_3;
+            this->id_4 = verification_key->id_4;
+            this->table_1 = verification_key->table_1;
+            this->table_2 = verification_key->table_2;
+            this->table_3 = verification_key->table_3;
+            this->table_4 = verification_key->table_4;
+            this->lagrange_first = verification_key->lagrange_first;
+            this->lagrange_last = verification_key->lagrange_last;
+
+            this->w_l = witness_commitments.w_l;
+            this->w_r = witness_commitments.w_r;
+            this->w_o = witness_commitments.w_o;
+            this->sorted_accum = witness_commitments.sorted_accum;
+            this->w_4 = witness_commitments.w_4;
+            this->z_perm = witness_commitments.z_perm;
+            this->z_lookup = witness_commitments.z_lookup;
+        }
     };
 
     using Transcript = proof_system::plonk::stdlib::recursion::honk::Transcript<CircuitBuilder>;
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.cpp
new file mode 100644
index 00000000000..8776989ee6d
--- /dev/null
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.cpp
@@ -0,0 +1,103 @@
+#include "barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.hpp"
+#include "barretenberg/commitment_schemes/zeromorph/zeromorph.hpp"
+#include "barretenberg/numeric/bitop/get_msb.hpp"
+#include "barretenberg/sumcheck/instance/verifier_instance.hpp"
+#include "barretenberg/transcript/transcript.hpp"
+
+namespace proof_system::plonk::stdlib::recursion::honk {
+
+template <typename Flavor>
+DeciderRecursiveVerifier_<Flavor>::DeciderRecursiveVerifier_(
+    Builder* builder, const std::shared_ptr<NativeVerificationKey>& native_verifier_key)
+    : key(std::make_shared<VerificationKey>(builder, native_verifier_key))
+    , builder(builder)
+{}
+
+/**
+ * @brief This function verifies an Ultra Honk proof for a given Flavor, produced for a relaxed instance (ϕ, \vec{β*},
+ * e*).
+ *
+ */
+template <typename Flavor>
+std::array<typename Flavor::GroupElement, 2> DeciderRecursiveVerifier_<Flavor>::verify_proof(const plonk::proof& proof)
+{
+    using Sumcheck = ::proof_system::honk::sumcheck::SumcheckVerifier<Flavor>;
+    using Curve = typename Flavor::Curve;
+    using ZeroMorph = ::proof_system::honk::pcs::zeromorph::ZeroMorphVerifier_<Curve>;
+    using VerifierCommitments = typename Flavor::VerifierCommitments;
+    using CommitmentLabels = typename Flavor::CommitmentLabels;
+    using RelationParams = ::proof_system::RelationParameters<FF>;
+    using Transcript = typename Flavor::Transcript;
+    using Instance = typename ::proof_system::honk::VerifierInstance_<Flavor>;
+
+    static constexpr size_t NUM_SUBRELATIONS = Flavor::NUM_SUBRELATIONS;
+    transcript = std::make_shared<Transcript>(builder, proof.proof_data);
+    auto inst = std::make_unique<Instance>();
+
+    inst->instance_size = transcript->template receive_from_prover<uint32_t>("instance_size");
+    inst->log_instance_size = static_cast<size_t>(numeric::get_msb(inst->instance_size));
+    inst->public_input_size = transcript->template receive_from_prover<uint32_t>("public_input_size");
+
+    for (size_t i = 0; i < inst->public_input_size; ++i) {
+        auto public_input_i = transcript->template receive_from_prover<FF>("public_input_" + std::to_string(i));
+        inst->public_inputs.emplace_back(public_input_i);
+    }
+
+    auto eta = transcript->template receive_from_prover<FF>("eta");
+    auto beta = transcript->template receive_from_prover<FF>("beta");
+    auto gamma = transcript->template receive_from_prover<FF>("gamma");
+    auto public_input_delta = transcript->template receive_from_prover<FF>("public_input_delta");
+    auto lookup_grand_product_delta = transcript->template receive_from_prover<FF>("lookup_grand_product_delta");
+    inst->relation_parameters =
+        RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
+
+    for (size_t idx = 0; idx < NUM_SUBRELATIONS - 1; idx++) {
+
+        inst->alphas[idx] = transcript->template receive_from_prover<FF>("alpha" + std::to_string(idx));
+    }
+
+    inst->target_sum = transcript->template receive_from_prover<FF>("target_sum");
+
+    inst->gate_challenges = std::vector<FF>(inst->log_instance_size);
+    for (size_t idx = 0; idx < inst->log_instance_size; idx++) {
+        inst->gate_challenges[idx] =
+            transcript->template receive_from_prover<FF>("gate_challenge_" + std::to_string(idx));
+    }
+    auto comm_view = inst->witness_commitments.get_all();
+    auto witness_labels = inst->commitment_labels.get_witness();
+    for (size_t idx = 0; idx < witness_labels.size(); idx++) {
+        comm_view[idx] = transcript->template receive_from_prover<Commitment>(witness_labels[idx]);
+    }
+
+    inst->verification_key = std::make_shared<VerificationKey>(inst->instance_size, inst->public_input_size);
+    auto vk_view = inst->verification_key->get_all();
+    auto vk_labels = inst->commitment_labels.get_precomputed();
+    for (size_t idx = 0; idx < vk_labels.size(); idx++) {
+        vk_view[idx] = transcript->template receive_from_prover<Commitment>(vk_labels[idx]);
+    }
+
+    VerifierCommitments commitments{ inst->verification_key, inst->witness_commitments };
+
+    auto sumcheck = Sumcheck(inst->log_instance_size, transcript, inst->target_sum);
+
+    auto [multivariate_challenge, claimed_evaluations, sumcheck_verified] =
+        sumcheck.verify(inst->relation_parameters, inst->alphas, inst->gate_challenges);
+
+    // Execute ZeroMorph rounds. See https://hackmd.io/dlf9xEwhTQyE3hiGbq4FsA?view for a complete description of the
+    // unrolled protocol.
+    auto pairing_points = ZeroMorph::verify(commitments.get_unshifted(),
+                                            commitments.get_to_be_shifted(),
+                                            claimed_evaluations.get_unshifted(),
+                                            claimed_evaluations.get_shifted(),
+                                            multivariate_challenge,
+                                            transcript);
+
+    return pairing_points;
+}
+
+template class DeciderRecursiveVerifier_<proof_system::honk::flavor::UltraRecursive_<UltraCircuitBuilder>>;
+template class DeciderRecursiveVerifier_<proof_system::honk::flavor::UltraRecursive_<GoblinUltraCircuitBuilder>>;
+template class DeciderRecursiveVerifier_<proof_system::honk::flavor::GoblinUltraRecursive_<UltraCircuitBuilder>>;
+template class DeciderRecursiveVerifier_<proof_system::honk::flavor::GoblinUltraRecursive_<GoblinUltraCircuitBuilder>>;
+
+} // namespace proof_system::plonk::stdlib::recursion::honk
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.hpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.hpp
new file mode 100644
index 00000000000..fe783b42a01
--- /dev/null
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.hpp
@@ -0,0 +1,39 @@
+#pragma once
+#include "barretenberg/flavor/goblin_ultra_recursive.hpp"
+#include "barretenberg/flavor/ultra_recursive.hpp"
+#include "barretenberg/plonk/proof_system/types/proof.hpp"
+#include "barretenberg/srs/global_crs.hpp"
+#include "barretenberg/stdlib/recursion/honk/transcript/transcript.hpp"
+#include "barretenberg/sumcheck/sumcheck.hpp"
+
+namespace proof_system::plonk::stdlib::recursion::honk {
+template <typename Flavor> class DeciderRecursiveVerifier_ {
+    using FF = typename Flavor::FF;
+    using Commitment = typename Flavor::Commitment;
+    using GroupElement = typename Flavor::GroupElement;
+    using VerificationKey = typename Flavor::VerificationKey;
+    using VerifierCommitmentKey = typename Flavor::VerifierCommitmentKey;
+    using NativeVerificationKey = typename Flavor::NativeVerificationKey;
+    using Builder = typename Flavor::CircuitBuilder;
+    using AlphaType = typename Flavor::AlphaType;
+    using PairingPoints = std::array<GroupElement, 2>;
+
+  public:
+    explicit DeciderRecursiveVerifier_(Builder* builder,
+                                       const std::shared_ptr<NativeVerificationKey>& native_verifier_key);
+
+    PairingPoints verify_proof(const plonk::proof& proof);
+
+    std::shared_ptr<VerificationKey> key;
+    std::map<std::string, Commitment> commitments;
+    std::shared_ptr<VerifierCommitmentKey> pcs_verification_key;
+    Builder* builder;
+    std::shared_ptr<Transcript<Builder>> transcript;
+};
+
+extern template class DeciderRecursiveVerifier_<proof_system::honk::flavor::UltraRecursive_<UltraCircuitBuilder>>;
+extern template class DeciderRecursiveVerifier_<proof_system::honk::flavor::UltraRecursive_<GoblinUltraCircuitBuilder>>;
+extern template class DeciderRecursiveVerifier_<proof_system::honk::flavor::GoblinUltraRecursive_<UltraCircuitBuilder>>;
+extern template class DeciderRecursiveVerifier_<
+    proof_system::honk::flavor::GoblinUltraRecursive_<GoblinUltraCircuitBuilder>>;
+} // namespace proof_system::plonk::stdlib::recursion::honk
\ No newline at end of file

From 8cb1d9e7164359769d0761c5834319bf8850f8e2 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Fri, 22 Dec 2023 13:20:46 +0000
Subject: [PATCH 69/86] fix idx overflow

---
 barretenberg/cpp/src/barretenberg/relations/utils.hpp | 11 +++++++----
 1 file changed, 7 insertions(+), 4 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
index 4b7f7ecba58..150223b6c4a 100644
--- a/barretenberg/cpp/src/barretenberg/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
@@ -16,6 +16,7 @@ template <typename Flavor> class RelationUtils {
     using AlphaType = typename Flavor::AlphaType;
 
     static constexpr size_t NUM_RELATIONS = Flavor::NUM_RELATIONS;
+    static constexpr size_t NUM_SUBRELATIONS = Flavor::NUM_SUBRELATIONS;
 
     /**
      * Utility methods for tuple of tuples of Univariates
@@ -74,9 +75,10 @@ template <typename Flavor> class RelationUtils {
         requires proof_system::IsFoldingFlavor<Flavor>
     {
         size_t idx = 0;
+        std::array<FF, NUM_SUBRELATIONS> tmp{ current_scalar };
+        std::copy(challenges.begin(), challenges.end(), tmp.begin() + 1);
         auto scale_by_challenges = [&]<size_t, size_t>(auto& element) {
-            element *= current_scalar;
-            current_scalar = challenges[idx];
+            element *= tmp[idx];
             idx++;
         };
         apply_to_tuple_of_tuples(tuple, scale_by_challenges);
@@ -191,10 +193,11 @@ template <typename Flavor> class RelationUtils {
         requires proof_system::IsFoldingFlavor<Flavor>
     {
         size_t idx = 0;
+        std::array<FF, NUM_SUBRELATIONS> tmp{ current_scalar };
+        std::copy(challenges.begin(), challenges.end(), tmp.begin() + 1);
         auto scale_by_challenges_and_accumulate = [&](auto& element) {
             for (auto& entry : element) {
-                result += entry * current_scalar;
-                current_scalar = challenges[idx];
+                result += entry * tmp[idx];
                 idx++;
             }
         };

From be3a2239b3067b33f96a7a7f32e247df313bbcb2 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Mon, 8 Jan 2024 15:21:51 +0000
Subject: [PATCH 70/86] debug

---
 .../stdlib/recursion/honk/verifier/goblin_verifier.test.cpp     | 2 +-
 .../stdlib/recursion/honk/verifier/verifier.test.cpp            | 1 +
 2 files changed, 2 insertions(+), 1 deletion(-)

diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/goblin_verifier.test.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/goblin_verifier.test.cpp
index ef0c98e5aea..5263c6b90c0 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/goblin_verifier.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/goblin_verifier.test.cpp
@@ -187,7 +187,7 @@ template <typename BuilderType> class GoblinRecursiveVerifierTest : public testi
         OuterBuilder outer_circuit;
         RecursiveVerifier verifier{ &outer_circuit, instance->verification_key };
         auto pairing_points = verifier.verify_proof(inner_proof);
-        info("Recursive Verifier: num gates = ", outer_circuit.num_gates);
+        info("Recursive Verifier Goblin: num gates = ", outer_circuit.num_gates);
 
         // Check for a failure flag in the recursive verifier circuit
         EXPECT_EQ(outer_circuit.failed(), false) << outer_circuit.err();
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/verifier.test.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/verifier.test.cpp
index 3b045a35a73..49b8882d759 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/verifier.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/verifier.test.cpp
@@ -170,6 +170,7 @@ template <typename BuilderType> class RecursiveVerifierTest : public testing::Te
         OuterBuilder outer_circuit;
         RecursiveVerifier verifier{ &outer_circuit, instance->verification_key };
         auto pairing_points = verifier.verify_proof(inner_proof);
+        info("Recursive Verifier Ultra: num gates = ", outer_circuit.num_gates);
 
         // Check for a failure flag in the recursive verifier circuit
         EXPECT_EQ(outer_circuit.failed(), false) << outer_circuit.err();

From deb239e8c6034bec5318cdd2d4124ecf9db70435 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Mon, 8 Jan 2024 17:08:04 +0000
Subject: [PATCH 71/86] remove computation of power polynomial at values on the
 verifier side

---
 barretenberg/cpp/src/barretenberg/polynomials/pow.hpp   | 8 +++-----
 barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp | 1 +
 2 files changed, 4 insertions(+), 5 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/polynomials/pow.hpp b/barretenberg/cpp/src/barretenberg/polynomials/pow.hpp
index fcfcfc255f0..7a95c26f292 100644
--- a/barretenberg/cpp/src/barretenberg/polynomials/pow.hpp
+++ b/barretenberg/cpp/src/barretenberg/polynomials/pow.hpp
@@ -76,9 +76,7 @@ template <typename FF> struct PowPolynomial {
 
     explicit PowPolynomial(const std::vector<FF>& betas)
         : betas(betas)
-    {
-        compute_pow_polynomial_at_values();
-    }
+    {}
 
     FF const& operator[](size_t idx) const { return pow_betas[idx]; }
 
@@ -103,13 +101,13 @@ template <typename FF> struct PowPolynomial {
     }
 
     /**
-     * @brief Given {β_0,...,β_{d-1}} compute pow(β) of size 2^d
+     * @brief Given \vec{β} = {β_0,...,β_{d-1}} compute pow_i(\vec{β}) for i=0,...,2^d
      *
      */
     void compute_pow_polynomial_at_values()
     {
         size_t pow_size = 1 << betas.size();
-        pow_betas = std::vector<FF>(1 << betas.size(), 0);
+        pow_betas = std::vector<FF>(pow_size, 0);
         for (size_t i = 0; i < pow_size; i++) {
             auto res = FF(1);
             for (size_t j = i, beta_idx = 0; j > 0; j >>= 1, beta_idx++) {
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
index 2cf017e4f33..6416ac3166e 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
@@ -89,6 +89,7 @@ template <typename Flavor> class SumcheckProver {
     {
 
         barretenberg::PowPolynomial<FF> pow_univariate(gate_challenges);
+        pow_univariate.compute_pow_polynomial_at_values();
 
         std::vector<FF> multivariate_challenge;
         multivariate_challenge.reserve(multivariate_d);

From ecf84cef4be3332cc99453f448b88b722595bc5b Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Mon, 8 Jan 2024 17:21:37 +0000
Subject: [PATCH 72/86] update tests and comments

---
 .../src/barretenberg/flavor/goblin_ultra_recursive.hpp |  6 +++---
 barretenberg/cpp/src/barretenberg/flavor/ultra.hpp     |  6 +++---
 .../cpp/src/barretenberg/flavor/ultra_recursive.hpp    |  4 ++--
 .../cpp/src/barretenberg/polynomials/pow.test.cpp      |  1 +
 .../cpp/src/barretenberg/protogalaxy/combiner.test.cpp | 10 ++++++----
 .../barretenberg/protogalaxy/protogalaxy_prover.cpp    |  3 ++-
 .../barretenberg/protogalaxy/protogalaxy_prover.hpp    |  5 ++---
 .../src/barretenberg/ultra_honk/protogalaxy.test.cpp   |  2 ++
 8 files changed, 21 insertions(+), 16 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
index 54a368bc269..094d0460e5c 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
@@ -74,9 +74,9 @@ class GoblinUltraRecursive {
     static constexpr size_t BATCHED_RELATION_PARTIAL_LENGTH = MAX_PARTIAL_RELATION_LENGTH + 1;
     static constexpr size_t NUM_RELATIONS = std::tuple_size<Relations>::value;
 
-    // For instances of this flavour, used in folding, we need a unique sumcheck batching challenges for each
-    // subrelation. This
-    // is because using powers of alpha would increase the degree of Protogalaxy polynomial $G$ (the combiner) to much.
+    // For instances of this flavour, used in folding, we need a unique sumcheck batching challenge for each
+    // subrelation. This is because using powers of alpha would increase the degree of Protogalaxy polynomial $G$ (the
+    // combiner) to much.
     static constexpr size_t NUM_SUBRELATIONS = compute_number_of_subrelations<Relations>();
     using AlphaType = std::array<FF, NUM_SUBRELATIONS - 1>;
 
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
index 7d2b6bb6302..75f7e3d4cf4 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
@@ -58,9 +58,9 @@ class Ultra {
     static constexpr size_t MAX_TOTAL_RELATION_LENGTH = compute_max_total_relation_length<Relations>();
     static_assert(MAX_TOTAL_RELATION_LENGTH == 12);
     static constexpr size_t NUM_SUBRELATIONS = compute_number_of_subrelations<Relations>();
-    // For instances of this flavour, used in folding, we need a unique sumcheck batching challenges for each
-    // subrelation. This
-    // is because using powers of alpha would increase the degree of Protogalaxy polynomial $G$ (the combiner) to much.
+    // For instances of this flavour, used in folding, we need a unique sumcheck batching challenge for each
+    // subrelation. This is because using powers of alpha would increase the degree of Protogalaxy polynomial $G$ (the
+    // combiner) too much.
     using AlphaType = std::array<FF, NUM_SUBRELATIONS - 1>;
 
     // BATCHED_RELATION_PARTIAL_LENGTH = algebraic degree of sumcheck relation *after* multiplying by the `pow_zeta`
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
index 042c4e4cec4..e44ed90cd37 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
@@ -87,8 +87,8 @@ template <typename BuilderType> class UltraRecursive_ {
     static constexpr size_t NUM_RELATIONS = std::tuple_size<Relations>::value;
 
     // For instances of this flavour, used in folding, we need a unique sumcheck batching challenges for each
-    // subrelation. This is because using powers of alpha would increase the degree of Protogalaxy polynomial $G$ (the
-    // combiner) to much.
+    // subrelation to avoid increasing the degree of Protogalaxy polynomial $G$ (the
+    // combiner) too much.
     static constexpr size_t NUM_SUBRELATIONS = compute_number_of_subrelations<Relations>();
     using AlphaType = std::array<FF, NUM_SUBRELATIONS - 1>;
 
diff --git a/barretenberg/cpp/src/barretenberg/polynomials/pow.test.cpp b/barretenberg/cpp/src/barretenberg/polynomials/pow.test.cpp
index cbafd648112..e45ebfb2920 100644
--- a/barretenberg/cpp/src/barretenberg/polynomials/pow.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/polynomials/pow.test.cpp
@@ -35,6 +35,7 @@ TEST(PowPolynomial, PowPolynomialsOnPowers)
 {
     auto betas = std::vector<FF>{ 2, 4, 16 };
     auto pow = PowPolynomial(betas);
+    pow.compute_pow_polynomial_at_values();
     auto expected_values = std::vector<FF>{ 1, 2, 4, 8, 16, 32, 64, 128 };
     EXPECT_EQ(expected_values, pow.pow_betas);
 }
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp
index a89c049a0ee..8ff572f47b9 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/combiner.test.cpp
@@ -53,8 +53,8 @@ TEST(Protogalaxy, CombinerOn2Instances)
 
             ProverInstances instances{ instance_data };
             instances.alphas.fill(Univariate<FF, 13>(FF(0))); // focus on the arithmetic relation only
-
-            auto result = prover.compute_combiner(instances, PowPolynomial(std::vector<FF>{ 2 }));
+            auto pow_polynomial = PowPolynomial(std::vector<FF>{ 2 });
+            auto result = prover.compute_combiner(instances, pow_polynomial);
             auto expected_result =
                 barretenberg::Univariate<FF, 13>(std::array<FF, 13>{ 87706,
                                                                      13644570,
@@ -131,7 +131,8 @@ TEST(Protogalaxy, CombinerOn2Instances)
             relation value:
                       0    0    0    0    0    0    0              0    0    6   18   36   60   90      */
 
-            auto result = prover.compute_combiner(instances, PowPolynomial(std::vector<FF>{ 2 }));
+            auto pow_polynomial = PowPolynomial(std::vector<FF>{ 2 });
+            auto result = prover.compute_combiner(instances, pow_polynomial);
             auto expected_result = barretenberg::Univariate<FF, 13>(
                 std::array<FF, 13>{ 0, 0, 12, 36, 72, 120, 180, 252, 336, 432, 540, 660, 792 });
 
@@ -181,7 +182,8 @@ TEST(Protogalaxy, CombinerOn4Instances)
         zero_all_selectors(instances[2]->prover_polynomials);
         zero_all_selectors(instances[3]->prover_polynomials);
 
-        auto result = prover.compute_combiner(instances, PowPolynomial(std::vector<FF>{ 2 }));
+        auto pow_polynomial = PowPolynomial(std::vector<FF>{ 2 });
+        auto result = prover.compute_combiner(instances, pow_polynomial);
         std::array<FF, 43> zeroes;
         std::fill(zeroes.begin(), zeroes.end(), 0);
         auto expected_result = barretenberg::Univariate<FF, 43>(zeroes);
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index 6c632f78841..6c2440091d0 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -292,7 +292,8 @@ FoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverInstanc
         update_gate_challenges(perturbator_challenge, accumulator->gate_challenges, deltas);
     combine_relation_parameters(instances);
     combine_alpha(instances);
-    auto combiner = compute_combiner(instances, PowPolynomial<FF>(instances.next_gate_challenges));
+    auto pow_polynomial = PowPolynomial<FF>(instances.next_gate_challenges);
+    auto combiner = compute_combiner(instances, pow_polynomial);
 
     auto compressed_perturbator = perturbator.evaluate(perturbator_challenge);
     auto combiner_quotient = compute_combiner_quotient(compressed_perturbator, combiner);
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index 6b368ad6c40..50842a1f58b 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -270,11 +270,10 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      * @brief Compute the combiner polynomial $G$ in the Protogalaxy paper.
      *
      */
-    ExtendedUnivariateWithRandomization compute_combiner(const ProverInstances& instances,
-                                                         const PowPolynomial<FF>& pow_betas)
+    ExtendedUnivariateWithRandomization compute_combiner(const ProverInstances& instances, PowPolynomial<FF>& pow_betas)
     {
         size_t common_instance_size = instances[0]->instance_size;
-
+        pow_betas.compute_pow_polynomial_at_values();
         // Determine number of threads for multithreading.
         // Note: Multithreading is "on" for every round but we reduce the number of threads from the max available based
         // on a specified minimum number of iterations per thread. This eventually leads to the use of a single thread.
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
index f0434cd0e2c..5e89b2e85ba 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
@@ -68,6 +68,7 @@ void check_accumulator_target_sum_manual(std::shared_ptr<Instance>& accumulator,
         accumulator->prover_polynomials, accumulator->alphas, accumulator->relation_parameters);
     // Construct pow(\vec{betas*}) as in the paper
     auto expected_pows = PowPolynomial(accumulator->gate_challenges);
+    expected_pows.compute_pow_polynomial_at_values();
 
     // Compute the corresponding target sum and create a dummy accumulator
     auto expected_target_sum = FF(0);
@@ -163,6 +164,7 @@ TEST_F(ProtoGalaxyTests, PerturbatorPolynomial)
 
     // Construct pow(\vec{betas}) as in the paper
     auto pow_beta = PowPolynomial(betas);
+    pow_beta.compute_pow_polynomial_at_values();
 
     // Compute the corresponding target sum and create a dummy accumulator
     auto target_sum = FF(0);

From ecdf4d38d1f4e9fca27f45df9b95358d6a3bafc5 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Wed, 10 Jan 2024 11:06:09 +0000
Subject: [PATCH 73/86] recursive decider verifier and folding wip

---
 .../src/barretenberg/flavor/goblin_ultra.hpp  |   2 +-
 .../flavor/goblin_ultra_recursive.hpp         |  11 +-
 .../barretenberg/flavor/ultra_recursive.hpp   |  23 +-
 .../verifier/decider_recursive_verifier.cpp   |  12 +-
 .../verifier/decider_recursive_verifier.hpp   |   5 +-
 .../honk/verifier/decider_verifier.test.cpp   |   0
 .../protogalaxy_recursive_verifier.cpp        | 284 ++++++++++++++++++
 .../protogalaxy_recursive_verifier.hpp        |  99 ++++++
 8 files changed, 419 insertions(+), 17 deletions(-)
 create mode 100644 barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_verifier.test.cpp
 create mode 100644 barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
 create mode 100644 barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp

diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
index a0e5f3bab8f..62d9e062502 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
@@ -454,7 +454,7 @@ class GoblinUltra {
         }
 
         VerifierCommitments_(const std::shared_ptr<VerificationKey>& verification_key,
-                             const WitnessCommitments& witness_commitments)
+                             const WitnessEntities<Commitment>& witness_commitments)
         {
             this->q_m = verification_key->q_m;
             this->q_l = verification_key->q_l;
diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
index 083ebf298aa..3b9463b4a7a 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
@@ -104,6 +104,12 @@ class GoblinUltraRecursive {
      */
     class VerificationKey : public VerificationKey_<GoblinUltra::PrecomputedEntities<Commitment>> {
       public:
+        VerificationKey(const size_t circuit_size, const size_t num_public_inputs)
+        {
+            this->circuit_size = circuit_size;
+            this->log_circuit_size = numeric::get_msb(circuit_size);
+            this->num_public_inputs = num_public_inputs;
+        };
         /**
          * @brief Construct a new Verification Key with stdlib types from a provided native verification
          * key
@@ -112,9 +118,10 @@ class GoblinUltraRecursive {
          * @param native_key Native verification key from which to extract the precomputed commitments
          */
         VerificationKey(CircuitBuilder* builder, const std::shared_ptr<NativeVerificationKey>& native_key)
-            : VerificationKey_<GoblinUltra::PrecomputedEntities<Commitment>>(native_key->circuit_size,
-                                                                             native_key->num_public_inputs)
         {
+            this->circuit_size = native_key->circuit_size;
+            this->log_circuit_size = numeric::get_msb(this->circuit_size);
+            this->num_public_inputs = native_key->num_public_inputs;
             this->q_m = Commitment::from_witness(builder, native_key->q_m);
             this->q_l = Commitment::from_witness(builder, native_key->q_l);
             this->q_r = Commitment::from_witness(builder, native_key->q_r);
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
index d26cccbe9a7..1ff2ba6fd41 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
@@ -235,6 +235,17 @@ template <typename BuilderType> class UltraRecursive_ {
 
             };
         };
+        RefVector<DataType> get_precomputed()
+        {
+            return { q_m,          q_c,   q_l,      q_r,     q_o,     q_4,     q_arith, q_sort,
+                     q_elliptic,   q_aux, q_lookup, sigma_1, sigma_2, sigma_3, sigma_4, id_1,
+                     id_2,         id_3,  id_4,     table_1, table_2, table_3, table_4, lagrange_first,
+                     lagrange_last
+
+            };
+        }
+
+        RefVector<DataType> get_witness() { return { w_l, w_r, w_o, w_4, sorted_accum, z_perm, z_lookup }; };
         RefVector<DataType> get_to_be_shifted()
         {
             return { table_1, table_2, table_3, table_4, w_l, w_r, w_o, w_4, sorted_accum, z_perm, z_lookup };
@@ -257,6 +268,12 @@ template <typename BuilderType> class UltraRecursive_ {
      */
     class VerificationKey : public VerificationKey_<PrecomputedEntities<Commitment>> {
       public:
+        VerificationKey(const size_t circuit_size, const size_t num_public_inputs)
+        {
+            this->circuit_size = circuit_size;
+            this->log_circuit_size = numeric::get_msb(circuit_size);
+            this->num_public_inputs = num_public_inputs;
+        };
         /**
          * @brief Construct a new Verification Key with stdlib types from a provided native verification key
          *
@@ -264,8 +281,10 @@ template <typename BuilderType> class UltraRecursive_ {
          * @param native_key Native verification key from which to extract the precomputed commitments
          */
         VerificationKey(CircuitBuilder* builder, const std::shared_ptr<NativeVerificationKey>& native_key)
-            : VerificationKey_<PrecomputedEntities<Commitment>>(native_key->circuit_size, native_key->num_public_inputs)
         {
+            this->circuit_size = native_key->circuit_size;
+            this->log_circuit_size = numeric::get_msb(this->circuit_size);
+            this->num_public_inputs = native_key->num_public_inputs;
             this->q_m = Commitment::from_witness(builder, native_key->q_m);
             this->q_l = Commitment::from_witness(builder, native_key->q_l);
             this->q_r = Commitment::from_witness(builder, native_key->q_r);
@@ -397,8 +416,6 @@ template <typename BuilderType> class UltraRecursive_ {
             this->q_elliptic = verification_key->q_elliptic;
             this->q_aux = verification_key->q_aux;
             this->q_lookup = verification_key->q_lookup;
-            this->q_poseidon2_external = verification_key->q_poseidon2_external;
-            this->q_poseidon2_internal = verification_key->q_poseidon2_internal;
             this->sigma_1 = verification_key->sigma_1;
             this->sigma_2 = verification_key->sigma_2;
             this->sigma_3 = verification_key->sigma_3;
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.cpp
index 8776989ee6d..847d1a087eb 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.cpp
@@ -25,8 +25,6 @@ std::array<typename Flavor::GroupElement, 2> DeciderRecursiveVerifier_<Flavor>::
     using Curve = typename Flavor::Curve;
     using ZeroMorph = ::proof_system::honk::pcs::zeromorph::ZeroMorphVerifier_<Curve>;
     using VerifierCommitments = typename Flavor::VerifierCommitments;
-    using CommitmentLabels = typename Flavor::CommitmentLabels;
-    using RelationParams = ::proof_system::RelationParameters<FF>;
     using Transcript = typename Flavor::Transcript;
     using Instance = typename ::proof_system::honk::VerifierInstance_<Flavor>;
 
@@ -34,9 +32,11 @@ std::array<typename Flavor::GroupElement, 2> DeciderRecursiveVerifier_<Flavor>::
     transcript = std::make_shared<Transcript>(builder, proof.proof_data);
     auto inst = std::make_unique<Instance>();
 
-    inst->instance_size = transcript->template receive_from_prover<uint32_t>("instance_size");
+    auto instance_size = transcript->template receive_from_prover<uint32_t>("instance_size");
+    auto public_input_size = transcript->template receive_from_prover<uint32_t>("public_input_size");
+    inst->instance_size = static_cast<size_t>(instance_size.get_value());
+    inst->public_input_size = static_cast<size_t>(public_input_size.get_value());
     inst->log_instance_size = static_cast<size_t>(numeric::get_msb(inst->instance_size));
-    inst->public_input_size = transcript->template receive_from_prover<uint32_t>("public_input_size");
 
     for (size_t i = 0; i < inst->public_input_size; ++i) {
         auto public_input_i = transcript->template receive_from_prover<FF>("public_input_" + std::to_string(i));
@@ -97,7 +97,5 @@ std::array<typename Flavor::GroupElement, 2> DeciderRecursiveVerifier_<Flavor>::
 
 template class DeciderRecursiveVerifier_<proof_system::honk::flavor::UltraRecursive_<UltraCircuitBuilder>>;
 template class DeciderRecursiveVerifier_<proof_system::honk::flavor::UltraRecursive_<GoblinUltraCircuitBuilder>>;
-template class DeciderRecursiveVerifier_<proof_system::honk::flavor::GoblinUltraRecursive_<UltraCircuitBuilder>>;
-template class DeciderRecursiveVerifier_<proof_system::honk::flavor::GoblinUltraRecursive_<GoblinUltraCircuitBuilder>>;
-
+template class DeciderRecursiveVerifier_<proof_system::honk::flavor::GoblinUltraRecursive>;
 } // namespace proof_system::plonk::stdlib::recursion::honk
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.hpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.hpp
index fe783b42a01..67a8d8cc783 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.hpp
@@ -2,7 +2,6 @@
 #include "barretenberg/flavor/goblin_ultra_recursive.hpp"
 #include "barretenberg/flavor/ultra_recursive.hpp"
 #include "barretenberg/plonk/proof_system/types/proof.hpp"
-#include "barretenberg/srs/global_crs.hpp"
 #include "barretenberg/stdlib/recursion/honk/transcript/transcript.hpp"
 #include "barretenberg/sumcheck/sumcheck.hpp"
 
@@ -33,7 +32,5 @@ template <typename Flavor> class DeciderRecursiveVerifier_ {
 
 extern template class DeciderRecursiveVerifier_<proof_system::honk::flavor::UltraRecursive_<UltraCircuitBuilder>>;
 extern template class DeciderRecursiveVerifier_<proof_system::honk::flavor::UltraRecursive_<GoblinUltraCircuitBuilder>>;
-extern template class DeciderRecursiveVerifier_<proof_system::honk::flavor::GoblinUltraRecursive_<UltraCircuitBuilder>>;
-extern template class DeciderRecursiveVerifier_<
-    proof_system::honk::flavor::GoblinUltraRecursive_<GoblinUltraCircuitBuilder>>;
+extern template class DeciderRecursiveVerifier_<proof_system::honk::flavor::GoblinUltraRecursive>;
 } // namespace proof_system::plonk::stdlib::recursion::honk
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_verifier.test.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_verifier.test.cpp
new file mode 100644
index 00000000000..e69de29bb2d
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
new file mode 100644
index 00000000000..f776755e4d4
--- /dev/null
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
@@ -0,0 +1,284 @@
+#include "protogalaxy_recursive_verifier.hpp"
+#include "barretenberg/proof_system/library/grand_product_delta.hpp"
+namespace proof_system::plonk::stdlib::recursion::honk {
+
+template <class VerifierInstances>
+void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::receive_accumulator(const std::shared_ptr<Instance>& inst,
+                                                                           const std::string& domain_separator)
+{
+    inst->instance_size = transcript->template receive_from_prover<uint32_t>(domain_separator + "_instance_size");
+    inst->log_instance_size = static_cast<size_t>(numeric::get_msb(inst->instance_size));
+    inst->public_input_size =
+        transcript->template receive_from_prover<uint32_t>(domain_separator + "_public_input_size");
+
+    for (size_t i = 0; i < inst->public_input_size; ++i) {
+        auto public_input_i =
+            transcript->template receive_from_prover<FF>(domain_separator + "_public_input_" + std::to_string(i));
+        inst->public_inputs.emplace_back(public_input_i);
+    }
+
+    auto eta = transcript->template receive_from_prover<FF>(domain_separator + "_eta");
+    auto beta = transcript->template receive_from_prover<FF>(domain_separator + "_beta");
+    auto gamma = transcript->template receive_from_prover<FF>(domain_separator + "_gamma");
+    auto public_input_delta = transcript->template receive_from_prover<FF>(domain_separator + "_public_input_delta");
+    auto lookup_grand_product_delta =
+        transcript->template receive_from_prover<FF>(domain_separator + "_lookup_grand_product_delta");
+    inst->relation_parameters =
+        RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
+
+    for (size_t idx = 0; idx < NUM_SUBRELATIONS - 1; idx++) {
+        inst->alphas[idx] =
+            transcript->template receive_from_prover<FF>(domain_separator + "_alpha_" + std::to_string(idx));
+    }
+
+    inst->target_sum = transcript->template receive_from_prover<FF>(domain_separator + "_target_sum");
+
+    inst->gate_challenges = std::vector<FF>(inst->log_instance_size);
+    for (size_t idx = 0; idx < inst->log_instance_size; idx++) {
+        inst->gate_challenges[idx] =
+            transcript->template receive_from_prover<FF>(domain_separator + "_gate_challenge_" + std::to_string(idx));
+    }
+    auto comm_view = inst->witness_commitments.get_all();
+    auto witness_labels = inst->commitment_labels.get_witness();
+    for (size_t idx = 0; idx < witness_labels.size(); idx++) {
+        comm_view[idx] =
+            transcript->template receive_from_prover<Commitment>(domain_separator + "_" + witness_labels[idx]);
+    }
+
+    inst->verification_key = std::make_shared<VerificationKey>(inst->instance_size, inst->public_input_size);
+    auto vk_view = inst->verification_key->get_all();
+    auto vk_labels = inst->commitment_labels.get_precomputed();
+    for (size_t idx = 0; idx < vk_labels.size(); idx++) {
+        vk_view[idx] = transcript->template receive_from_prover<Commitment>(domain_separator + "_" + vk_labels[idx]);
+    }
+}
+
+template <class VerifierInstances>
+void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::receive_and_finalise_instance(
+    const std::shared_ptr<Instance>& inst, const std::string& domain_separator)
+{
+    inst->instance_size = transcript->template receive_from_prover<uint32_t>(domain_separator + "_instance_size");
+    inst->log_instance_size = static_cast<size_t>(numeric::get_msb(inst->instance_size));
+    inst->public_input_size =
+        transcript->template receive_from_prover<uint32_t>(domain_separator + "_public_input_size");
+
+    for (size_t i = 0; i < inst->public_input_size; ++i) {
+        auto public_input_i =
+            transcript->template receive_from_prover<FF>(domain_separator + "_public_input_" + std::to_string(i));
+        inst->public_inputs.emplace_back(public_input_i);
+    }
+
+    inst->pub_inputs_offset =
+        transcript->template receive_from_prover<uint32_t>(domain_separator + "_pub_inputs_offset");
+
+    auto labels = inst->commitment_labels;
+    auto& witness_commitments = inst->witness_commitments;
+    witness_commitments.w_l = transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.w_l);
+    witness_commitments.w_r = transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.w_r);
+    witness_commitments.w_o = transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.w_o);
+
+    auto eta = transcript->get_challenge(domain_separator + "_eta");
+    witness_commitments.sorted_accum =
+        transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.sorted_accum);
+    witness_commitments.w_4 = transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.w_4);
+
+    auto [beta, gamma] = transcript->get_challenges(domain_separator + "_beta", domain_separator + "_gamma");
+    witness_commitments.z_perm =
+        transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.z_perm);
+    witness_commitments.z_lookup =
+        transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.z_lookup);
+
+    const FF public_input_delta = compute_public_input_delta<Flavor>(
+        inst->public_inputs, beta, gamma, inst->instance_size, inst->pub_inputs_offset);
+    const FF lookup_grand_product_delta = compute_lookup_grand_product_delta<FF>(beta, gamma, inst->instance_size);
+    inst->relation_parameters =
+        RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
+
+    for (size_t idx = 0; idx < NUM_SUBRELATIONS - 1; idx++) {
+        inst->alphas[idx] = transcript->get_challenge(domain_separator + "_alpha_" + std::to_string(idx));
+    }
+
+    inst->verification_key = std::make_shared<VerificationKey>(inst->instance_size, inst->public_input_size);
+    auto vk_view = inst->verification_key->get_all();
+    auto vk_labels = labels.get_precomputed();
+    for (size_t idx = 0; idx < vk_labels.size(); idx++) {
+        vk_view[idx] = transcript->template receive_from_prover<Commitment>(domain_separator + "_" + vk_labels[idx]);
+    }
+}
+
+// TODO(https://github.com/AztecProtocol/barretenberg/issues/795): The rounds prior to actual verifying are common
+// between decider and folding verifier and could be somehow shared so we do not duplicate code so much.
+template <class VerifierInstances>
+void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::prepare_for_folding(const std::vector<uint8_t>& fold_data)
+{
+    transcript = std::make_shared<Transcript>(fold_data);
+    auto index = 0;
+    auto inst = instances[0];
+    auto domain_separator = std::to_string(index);
+    inst->is_accumulator = transcript->template receive_from_prover<bool>(domain_separator + "is_accumulator");
+    if (inst->is_accumulator) {
+        receive_accumulator(inst, domain_separator);
+    } else {
+        // This is the first round of folding and we need to generate some gate challenges.
+        // TODO(https://github.com/AztecProtocol/barretenberg/issues/740): implement option 2 to make this more
+        // efficient by avoiding the computation of the perturbator
+        receive_and_finalise_instance(inst, domain_separator);
+        inst->target_sum = 0;
+        auto beta = transcript->get_challenge(domain_separator + "_initial_gate_challenge");
+        std::vector<FF> gate_challenges(inst->log_instance_size);
+        gate_challenges[0] = beta;
+        for (size_t i = 1; i < inst->log_instance_size; i++) {
+            gate_challenges[i] = gate_challenges[i - 1].sqr();
+        }
+        inst->gate_challenges = gate_challenges;
+    }
+    index++;
+
+    for (auto it = instances.begin() + 1; it != instances.end(); it++, index++) {
+        auto inst = *it;
+        auto domain_separator = std::to_string(index);
+        receive_and_finalise_instance(inst, domain_separator);
+    }
+}
+
+template <class VerifierInstances>
+bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std::vector<uint8_t> fold_data)
+{
+    prepare_for_folding(fold_data);
+
+    auto delta = transcript->get_challenge("delta");
+    auto accumulator = get_accumulator();
+    auto deltas = compute_round_challenge_pows(accumulator->log_instance_size, delta);
+
+    std::vector<FF> perturbator_coeffs(accumulator->log_instance_size + 1);
+    for (size_t idx = 0; idx <= accumulator->log_instance_size; idx++) {
+        perturbator_coeffs[idx] = transcript->template receive_from_prover<FF>("perturbator_" + std::to_string(idx));
+    }
+
+    if (perturbator_coeffs[0] != accumulator->target_sum) {
+        return false;
+    }
+
+    auto perturbator = Polynomial<FF>(perturbator_coeffs);
+    FF perturbator_challenge = transcript->get_challenge("perturbator_challenge");
+    auto perturbator_at_challenge = perturbator.evaluate(perturbator_challenge);
+
+    // The degree of K(X) is dk - k - 1 = k(d - 1) - 1. Hence we need  k(d - 1) evaluations to represent it.
+    std::array<FF, VerifierInstances::BATCHED_EXTENDED_LENGTH - VerifierInstances::NUM> combiner_quotient_evals;
+    for (size_t idx = 0; idx < VerifierInstances::BATCHED_EXTENDED_LENGTH - VerifierInstances::NUM; idx++) {
+        combiner_quotient_evals[idx] = transcript->template receive_from_prover<FF>(
+            "combiner_quotient_" + std::to_string(idx + VerifierInstances::NUM));
+    }
+    Univariate<FF, VerifierInstances::BATCHED_EXTENDED_LENGTH, VerifierInstances::NUM> combiner_quotient(
+        combiner_quotient_evals);
+    FF combiner_challenge = transcript->get_challenge("combiner_quotient_challenge");
+    auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge);
+
+    auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
+    auto lagranges = std::vector<FF>{ FF(1) - combiner_challenge, combiner_challenge };
+
+    // Compute next folding parameters and verify against the ones received from the prover
+    auto expected_next_target_sum =
+        perturbator_at_challenge * lagranges[0] + vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
+    auto next_target_sum = transcript->template receive_from_prover<FF>("next_target_sum");
+    bool verified = (expected_next_target_sum == next_target_sum);
+    auto expected_betas_star = update_gate_challenges(perturbator_challenge, accumulator->gate_challenges, deltas);
+    for (size_t idx = 0; idx < accumulator->log_instance_size; idx++) {
+        auto beta_star = transcript->template receive_from_prover<FF>("next_gate_challenge_" + std::to_string(idx));
+        verified = verified & (expected_betas_star[idx] == beta_star);
+    }
+
+    // Compute ϕ and verify against the data received from the prover
+    WitnessCommitments acc_witness_commitments;
+    auto witness_labels = commitment_labels.get_witness();
+    size_t comm_idx = 0;
+    for (auto& expected_comm : acc_witness_commitments.get_all()) {
+        expected_comm = Commitment::infinity();
+        size_t inst = 0;
+        for (auto& instance : instances) {
+            expected_comm = expected_comm + instance->witness_commitments.get_all()[comm_idx] * lagranges[inst];
+            inst++;
+        }
+        auto comm = transcript->template receive_from_prover<Commitment>("next_" + witness_labels[comm_idx]);
+        verified = verified & (comm == expected_comm);
+        comm_idx++;
+    }
+
+    std::vector<FF> folded_public_inputs(instances[0]->public_inputs.size(), 0);
+    size_t el_idx = 0;
+    for (auto& expected_el : folded_public_inputs) {
+        size_t inst = 0;
+        for (auto& instance : instances) {
+            expected_el += instance->public_inputs[el_idx] * lagranges[inst];
+            inst++;
+        }
+        auto el = transcript->template receive_from_prover<FF>("next_public_input" + std::to_string(el_idx));
+        verified = verified & (el == expected_el);
+        el_idx++;
+    }
+
+    AlphaType expected_alphas;
+    size_t alpha_idx = 0;
+    for (auto& alpha : expected_alphas) {
+        alpha = FF(0);
+        size_t instance_idx = 0;
+        for (auto& instance : instances) {
+            alpha += instance->alphas[alpha_idx] * lagranges[instance_idx];
+            instance_idx++;
+        }
+        auto next_alpha = transcript->template receive_from_prover<FF>("next_alpha_" + std::to_string(alpha_idx));
+        verified = verified & (alpha == next_alpha);
+        alpha_idx++;
+    }
+    auto expected_parameters = proof_system::RelationParameters<FF>{};
+    for (size_t inst_idx = 0; inst_idx < VerifierInstances::NUM; inst_idx++) {
+        auto instance = instances[inst_idx];
+        expected_parameters.eta += instance->relation_parameters.eta * lagranges[inst_idx];
+        expected_parameters.beta += instance->relation_parameters.beta * lagranges[inst_idx];
+        expected_parameters.gamma += instance->relation_parameters.gamma * lagranges[inst_idx];
+        expected_parameters.public_input_delta +=
+            instance->relation_parameters.public_input_delta * lagranges[inst_idx];
+        expected_parameters.lookup_grand_product_delta +=
+            instance->relation_parameters.lookup_grand_product_delta * lagranges[inst_idx];
+    }
+
+    auto next_eta = transcript->template receive_from_prover<FF>("next_eta");
+    verified = verified & (next_eta == expected_parameters.eta);
+
+    auto next_beta = transcript->template receive_from_prover<FF>("next_beta");
+    verified = verified & (next_beta == expected_parameters.beta);
+
+    auto next_gamma = transcript->template receive_from_prover<FF>("next_gamma");
+    verified = verified & (next_gamma == expected_parameters.gamma);
+
+    auto next_public_input_delta = transcript->template receive_from_prover<FF>("next_public_input_delta");
+    verified = verified & (next_public_input_delta == expected_parameters.public_input_delta);
+
+    auto next_lookup_grand_product_delta =
+        transcript->template receive_from_prover<FF>("next_lookup_grand_product_delta");
+    verified = verified & (next_lookup_grand_product_delta == expected_parameters.lookup_grand_product_delta);
+
+    auto acc_vk = std::make_shared<VerificationKey>(instances[0]->instance_size, instances[0]->public_input_size);
+    auto vk_labels = commitment_labels.get_precomputed();
+    size_t vk_idx = 0;
+    for (auto& expected_vk : acc_vk->get_all()) {
+        size_t inst = 0;
+        expected_vk = Commitment::infinity();
+        for (auto& instance : instances) {
+            expected_vk = expected_vk + instance->verification_key->get_all()[vk_idx] * lagranges[inst];
+            inst++;
+        }
+        auto vk = transcript->template receive_from_prover<Commitment>("next_" + vk_labels[vk_idx]);
+        verified = verified & (vk == expected_vk);
+        vk_idx++;
+    }
+
+    return verified;
+}
+
+template class ProtoGalaxyRecursiveVerifier_<
+    VerifierInstances_<proof_system::honk::flavor::UltraRecursive_<UltraCircuitBuilder>, 2>>;
+template class ProtoGalaxyRecursiveVerifier_<
+    VerifierInstances_<proof_system::honk::flavor::UltraRecursive_<GoblinUltraCircuitBuilder>, 2>>;
+template class ProtoGalaxyRecursiveVerifier_<proof_system::honk::flavor::GoblinUltraRecursive, 2>;
+} // namespace proof_system::plonk::stdlib::recursion::honk
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp
new file mode 100644
index 00000000000..b67ba4a5a8f
--- /dev/null
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp
@@ -0,0 +1,99 @@
+#pragma once
+#include "barretenberg/flavor/flavor.hpp"
+#include "barretenberg/flavor/goblin_ultra.hpp"
+#include "barretenberg/flavor/ultra.hpp"
+#include "barretenberg/plonk/proof_system/types/proof.hpp"
+#include "barretenberg/protogalaxy/folding_result.hpp"
+#include "barretenberg/sumcheck/instance/instances.hpp"
+#include "barretenberg/transcript/transcript.hpp"
+
+namespace proof_system::plonk::stdlib::recursion::honk {
+template <class VerifierInstances> class ProtoGalaxyRecursiveVerifier_ {
+  public:
+    using Flavor = typename VerifierInstances::Flavor;
+    using Transcript = typename Flavor::Transcript;
+    using FF = typename Flavor::FF;
+    using Commitment = typename Flavor::Commitment;
+    using GroupElement = typename Flavor::GroupElement;
+    using Instance = typename VerifierInstances::Instance;
+    using VerificationKey = typename Flavor::VerificationKey;
+    using WitnessCommitments = typename Flavor::WitnessCommitments;
+    using CommitmentLabels = typename Flavor::CommitmentLabels;
+    using Builder = typename Flavor::CircuitBuilder;
+    using AlphaType = typename Flavor::AlphaType;
+    using PairingPoints = std::array<GroupElement, 2>;
+
+    static constexpr size_t NUM_SUBRELATIONS = Flavor::NUM_SUBRELATIONS;
+
+    VerifierInstances instances;
+
+    std::shared_ptr<Transcript> transcript = std::make_shared<Transcript>();
+
+    CommitmentLabels commitment_labels;
+
+    ProtoGalaxyRecursiveVerifier_(VerifierInstances insts)
+        : instances(insts){};
+    ~ProtoGalaxyRecursiveVerifier_() = default;
+    /**
+     * @brief Given a new round challenge δ for each iteration of the full ProtoGalaxy protocol, compute the vector
+     * [δ, δ^2,..., δ^t] where t = logn and n is the size of the instance.
+     */
+    static std::vector<FF> compute_round_challenge_pows(size_t log_instance_size, FF round_challenge)
+    {
+        std::vector<FF> pows(log_instance_size);
+        pows[0] = round_challenge;
+        for (size_t i = 1; i < log_instance_size; i++) {
+            pows[i] = pows[i - 1].sqr();
+        }
+        return pows;
+    }
+
+    static std::vector<FF> update_gate_challenges(const FF perturbator_challenge,
+                                                  const std::vector<FF>& gate_challenges,
+                                                  const std::vector<FF>& round_challenges)
+    {
+        auto log_instance_size = gate_challenges.size();
+        std::vector<FF> next_gate_challenges(log_instance_size);
+
+        for (size_t idx = 0; idx < log_instance_size; idx++) {
+            next_gate_challenges[idx] = gate_challenges[idx] + perturbator_challenge * round_challenges[idx];
+        }
+        return next_gate_challenges;
+    }
+
+    std::shared_ptr<Instance> get_accumulator() { return instances[0]; }
+
+    /**
+     * @brief Instatiate the instances and the transcript.
+     *
+     * @param fold_data The data transmitted via the transcript by the prover.
+     */
+    void prepare_for_folding(const std::vector<uint8_t>&);
+
+    /**
+     * @brief Instantiatied the accumulator (i.e. the relaxed instance) from the transcript.
+     *
+     */
+    void receive_accumulator(const std::shared_ptr<Instance>&, const std::string&);
+
+    /**
+     * @brief Process the public data ϕ for the Instances to be folded.
+     *
+     */
+    void receive_and_finalise_instance(const std::shared_ptr<Instance>&, const std::string&);
+
+    /**
+     * @brief Run the folding protocol on the verifier side to establish whether the public data ϕ of the new
+     * accumulator, received from the prover is the same as that produced by the verifier.
+     *
+     */
+    bool verify_folding_proof(const plonk::proof& proof);
+};
+
+extern template class ProtoGalaxyRecursiveVerifier_<
+    proof_system::honk::VerifierInstances_<proof_system::honk::flavor::UltraRecursive_<UltraCircuitBuilder>, 2>>;
+extern template class ProtoGalaxyRecursiveVerifier_<
+    proof_system::honk::VerifierInstances_<proof_system::honk::flavor::UltraRecursive_<GoblinUltraCircuitBuilder>, 2>>;
+extern template class ProtoGalaxyRecursiveVerifier_<
+    proof_system::honk::VerifierInstances_<proof_system::honk::flavor::GoblinUltraRecursive, 2>>;
+} // namespace proof_system::plonk::stdlib::recursion::honk
\ No newline at end of file

From 65c6aeb60f2f79f6c4d94960741a0a750acf8f24 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Wed, 10 Jan 2024 16:08:42 +0000
Subject: [PATCH 74/86] resolve review comments

---
 .../src/barretenberg/common/thread_utils.hpp  |  1 +
 .../cpp/src/barretenberg/flavor/ecc_vm.hpp    |  2 +-
 .../flavor/generated/AvmMini_flavor.hpp       |  2 +-
 .../flavor/generated/Toy_flavor.hpp           |  2 +-
 .../barretenberg/flavor/goblin_translator.hpp |  2 +-
 .../src/barretenberg/flavor/goblin_ultra.hpp  |  2 +-
 .../flavor/goblin_ultra_recursive.hpp         |  2 +-
 .../cpp/src/barretenberg/flavor/ultra.hpp     |  2 +-
 .../barretenberg/flavor/ultra_recursive.hpp   |  2 +-
 .../cpp/src/barretenberg/polynomials/pow.hpp  | 82 +++++++++++++------
 .../src/barretenberg/polynomials/pow.test.cpp |  2 +-
 .../protogalaxy/decider_prover.cpp            |  5 +-
 .../protogalaxy/decider_prover.hpp            |  2 +-
 .../protogalaxy/decider_verifier.cpp          |  1 -
 .../protogalaxy/protogalaxy_prover.cpp        |  1 -
 .../protogalaxy/protogalaxy_prover.hpp        | 12 +--
 .../protogalaxy/protogalaxy_verifier.cpp      |  7 +-
 .../protogalaxy/protogalaxy_verifier.hpp      |  2 +-
 .../cpp/src/barretenberg/relations/utils.hpp  | 16 ++--
 .../verifier/ultra_recursive_verifier.cpp     |  2 +-
 .../verifier/ultra_recursive_verifier.hpp     |  2 +-
 .../sumcheck/instance/instances.hpp           |  4 +-
 .../sumcheck/instance/prover_instance.hpp     |  5 +-
 .../sumcheck/instance/verifier_instance.hpp   |  4 +-
 .../src/barretenberg/sumcheck/sumcheck.hpp    | 11 ++-
 .../barretenberg/sumcheck/sumcheck.test.cpp   | 10 +--
 .../barretenberg/sumcheck/sumcheck_round.hpp  | 15 ++--
 .../sumcheck/sumcheck_round.test.cpp          |  8 +-
 .../ultra_honk/protogalaxy.test.cpp           |  8 +-
 .../barretenberg/ultra_honk/sumcheck.test.cpp |  6 +-
 .../barretenberg/ultra_honk/ultra_prover.cpp  |  2 +-
 .../barretenberg/ultra_honk/ultra_prover.hpp  |  2 +-
 .../ultra_honk/ultra_verifier.cpp             |  2 +-
 .../ultra_honk/ultra_verifier.hpp             |  2 +-
 34 files changed, 130 insertions(+), 100 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/common/thread_utils.hpp b/barretenberg/cpp/src/barretenberg/common/thread_utils.hpp
index 55ee79ff1ab..e6551665479 100644
--- a/barretenberg/cpp/src/barretenberg/common/thread_utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/common/thread_utils.hpp
@@ -1,3 +1,4 @@
+#pragma once
 #include "thread.hpp"
 
 namespace barretenberg::thread_utils {
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ecc_vm.hpp b/barretenberg/cpp/src/barretenberg/flavor/ecc_vm.hpp
index e3537e0f1f8..ede2d100ce8 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ecc_vm.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ecc_vm.hpp
@@ -45,7 +45,7 @@ template <typename CycleGroup_T, typename Curve_T, typename PCS_T> class ECCVMBa
     using CommitmentHandle = typename G1::affine_element;
     using CommitmentKey = pcs::CommitmentKey<Curve>;
     using VerifierCommitmentKey = pcs::VerifierCommitmentKey<Curve>;
-    using AlphaType = FF;
+    using RelationSeparator = FF;
 
     static constexpr size_t NUM_WIRES = 74;
 
diff --git a/barretenberg/cpp/src/barretenberg/flavor/generated/AvmMini_flavor.hpp b/barretenberg/cpp/src/barretenberg/flavor/generated/AvmMini_flavor.hpp
index 1dc4092ac12..e9355c1f3f9 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/generated/AvmMini_flavor.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/generated/AvmMini_flavor.hpp
@@ -33,7 +33,7 @@ class AvmMiniFlavor {
     using CommitmentHandle = G1::affine_element;
     using CommitmentKey = pcs::CommitmentKey<Curve>;
     using VerifierCommitmentKey = pcs::VerifierCommitmentKey<Curve>;
-    using AlphaType = FF;
+    using RelationSeparator = FF;
 
     static constexpr size_t NUM_PRECOMPUTED_ENTITIES = 2;
     static constexpr size_t NUM_WITNESS_ENTITIES = 38;
diff --git a/barretenberg/cpp/src/barretenberg/flavor/generated/Toy_flavor.hpp b/barretenberg/cpp/src/barretenberg/flavor/generated/Toy_flavor.hpp
index 4703b2c87c3..fdb992586b7 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/generated/Toy_flavor.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/generated/Toy_flavor.hpp
@@ -34,7 +34,7 @@ class ToyFlavor {
     using CommitmentHandle = G1::affine_element;
     using CommitmentKey = pcs::CommitmentKey<Curve>;
     using VerifierCommitmentKey = pcs::VerifierCommitmentKey<Curve>;
-    using AlphaType = FF;
+    using RelationSeparator = FF;
 
     static constexpr size_t NUM_PRECOMPUTED_ENTITIES = 1;
     static constexpr size_t NUM_WITNESS_ENTITIES = 16;
diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_translator.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_translator.hpp
index 53fe5e871df..8d70d97653b 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_translator.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_translator.hpp
@@ -34,7 +34,7 @@ class GoblinTranslator {
     using BF = Curve::BaseField;
     using Polynomial = barretenberg::Polynomial<FF>;
     using PolynomialHandle = std::span<FF>;
-    using AlphaType = FF;
+    using RelationSeparator = FF;
 
     // The size of the circuit which is filled with non-zero values for most polynomials. Most relations (everything
     // except for Permutation and GenPermSort) can be evaluated just on the first chunk
diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
index 4934a1a6564..240f8f7552e 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
@@ -80,7 +80,7 @@ class GoblinUltra {
     // subrelation. This
     // is because using powers of alpha would increase the degree of Protogalaxy polynomial $G$ (the combiner) to much.
     static constexpr size_t NUM_SUBRELATIONS = compute_number_of_subrelations<Relations>();
-    using AlphaType = std::array<FF, NUM_SUBRELATIONS - 1>;
+    using RelationSeparator = std::array<FF, NUM_SUBRELATIONS - 1>;
 
     template <size_t NUM_INSTANCES>
     using ProtogalaxyTupleOfTuplesOfUnivariates =
diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
index 094d0460e5c..ee257322187 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
@@ -78,7 +78,7 @@ class GoblinUltraRecursive {
     // subrelation. This is because using powers of alpha would increase the degree of Protogalaxy polynomial $G$ (the
     // combiner) to much.
     static constexpr size_t NUM_SUBRELATIONS = compute_number_of_subrelations<Relations>();
-    using AlphaType = std::array<FF, NUM_SUBRELATIONS - 1>;
+    using RelationSeparator = std::array<FF, NUM_SUBRELATIONS - 1>;
 
     // define the container for storing the univariate contribution from each relation in Sumcheck
     using SumcheckTupleOfTuplesOfUnivariates = decltype(create_sumcheck_tuple_of_tuples_of_univariates<Relations>());
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
index 75f7e3d4cf4..de77c42235d 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
@@ -61,7 +61,7 @@ class Ultra {
     // For instances of this flavour, used in folding, we need a unique sumcheck batching challenge for each
     // subrelation. This is because using powers of alpha would increase the degree of Protogalaxy polynomial $G$ (the
     // combiner) too much.
-    using AlphaType = std::array<FF, NUM_SUBRELATIONS - 1>;
+    using RelationSeparator = std::array<FF, NUM_SUBRELATIONS - 1>;
 
     // BATCHED_RELATION_PARTIAL_LENGTH = algebraic degree of sumcheck relation *after* multiplying by the `pow_zeta`
     // random polynomial e.g. For \sum(x) [A(x) * B(x) + C(x)] * PowZeta(X), relation length = 2 and random relation
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
index e44ed90cd37..19fced27b74 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
@@ -90,7 +90,7 @@ template <typename BuilderType> class UltraRecursive_ {
     // subrelation to avoid increasing the degree of Protogalaxy polynomial $G$ (the
     // combiner) too much.
     static constexpr size_t NUM_SUBRELATIONS = compute_number_of_subrelations<Relations>();
-    using AlphaType = std::array<FF, NUM_SUBRELATIONS - 1>;
+    using RelationSeparator = std::array<FF, NUM_SUBRELATIONS - 1>;
 
     // define the container for storing the univariate contribution from each relation in Sumcheck
     using SumcheckTupleOfTuplesOfUnivariates = decltype(create_sumcheck_tuple_of_tuples_of_univariates<Relations>());
diff --git a/barretenberg/cpp/src/barretenberg/polynomials/pow.hpp b/barretenberg/cpp/src/barretenberg/polynomials/pow.hpp
index 7a95c26f292..a61bc21ba85 100644
--- a/barretenberg/cpp/src/barretenberg/polynomials/pow.hpp
+++ b/barretenberg/cpp/src/barretenberg/polynomials/pow.hpp
@@ -1,23 +1,28 @@
 #pragma once
-
+#include "barretenberg/common/thread.hpp"
+#include "barretenberg/common/thread_utils.hpp"
 #include <cstddef>
 #include <vector>
 namespace barretenberg {
 
 /**
  * @brief Succinct representation of the `pow` polynomial that can be partially evaluated variable-by-variable.
- * pow(X_0,X_1,..,X_d) = \prod_{0≤l<d} ((1−X_l) + X_l⋅β_l) for a vector {β_0,...,β_{d-1}}
+ * pow_{\vec{β}}(X_0,X_1,..,X_{d-1}) = \prod_{0≤l<d} ((1−X_l) + X_l⋅β_l) for a vector {β_0,...,β_{d-1}} used in
+sumcheck.
  *
  * @details Let
  * - d be the number of variables
  * - l be the current Sumcheck round ( l ∈ {0, …, d-1} )
- * - u_0, ..., u_l-1 the challenges sent by the verifier in rounds 0 to l-1.
+ * - u_0, ..., u_{l-1} the challenges sent by the verifier in rounds 0 to l-1.
+ *
+ * - pow_\vec{β}(X) = ∏_{0≤l<d} ((1−X_l) + X_l⋅β_l) is the multilinear polynomial whose evaluation at the i-th index
+ *   of the full hypercube equals pow_\vec{β}(i) = ∏_{j ∈ bin(i), j = 1} β_j. More explicitly, the product contains the
+elements of \vec{β} whose
+ *  indexes j represent bits set to 1 in the binary representation of i.
  *
- * - pow(X) = ∏_{0≤l<d} ((1−X_l) + X_l⋅β_l) is the multilinear polynomial whose evaluation at the i-th index
- *   of the full hypercube, equals ∏ β_j where j are the bits set to 1 in the binary representation of i
- *   We can also see it as the multi-linear extension of the vector \vec{β}.
+ *  We can also see it as the multi-linear extension of the vector \vec{β}.
  *
- * - At round l, we iterate over all remaining vertices (i_{l+1}, ..., i_{d-1}) ∈ {0,1}^{d-l-1}.
+ * - At round l, we iterate over a boolean hypercube of dimension (d-l).
  *   Let i = ∑_{l<k<d} i_k ⋅ 2^{k-(l+1)} be the index of the current edge over which we are evaluating the relation.
  *   We define the edge univariate for the pow polynomial as powˡᵢ( X_l ) and it can be represented as:
  *
@@ -28,12 +33,15 @@ namespace barretenberg {
  *                             ⋅( (1−X_l) + X_l⋅β_l )
  *                    ∏_{l<k<d} ( (1-i_k) + i_k⋅β_k )
  *
- *  Note that as we iterate over the remaining vertices(i_{l+1}, ..., i_{d-1}) ∈ {0,1}^{d-l-1}, the subproducts
- * ∏_{l<k<d} ( (1-i_k) + i_k⋅β_k ) represents the terms of pow(\vec{β}) that do not contain β_0,...,β_l. These are
- * periodic in pow(\vec{β}), their periodicity being 2*(l+1).
+ * Product ∏_{0≤k<l} ( (1-u_k) + u_k⋅β_k ) gets updated at each round of sumcheck and represents the
+ * partial_evaluation_result in the implementation. This is the pow polynomial, partially evaluated in the first l-1
+ * variables as (X_{0}, ..., X_{l-1}) = (u_{0}, ...,u_{l-1}).
+ *
+ * As we iterate over the other points in the boolean hypercube (i_{l+1}, ..., i_{d-1}) ∈ {0,1}^{d-l-1},
+ * the subproducts
+ * ∏_{l<k<d} ( (1-i_k) + i_k⋅β_k ) represent the terms of pow(\vec{β}) that do not contain β_0,...,β_l. These appear in
+ * the set {pow_\vec{β}(i)| i =0,..,2^{d}-1} at indices 2^{l+1} * p where p ≥ 1 and 2^{l+1} * p < 2^d
  *
- * This is the pow polynomial,partially evaluated in the first l-1 variables as (X_{0}, ..., X_{l-1}) = (u_{0}, ...,
- * u_{l-1}).
  *
  * - Sˡᵢ( X_l ) is the univariate of the full relation at edge pair i
  * i.e. it is the alpha-linear-combination of the relations evaluated in the edge at index i.
@@ -64,14 +72,23 @@ namespace barretenberg {
  */
 
 template <typename FF> struct PowPolynomial {
+
+    // \vec{β} = {β_0, β_1,.., β_{d-1}}
     std::vector<FF> betas;
+
+    // The values of pow_\vec{β}(i) for i=0,..,2^d - 1 for the given \vec{β}
     std::vector<FF> pow_betas;
+
+    // At round l of sumcheck this will point to the l-th element in \vec{β}
     size_t current_element_idx = 0;
+
+    // At round l of sumcheck, the periodicity represents the fixed interval at which elements not containing either of
+    // β_0,..,β_l appear in pow_betas
     size_t periodicity = 2;
 
-    // The constant c_l obtained by partially evaluating one variable in the power polynomial at each round. At the
+    // The value c_l obtained by partially evaluating one variable in the power polynomial at each round. At the
     // end of round l in the sumcheck protocol, variable X_l is replaced by a verifier challenge u_l. The partial
-    // evaluation constant is updated to represent pow(u_0,.., u_l) = \prod_{0 ≤ k < l} ( (1-u_k) + u_k⋅β_k).
+    // evaluation result is updated to represent pow(u_0,.., u_{l-1}) = \prod_{0 ≤ k < l} ( (1-u_k) + u_k⋅β_k).
     FF partial_evaluation_result = FF(1);
 
     explicit PowPolynomial(const std::vector<FF>& betas)
@@ -88,7 +105,7 @@ template <typename FF> struct PowPolynomial {
     FF univariate_eval(FF challenge) const { return (FF(1) + (challenge * (betas[current_element_idx] - FF(1)))); };
 
     /**
-     * @brief Parially evaluate the pow polynomial in X_l and updating the constant c_l -> c_{l+1}.
+     * @brief Parially evaluate the pow polynomial in X_l and updating the value c_l -> c_{l+1}.
      *
      * @param challenge l-th verifier challenge u_l
      */
@@ -101,22 +118,37 @@ template <typename FF> struct PowPolynomial {
     }
 
     /**
-     * @brief Given \vec{β} = {β_0,...,β_{d-1}} compute pow_i(\vec{β}) for i=0,...,2^d
+     * @brief Given \vec{β} = {β_0,...,β_{d-1}} compute pow_\vec{β}(i) for i=0,...,2^{d}-1
      *
      */
-    void compute_pow_polynomial_at_values()
+    void compute_values()
     {
         size_t pow_size = 1 << betas.size();
-        pow_betas = std::vector<FF>(pow_size, 0);
-        for (size_t i = 0; i < pow_size; i++) {
-            auto res = FF(1);
-            for (size_t j = i, beta_idx = 0; j > 0; j >>= 1, beta_idx++) {
-                if ((j & 1) == 1) {
-                    res *= betas[beta_idx];
+        pow_betas = std::vector<FF>(pow_size);
+
+        // Determine number of threads for multithreading.
+        // Note: Multithreading is "on" for every round but we reduce the number of threads from the max available based
+        // on a specified minimum number of iterations per thread. This eventually leads to the use of a single thread.
+        // For now we use a power of 2 number of threads simply to ensure the round size is evenly divided.
+        size_t max_num_threads = get_num_cpus_pow2(); // number of available threads (power of 2)
+        size_t min_iterations_per_thread = 1 << 6; // min number of iterations for which we'll spin up a unique thread
+        size_t desired_num_threads = pow_size / min_iterations_per_thread;
+        size_t num_threads = std::min(desired_num_threads, max_num_threads); // fewer than max if justified
+        num_threads = num_threads > 0 ? num_threads : 1;                     // ensure num threads is >= 1
+        size_t iterations_per_thread = pow_size / num_threads;               // actual iterations per thread
+        parallel_for(num_threads, [&](size_t thread_idx) {
+            size_t start = thread_idx * iterations_per_thread;
+            size_t end = (thread_idx + 1) * iterations_per_thread;
+            for (size_t i = start; i < end; i++) {
+                auto res = FF(1);
+                for (size_t j = i, beta_idx = 0; j > 0; j >>= 1, beta_idx++) {
+                    if ((j & 1) == 1) {
+                        res *= betas[beta_idx];
+                    }
                 }
+                pow_betas[i] = res;
             }
-            pow_betas[i] = res;
-        }
+        });
     }
 };
 } // namespace barretenberg
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/polynomials/pow.test.cpp b/barretenberg/cpp/src/barretenberg/polynomials/pow.test.cpp
index e45ebfb2920..f73edfeea0d 100644
--- a/barretenberg/cpp/src/barretenberg/polynomials/pow.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/polynomials/pow.test.cpp
@@ -35,7 +35,7 @@ TEST(PowPolynomial, PowPolynomialsOnPowers)
 {
     auto betas = std::vector<FF>{ 2, 4, 16 };
     auto pow = PowPolynomial(betas);
-    pow.compute_pow_polynomial_at_values();
+    pow.compute_values();
     auto expected_values = std::vector<FF>{ 1, 2, 4, 8, 16, 32, 64, 128 };
     EXPECT_EQ(expected_values, pow.pow_betas);
 }
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
index b1344796302..eea53966aa6 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.cpp
@@ -6,7 +6,7 @@ namespace proof_system::honk {
 /**
  * Create DeciderProver_ from an accumulator.
  *
- * @param accumulator Relaxed instance (ϕ, ω' \vec{β*}, e*) whose proof we want to generate, produced by Protogalaxy
+ * @param accumulator Relaxed instance (ϕ, ω, \vec{β}, e) whose proof we want to generate, produced by Protogalaxy
  * folding prover
  *
  * @tparam a type of UltraFlavor
@@ -21,7 +21,8 @@ DeciderProver_<Flavor>::DeciderProver_(const std::shared_ptr<Instance>& inst,
 {}
 
 /**
- * @brief Add  ϕ, \vec{β*}, e* to the transcript.
+ * @brief Add  ϕ, \vec{β}, e to the transcript. These are produced in the last round of folding that was carried out
+ * before deciding.
  */
 template <UltraFlavor Flavor> void DeciderProver_<Flavor>::execute_preamble_round()
 {
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.hpp
index 2f9aee6c4dd..b11af2419a6 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_prover.hpp
@@ -21,7 +21,7 @@ template <UltraFlavor Flavor> class DeciderProver_ {
     using Curve = typename Flavor::Curve;
     using Instance = ProverInstance_<Flavor>;
     using Transcript = typename Flavor::Transcript;
-    using AlphaType = typename Flavor::AlphaType;
+    using RelationSeparator = typename Flavor::RelationSeparator;
 
   public:
     explicit DeciderProver_(const std::shared_ptr<Instance>&,
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
index 7dd44d52b5f..8b533659270 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/decider_verifier.cpp
@@ -90,7 +90,6 @@ template <typename Flavor> bool DeciderVerifier_<Flavor>::verify_proof(const plo
 
     // If Sumcheck did not verify, return false
     if (sumcheck_verified.has_value() && !sumcheck_verified.value()) {
-        info("at sumcheck");
         return false;
     }
 
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index 6c2440091d0..e735ff8cab9 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -179,7 +179,6 @@ std::shared_ptr<typename ProverInstances::Instance> ProtoGalaxyProver_<ProverIns
     for (auto& polynomial : acc_prover_polynomials.get_all()) {
         polynomial = typename Flavor::Polynomial(instances[0]->instance_size);
     }
-    // WORKTODO return storage
 
     // Fold the prover polynomials
     auto acc_poly_views = acc_prover_polynomials.get_all();
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
index 50842a1f58b..0270afc5b24 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.hpp
@@ -23,8 +23,8 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     using RowEvaluations = typename Flavor::AllValues;
     using ProverPolynomials = typename Flavor::ProverPolynomials;
     using Relations = typename Flavor::Relations;
-    using AlphaType = typename Flavor::AlphaType;
-    using CombinedAlphaType = typename ProverInstances::AlphaType;
+    using RelationSeparator = typename Flavor::RelationSeparator;
+    using CombinedRelationSeparator = typename ProverInstances::RelationSeparator;
     using VerificationKey = typename Flavor::VerificationKey;
     using CommitmentKey = typename Flavor::CommitmentKey;
     using WitnessCommitments = typename Flavor::WitnessCommitments;
@@ -132,7 +132,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
      * each subrelation is independently valid in Honk - from the Plonk paper, DO NOT confuse with α in ProtoGalaxy),
      */
     static std::vector<FF> compute_full_honk_evaluations(const ProverPolynomials& instance_polynomials,
-                                                         const AlphaType& alpha,
+                                                         const RelationSeparator& alpha,
                                                          const RelationParameters<FF>& relation_parameters)
     {
         auto instance_size = instance_polynomials.get_polynomial_size();
@@ -273,7 +273,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
     ExtendedUnivariateWithRandomization compute_combiner(const ProverInstances& instances, PowPolynomial<FF>& pow_betas)
     {
         size_t common_instance_size = instances[0]->instance_size;
-        pow_betas.compute_pow_polynomial_at_values();
+        pow_betas.compute_values();
         // Determine number of threads for multithreading.
         // Note: Multithreading is "on" for every round but we reduce the number of threads from the max available based
         // on a specified minimum number of iterations per thread. This eventually leads to the use of a single thread.
@@ -311,7 +311,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
                 accumulate_relation_univariates(
                     thread_univariate_accumulators[thread_idx],
                     extended_univariates[thread_idx],
-                    instances.relation_parameters, // these para meters have already been folded
+                    instances.relation_parameters, // these parameters have already been folded
                     pow_challenge);
             }
         });
@@ -324,7 +324,7 @@ template <class ProverInstances_> class ProtoGalaxyProver_ {
         return batch_over_relations(univariate_accumulators, instances.alphas);
     }
     static ExtendedUnivariateWithRandomization batch_over_relations(TupleOfTuplesOfUnivariates& univariate_accumulators,
-                                                                    const CombinedAlphaType& alpha)
+                                                                    const CombinedRelationSeparator& alpha)
     {
 
         // First relation does not get multiplied by a batching challenge
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index 6a340f07015..871d4ad23b8 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -217,10 +217,8 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
         el_idx++;
     }
 
-    AlphaType expected_alphas;
-    size_t alpha_idx = 0;
-    for (auto& alpha : expected_alphas) {
-        alpha = FF(0);
+    for (size_t alpha_idx = 0; alpha_idx < NUM_SUBRELATIONS - 1; alpha_idx++) {
+        FF alpha(0);
         size_t instance_idx = 0;
         for (auto& instance : instances) {
             alpha += instance->alphas[alpha_idx] * lagranges[instance_idx];
@@ -228,7 +226,6 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
         }
         auto next_alpha = transcript->template receive_from_prover<FF>("next_alpha_" + std::to_string(alpha_idx));
         verified = verified & (alpha == next_alpha);
-        alpha_idx++;
     }
     auto expected_parameters = proof_system::RelationParameters<FF>{};
     for (size_t inst_idx = 0; inst_idx < VerifierInstances::NUM; inst_idx++) {
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
index 9c2660699ae..b1ac218c32f 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.hpp
@@ -17,7 +17,7 @@ template <class VerifierInstances> class ProtoGalaxyVerifier_ {
     using VerificationKey = typename Flavor::VerificationKey;
     using WitnessCommitments = typename Flavor::WitnessCommitments;
     using CommitmentLabels = typename Flavor::CommitmentLabels;
-    using AlphaType = typename Flavor::AlphaType;
+    using RelationSeparator = typename Flavor::RelationSeparator;
 
     static constexpr size_t NUM_SUBRELATIONS = Flavor::NUM_SUBRELATIONS;
 
diff --git a/barretenberg/cpp/src/barretenberg/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
index 150223b6c4a..894a28b9d72 100644
--- a/barretenberg/cpp/src/barretenberg/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
@@ -13,7 +13,7 @@ template <typename Flavor> class RelationUtils {
     using Relations = typename Flavor::Relations;
     using PolynomialEvaluations = typename Flavor::AllValues;
     using RelationEvaluations = typename Flavor::TupleOfArraysOfValues;
-    using AlphaType = typename Flavor::AlphaType;
+    using RelationSeparator = typename Flavor::RelationSeparator;
 
     static constexpr size_t NUM_RELATIONS = Flavor::NUM_RELATIONS;
     static constexpr size_t NUM_SUBRELATIONS = Flavor::NUM_SUBRELATIONS;
@@ -71,7 +71,7 @@ template <typename Flavor> class RelationUtils {
      * scaled)
      * @param current_scalar power of the challenge
      */
-    static void scale_univariates(auto& tuple, const AlphaType& challenges, FF& current_scalar)
+    static void scale_univariates(auto& tuple, const RelationSeparator& challenges, FF& current_scalar)
         requires proof_system::IsFoldingFlavor<Flavor>
     {
         size_t idx = 0;
@@ -91,7 +91,7 @@ template <typename Flavor> class RelationUtils {
      * @param challenge
      * @param current_scalar power of the challenge
      */
-    static void scale_univariates(auto& tuple, const AlphaType& challenge, FF& current_scalar)
+    static void scale_univariates(auto& tuple, const RelationSeparator& challenge, FF& current_scalar)
         requires(!proof_system::IsFoldingFlavor<Flavor>)
     {
         auto scale_by_consecutive_powers_of_challenge = [&]<size_t, size_t>(auto& element) {
@@ -113,8 +113,7 @@ template <typename Flavor> class RelationUtils {
     template <typename... T>
     static constexpr void add_tuples(std::tuple<T...>& tuple_1, const std::tuple<T...>& tuple_2)
     {
-        auto add_tuples_helper = [&]<std::size_t... I>(std::index_sequence<I...>)
-        {
+        auto add_tuples_helper = [&]<std::size_t... I>(std::index_sequence<I...>) {
             ((std::get<I>(tuple_1) += std::get<I>(tuple_2)), ...);
         };
 
@@ -189,7 +188,10 @@ template <typename Flavor> class RelationUtils {
      * scaled)
      * @param result Batched result
      */
-    static void scale_and_batch_elements(auto& tuple, const AlphaType& challenges, FF current_scalar, FF& result)
+    static void scale_and_batch_elements(auto& tuple,
+                                         const RelationSeparator& challenges,
+                                         FF current_scalar,
+                                         FF& result)
         requires proof_system::IsFoldingFlavor<Flavor>
     {
         size_t idx = 0;
@@ -208,7 +210,7 @@ template <typename Flavor> class RelationUtils {
      * @brief Scale elements by consecutive powers of the challenge then sum
      * @param result Batched result
      */
-    static void scale_and_batch_elements(auto& tuple, const AlphaType& challenge, FF current_scalar, FF& result)
+    static void scale_and_batch_elements(auto& tuple, const RelationSeparator& challenge, FF current_scalar, FF& result)
         requires(!proof_system::IsFoldingFlavor<Flavor>)
     {
         auto scale_by_challenge_and_accumulate = [&](auto& element) {
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp
index 941f3385b66..fb9719d6cde 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.cpp
@@ -104,7 +104,7 @@ std::array<typename Flavor::GroupElement, 2> UltraRecursiveVerifier_<Flavor>::ve
     // multivariate evaluations at u
     const size_t log_circuit_size = numeric::get_msb(static_cast<uint32_t>(circuit_size.get_value()));
     auto sumcheck = Sumcheck(log_circuit_size, transcript);
-    AlphaType alpha;
+    RelationSeparator alpha;
     for (size_t idx = 0; idx < alpha.size(); idx++) {
         alpha[idx] = transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
     }
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.hpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.hpp
index 4fb00993194..6077fe241ef 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.hpp
@@ -15,7 +15,7 @@ template <typename Flavor> class UltraRecursiveVerifier_ {
     using NativeVerificationKey = typename Flavor::NativeVerificationKey;
     using VerifierCommitmentKey = typename Flavor::VerifierCommitmentKey;
     using Builder = typename Flavor::CircuitBuilder;
-    using AlphaType = typename Flavor::AlphaType;
+    using RelationSeparator = typename Flavor::RelationSeparator;
     using PairingPoints = std::array<GroupElement, 2>;
 
     explicit UltraRecursiveVerifier_(Builder* builder,
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
index d4a78800349..405f49d8463 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/instances.hpp
@@ -18,10 +18,10 @@ template <typename Flavor_, size_t NUM_> struct ProverInstances_ {
     static constexpr size_t EXTENDED_LENGTH = (Flavor::MAX_TOTAL_RELATION_LENGTH - 1) * (NUM - 1) + 1;
     static constexpr size_t BATCHED_EXTENDED_LENGTH = (Flavor::MAX_TOTAL_RELATION_LENGTH - 1 + NUM - 1) * (NUM - 1) + 1;
     using RelationParameters = proof_system::RelationParameters<Univariate<FF, EXTENDED_LENGTH>>;
-    using AlphaType = std::array<Univariate<FF, BATCHED_EXTENDED_LENGTH>, NUM_SUBRELATIONS - 1>;
+    using RelationSeparator = std::array<Univariate<FF, BATCHED_EXTENDED_LENGTH>, NUM_SUBRELATIONS - 1>;
     ArrayType _data;
     RelationParameters relation_parameters;
-    AlphaType alphas;
+    RelationSeparator alphas;
     std::vector<FF> next_gate_challenges;
 
     std::shared_ptr<Instance> const& operator[](size_t idx) const { return _data[idx]; }
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
index 878e629d6f2..d7f14f6c9f1 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/prover_instance.hpp
@@ -26,7 +26,7 @@ template <class Flavor> class ProverInstance_ {
     using Polynomial = typename Flavor::Polynomial;
     using WitnessCommitments = typename Flavor::WitnessCommitments;
     using CommitmentLabels = typename Flavor::CommitmentLabels;
-    using AlphaType = typename Flavor::AlphaType;
+    using RelationSeparator = typename Flavor::RelationSeparator;
 
   public:
     std::shared_ptr<ProvingKey> proving_key;
@@ -36,7 +36,6 @@ template <class Flavor> class ProverInstance_ {
     WitnessCommitments witness_commitments;
     CommitmentLabels commitment_labels;
 
-    // This is meh here
     std::array<Polynomial, 4> sorted_polynomials;
 
     // The number of public inputs has to be the same for all instances because they are
@@ -46,7 +45,7 @@ template <class Flavor> class ProverInstance_ {
     // non-zero  for Instances constructed from circuits, this concept doesn't exist for accumulated
     // instances
     size_t pub_inputs_offset = 0;
-    AlphaType alphas;
+    RelationSeparator alphas;
     proof_system::RelationParameters<FF> relation_parameters;
     std::vector<uint32_t> recursive_proof_public_input_indices;
 
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
index 304abca199c..23faaf87875 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/instance/verifier_instance.hpp
@@ -9,7 +9,7 @@ template <class Flavor> class VerifierInstance_ {
     using VerificationKey = typename Flavor::VerificationKey;
     using WitnessCommitments = typename Flavor::WitnessCommitments;
     using CommitmentLabels = typename Flavor::CommitmentLabels;
-    using AlphaType = typename Flavor::AlphaType;
+    using RelationSeparator = typename Flavor::RelationSeparator;
 
     std::shared_ptr<VerificationKey> verification_key;
     std::vector<FF> public_inputs;
@@ -18,7 +18,7 @@ template <class Flavor> class VerifierInstance_ {
     size_t instance_size;
     size_t log_instance_size;
     RelationParameters<FF> relation_parameters;
-    AlphaType alphas;
+    RelationSeparator alphas;
     bool is_accumulator = false;
 
     // The folding parameters (\vec{β}, e) which are set for accumulators (i.e. relaxed instances).
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
index 6416ac3166e..9971f0447ea 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.hpp
@@ -16,7 +16,7 @@ template <typename Flavor> class SumcheckProver {
     using ClaimedEvaluations = typename Flavor::AllValues;
     using Transcript = typename Flavor::Transcript;
     using Instance = ProverInstance_<Flavor>;
-    using AlphaType = typename Flavor::AlphaType;
+    using RelationSeparator = typename Flavor::RelationSeparator;
 
     const size_t multivariate_n;
     const size_t multivariate_d;
@@ -84,12 +84,12 @@ template <typename Flavor> class SumcheckProver {
      */
     SumcheckOutput<Flavor> prove(ProverPolynomials& full_polynomials,
                                  const proof_system::RelationParameters<FF>& relation_parameters,
-                                 const AlphaType alpha,
+                                 const RelationSeparator alpha,
                                  const std::vector<FF>& gate_challenges)
     {
 
         barretenberg::PowPolynomial<FF> pow_univariate(gate_challenges);
-        pow_univariate.compute_pow_polynomial_at_values();
+        pow_univariate.compute_values();
 
         std::vector<FF> multivariate_challenge;
         multivariate_challenge.reserve(multivariate_d);
@@ -178,13 +178,12 @@ template <typename Flavor> class SumcheckVerifier {
     using FF = typename Flavor::FF;
     using ClaimedEvaluations = typename Flavor::AllValues;
     using Transcript = typename Flavor::Transcript;
-    using AlphaType = typename Flavor::AlphaType;
+    using RelationSeparator = typename Flavor::RelationSeparator;
 
     static constexpr size_t BATCHED_RELATION_PARTIAL_LENGTH = Flavor::BATCHED_RELATION_PARTIAL_LENGTH;
     static constexpr size_t NUM_POLYNOMIALS = Flavor::NUM_ALL_ENTITIES;
 
     const size_t multivariate_d;
-    FF targegt_sum = 0;
     std::shared_ptr<Transcript> transcript;
     SumcheckVerifierRound<Flavor> round;
 
@@ -204,7 +203,7 @@ template <typename Flavor> class SumcheckVerifier {
      * @param transcript
      */
     SumcheckOutput<Flavor> verify(const proof_system::RelationParameters<FF>& relation_parameters,
-                                  AlphaType alpha,
+                                  RelationSeparator alpha,
                                   const std::vector<FF>& gate_challenges)
     {
         bool verified(true);
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp
index 0801391d9d5..6f83fbdd2e6 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck.test.cpp
@@ -17,7 +17,7 @@ using namespace proof_system::honk::sumcheck;
 using Flavor = proof_system::honk::flavor::Ultra;
 using FF = typename Flavor::FF;
 using ProverPolynomials = typename Flavor::ProverPolynomials;
-using AlphaType = Flavor::AlphaType;
+using RelationSeparator = Flavor::RelationSeparator;
 const size_t NUM_POLYNOMIALS = Flavor::NUM_ALL_ENTITIES;
 
 namespace test_sumcheck_round {
@@ -67,7 +67,7 @@ TEST_F(SumcheckTests, PolynomialNormalization)
     auto transcript = Flavor::Transcript::prover_init_empty();
 
     auto sumcheck = SumcheckProver<Flavor>(multivariate_n, transcript);
-    AlphaType alpha;
+    RelationSeparator alpha;
     for (size_t idx = 0; idx < alpha.size(); idx++) {
         alpha[idx] = transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
     }
@@ -139,7 +139,7 @@ TEST_F(SumcheckTests, Prover)
 
     auto sumcheck = SumcheckProver<Flavor>(multivariate_n, transcript);
 
-    AlphaType alpha;
+    RelationSeparator alpha;
     for (size_t idx = 0; idx < alpha.size(); idx++) {
         alpha[idx] = transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
     }
@@ -222,7 +222,7 @@ TEST_F(SumcheckTests, ProverAndVerifierSimple)
         auto prover_transcript = Flavor::Transcript::prover_init_empty();
         auto sumcheck_prover = SumcheckProver<Flavor>(multivariate_n, prover_transcript);
 
-        AlphaType prover_alpha;
+        RelationSeparator prover_alpha;
         for (size_t idx = 0; idx < prover_alpha.size(); idx++) {
             prover_alpha[idx] = prover_transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
         }
@@ -236,7 +236,7 @@ TEST_F(SumcheckTests, ProverAndVerifierSimple)
         auto verifier_transcript = Flavor::Transcript::verifier_init_empty(prover_transcript);
 
         auto sumcheck_verifier = SumcheckVerifier<Flavor>(multivariate_d, verifier_transcript);
-        AlphaType verifier_alpha;
+        RelationSeparator verifier_alpha;
         for (size_t idx = 0; idx < verifier_alpha.size(); idx++) {
             verifier_alpha[idx] = verifier_transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
         }
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
index 44ed5a4b975..0d6bc37092c 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.hpp
@@ -55,7 +55,7 @@ template <typename Flavor> class SumcheckProverRound {
     using Utils = barretenberg::RelationUtils<Flavor>;
     using Relations = typename Flavor::Relations;
     using SumcheckTupleOfTuplesOfUnivariates = typename Flavor::SumcheckTupleOfTuplesOfUnivariates;
-    using AlphaType = typename Flavor::AlphaType;
+    using RelationSeparator = typename Flavor::RelationSeparator;
 
   public:
     using FF = typename Flavor::FF;
@@ -105,10 +105,11 @@ template <typename Flavor> class SumcheckProverRound {
         ProverPolynomialsOrPartiallyEvaluatedMultivariates& polynomials,
         const proof_system::RelationParameters<FF>& relation_parameters,
         const barretenberg::PowPolynomial<FF>& pow_polynomial,
-        const AlphaType alpha)
+        const RelationSeparator alpha)
     {
-        // Compute the constant contribution of pow polynomials for each edge as the product of c_l and the elements of
-        // pow(\vec{β}) not containing β_0,..., β_l
+        // Compute the constant contribution of pow polynomials for each edge. This is  the product of the partial
+        // evaluation result c_l (i.e. pow(u_0,...,u_{l-1})) where u_0,...,u_{l-1} are the verifier challenges from
+        // previous rounds) and the elements of pow(\vec{β}) not containing β_0,..., β_l.
         std::vector<FF> pow_challenges(round_size >> 1);
         pow_challenges[0] = pow_polynomial.partial_evaluation_result;
         for (size_t i = 1; i < (round_size >> 1); ++i) {
@@ -167,7 +168,7 @@ template <typename Flavor> class SumcheckProverRound {
      */
     template <typename ExtendedUnivariate, typename ContainerOverSubrelations>
     static ExtendedUnivariate batch_over_relations(ContainerOverSubrelations& univariate_accumulators,
-                                                   const AlphaType& challenge,
+                                                   const RelationSeparator& challenge,
                                                    const barretenberg::PowPolynomial<FF>& pow_polynomial)
     {
         auto running_challenge = FF(1);
@@ -255,7 +256,7 @@ template <typename Flavor> class SumcheckVerifierRound {
     using Utils = barretenberg::RelationUtils<Flavor>;
     using Relations = typename Flavor::Relations;
     using TupleOfArraysOfValues = typename Flavor::TupleOfArraysOfValues;
-    using AlphaType = typename Flavor::AlphaType;
+    using RelationSeparator = typename Flavor::RelationSeparator;
 
   public:
     using FF = typename Flavor::FF;
@@ -325,7 +326,7 @@ template <typename Flavor> class SumcheckVerifierRound {
     FF compute_full_honk_relation_purported_value(ClaimedEvaluations purported_evaluations,
                                                   const proof_system::RelationParameters<FF>& relation_parameters,
                                                   const barretenberg::PowPolynomial<FF>& pow_polynomial,
-                                                  const AlphaType alpha)
+                                                  const RelationSeparator alpha)
     {
         Utils::template accumulate_relation_evaluations<>(
             purported_evaluations, relation_evaluations, relation_parameters, pow_polynomial.partial_evaluation_result);
diff --git a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.test.cpp b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.test.cpp
index e6a6ca28121..c2e62341a53 100644
--- a/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/sumcheck/sumcheck_round.test.cpp
@@ -24,7 +24,7 @@ TEST(SumcheckRound, SumcheckTupleOfTuplesOfUnivariates)
 {
     using Flavor = proof_system::honk::flavor::Ultra;
     using FF = typename Flavor::FF;
-    using AlphaType = typename Flavor::AlphaType;
+    using RelationSeparator = typename Flavor::RelationSeparator;
 
     // Define three linear univariates of different sizes
     Univariate<FF, 3> univariate_1({ 1, 2, 3 });
@@ -36,7 +36,7 @@ TEST(SumcheckRound, SumcheckTupleOfTuplesOfUnivariates)
     auto tuple_of_tuples = std::make_tuple(std::make_tuple(univariate_1), std::make_tuple(univariate_2, univariate_3));
 
     // Use scale_univariate_accumulators to scale by challenge powers
-    AlphaType challenge = {};
+    RelationSeparator challenge = {};
     challenge[0] = 5;
     challenge[1] = challenge[0].sqr();
     FF running_challenge = 1;
@@ -76,7 +76,7 @@ TEST(SumcheckRound, TuplesOfEvaluationArrays)
     using Flavor = proof_system::honk::flavor::Ultra;
     using Utils = barretenberg::RelationUtils<Flavor>;
     using FF = typename Flavor::FF;
-    using AlphaType = typename Flavor::AlphaType;
+    using RelationSeparator = typename Flavor::RelationSeparator;
 
     // Define two arrays of arbitrary elements
     std::array<FF, 1> evaluations_1 = { 4 };
@@ -87,7 +87,7 @@ TEST(SumcheckRound, TuplesOfEvaluationArrays)
 
     // Use scale_and_batch_elements to scale by challenge powers
     FF running_challenge = 1;
-    AlphaType challenge = {};
+    RelationSeparator challenge = {};
     challenge[0] = 5;
     challenge[1] = challenge[0].sqr();
     FF result = 0;
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
index 5e89b2e85ba..f2f14ab43b4 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
@@ -68,7 +68,7 @@ void check_accumulator_target_sum_manual(std::shared_ptr<Instance>& accumulator,
         accumulator->prover_polynomials, accumulator->alphas, accumulator->relation_parameters);
     // Construct pow(\vec{betas*}) as in the paper
     auto expected_pows = PowPolynomial(accumulator->gate_challenges);
-    expected_pows.compute_pow_polynomial_at_values();
+    expected_pows.compute_values();
 
     // Compute the corresponding target sum and create a dummy accumulator
     auto expected_target_sum = FF(0);
@@ -140,7 +140,7 @@ TEST_F(ProtoGalaxyTests, PerturbatorCoefficients)
 
 TEST_F(ProtoGalaxyTests, PerturbatorPolynomial)
 {
-    using AlphaType = Flavor::AlphaType;
+    using RelationSeparator = Flavor::RelationSeparator;
     const size_t log_instance_size(3);
     const size_t instance_size(1 << log_instance_size);
 
@@ -150,7 +150,7 @@ TEST_F(ProtoGalaxyTests, PerturbatorPolynomial)
     }
     auto full_polynomials = construct_ultra_full_polynomials(random_polynomials);
     auto relation_parameters = proof_system::RelationParameters<FF>::get_random();
-    AlphaType alphas;
+    RelationSeparator alphas;
     for (auto& alpha : alphas) {
         alpha = FF::random_element();
     }
@@ -164,7 +164,7 @@ TEST_F(ProtoGalaxyTests, PerturbatorPolynomial)
 
     // Construct pow(\vec{betas}) as in the paper
     auto pow_beta = PowPolynomial(betas);
-    pow_beta.compute_pow_polynomial_at_values();
+    pow_beta.compute_values();
 
     // Compute the corresponding target sum and create a dummy accumulator
     auto target_sum = FF(0);
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/sumcheck.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/sumcheck.test.cpp
index 545a587c28a..09815ab23c8 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/sumcheck.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/sumcheck.test.cpp
@@ -36,7 +36,7 @@ TEST_F(SumcheckTestsRealCircuit, Ultra)
     using Flavor = flavor::Ultra;
     using FF = typename Flavor::FF;
     using Transcript = typename Flavor::Transcript;
-    using AlphaType = typename Flavor::AlphaType;
+    using RelationSeparator = typename Flavor::RelationSeparator;
 
     // Create a composer and a dummy circuit with a few gates
     auto builder = proof_system::UltraCircuitBuilder();
@@ -165,7 +165,7 @@ TEST_F(SumcheckTestsRealCircuit, Ultra)
     auto circuit_size = instance->proving_key->circuit_size;
     auto log_circuit_size = numeric::get_msb(circuit_size);
 
-    AlphaType prover_alphas;
+    RelationSeparator prover_alphas;
     for (size_t idx = 0; idx < prover_alphas.size(); idx++) {
         prover_alphas[idx] = prover_transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
     }
@@ -183,7 +183,7 @@ TEST_F(SumcheckTestsRealCircuit, Ultra)
     auto verifier_transcript = Transcript::verifier_init_empty(prover_transcript);
 
     auto sumcheck_verifier = SumcheckVerifier<Flavor>(log_circuit_size, verifier_transcript);
-    AlphaType verifier_alphas;
+    RelationSeparator verifier_alphas;
     for (size_t idx = 0; idx < verifier_alphas.size(); idx++) {
         verifier_alphas[idx] = verifier_transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
     }
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp
index c9c7f258972..44b95ce6bf4 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.cpp
@@ -149,7 +149,7 @@ template <UltraFlavor Flavor> void UltraProver_<Flavor>::execute_relation_check_
     using Sumcheck = sumcheck::SumcheckProver<Flavor>;
     auto circuit_size = instance->proving_key->circuit_size;
     auto sumcheck = Sumcheck(circuit_size, transcript);
-    AlphaType alphas;
+    RelationSeparator alphas;
     for (size_t idx = 0; idx < alphas.size(); idx++) {
         alphas[idx] = transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
     }
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.hpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.hpp
index 06d9c324555..2ba793fa0bc 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.hpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_prover.hpp
@@ -20,7 +20,7 @@ template <UltraFlavor Flavor> class UltraProver_ {
     using Curve = typename Flavor::Curve;
     using Instance = ProverInstance_<Flavor>;
     using Transcript = typename Flavor::Transcript;
-    using AlphaType = typename Flavor::AlphaType;
+    using RelationSeparator = typename Flavor::RelationSeparator;
 
   public:
     explicit UltraProver_(const std::shared_ptr<Instance>&,
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp
index 62689f5ae19..637806709e8 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.cpp
@@ -132,7 +132,7 @@ template <typename Flavor> bool UltraVerifier_<Flavor>::verify_proof(const plonk
     // Execute Sumcheck Verifier
     const size_t log_circuit_size = numeric::get_msb(circuit_size);
     auto sumcheck = SumcheckVerifier<Flavor>(log_circuit_size, transcript);
-    AlphaType alphas;
+    RelationSeparator alphas;
     for (size_t idx = 0; idx < alphas.size(); idx++) {
         alphas[idx] = transcript->get_challenge("Sumcheck:alpha_" + std::to_string(idx));
     }
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.hpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.hpp
index d19e57cfbfe..8e6c02d0b68 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_verifier.hpp
@@ -12,7 +12,7 @@ template <typename Flavor> class UltraVerifier_ {
     using VerificationKey = typename Flavor::VerificationKey;
     using VerifierCommitmentKey = typename Flavor::VerifierCommitmentKey;
     using Transcript = typename Flavor::Transcript;
-    using AlphaType = typename Flavor::AlphaType;
+    using RelationSeparator = typename Flavor::RelationSeparator;
 
   public:
     explicit UltraVerifier_(const std::shared_ptr<Transcript>& transcript,

From 4f7d79bbb772d59e9df88b08d4847886e658a53d Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Wed, 10 Jan 2024 16:32:58 +0000
Subject: [PATCH 75/86] fix formatting

---
 barretenberg/cpp/src/barretenberg/relations/utils.hpp | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/barretenberg/cpp/src/barretenberg/relations/utils.hpp b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
index 894a28b9d72..b3a0750617e 100644
--- a/barretenberg/cpp/src/barretenberg/relations/utils.hpp
+++ b/barretenberg/cpp/src/barretenberg/relations/utils.hpp
@@ -113,7 +113,8 @@ template <typename Flavor> class RelationUtils {
     template <typename... T>
     static constexpr void add_tuples(std::tuple<T...>& tuple_1, const std::tuple<T...>& tuple_2)
     {
-        auto add_tuples_helper = [&]<std::size_t... I>(std::index_sequence<I...>) {
+        auto add_tuples_helper = [&]<std::size_t... I>(std::index_sequence<I...>)
+        {
             ((std::get<I>(tuple_1) += std::get<I>(tuple_2)), ...);
         };
 

From 9ac85f110a0534eb6fde46a20fab13f9e25e611f Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Thu, 11 Jan 2024 13:16:58 +0000
Subject: [PATCH 76/86] doesn't work

---
 .../ecc/fields/field_declarations.hpp         |  7 ++
 .../flavor/goblin_ultra_recursive.hpp         |  4 +-
 .../barretenberg/flavor/ultra_recursive.hpp   |  4 +
 .../stdlib/primitives/curves/bn254.hpp        |  2 +
 .../verifier/decider_recursive_verifier.hpp   |  2 +-
 .../protogalaxy_recursive_verifier.cpp        | 91 +++++++++++--------
 .../protogalaxy_recursive_verifier.hpp        | 16 ++--
 .../barretenberg/stdlib/utility/utility.hpp   |  4 +
 8 files changed, 81 insertions(+), 49 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/ecc/fields/field_declarations.hpp b/barretenberg/cpp/src/barretenberg/ecc/fields/field_declarations.hpp
index a0446f17916..0c580dd304c 100644
--- a/barretenberg/cpp/src/barretenberg/ecc/fields/field_declarations.hpp
+++ b/barretenberg/cpp/src/barretenberg/ecc/fields/field_declarations.hpp
@@ -112,6 +112,13 @@ template <class Params_> struct alignas(32) field {
         *this = field(value);
     }
 
+    constexpr explicit operator bool() const
+    {
+        field out = from_montgomery_form();
+        ASSERT(out.data[0] == 0 || out.data[0] == 1);
+        return static_cast<bool>(out.data[0]);
+    }
+
     constexpr explicit operator uint32_t() const
     {
         field out = from_montgomery_form();
diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
index e15f73318d7..d9f275bf21b 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
@@ -67,12 +67,14 @@ class GoblinUltraRecursive {
     using Relations = GoblinUltra::Relations_<FF>;
 
     static constexpr size_t MAX_PARTIAL_RELATION_LENGTH = compute_max_partial_relation_length<Relations>();
+    static constexpr size_t MAX_TOTAL_RELATION_LENGTH = compute_max_total_relation_length<Relations>();
 
     // BATCHED_RELATION_PARTIAL_LENGTH = algebraic degree of sumcheck relation *after* multiplying by the `pow_zeta`
     // random polynomial e.g. For \sum(x) [A(x) * B(x) + C(x)] * PowZeta(X), relation length = 2 and random relation
     // length = 3
     static constexpr size_t BATCHED_RELATION_PARTIAL_LENGTH = MAX_PARTIAL_RELATION_LENGTH + 1;
-    static constexpr size_t NUM_RELATIONS = std::tuple_size<Relations>::value;
+    static constexpr size_t BATCHED_RELATION_TOTAL_LENGTH = MAX_TOTAL_RELATION_LENGTH + 1;
+    static constexpr size_t NUM_RELATIONS = std::tuple_size_v<Relations>;
 
     // For instances of this flavour, used in folding, we need a unique sumcheck batching challenge for each
     // subrelation. This is because using powers of alpha would increase the degree of Protogalaxy polynomial $G$ (the
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
index 125dc755c3d..a8ad134fc4e 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
@@ -79,11 +79,15 @@ template <typename BuilderType> class UltraRecursive_ {
                                  proof_system::AuxiliaryRelation<FF>>;
 
     static constexpr size_t MAX_PARTIAL_RELATION_LENGTH = compute_max_partial_relation_length<Relations>();
+    static_assert(MAX_PARTIAL_RELATION_LENGTH == 6);
+    static constexpr size_t MAX_TOTAL_RELATION_LENGTH = compute_max_total_relation_length<Relations>();
+    static_assert(MAX_TOTAL_RELATION_LENGTH == 12);
 
     // BATCHED_RELATION_PARTIAL_LENGTH = algebraic degree of sumcheck relation *after* multiplying by the `pow_zeta`
     // random polynomial e.g. For \sum(x) [A(x) * B(x) + C(x)] * PowZeta(X), relation length = 2 and random relation
     // length = 3
     static constexpr size_t BATCHED_RELATION_PARTIAL_LENGTH = MAX_PARTIAL_RELATION_LENGTH + 1;
+    static constexpr size_t BATCHED_RELATION_TOTAL_LENGTH = MAX_TOTAL_RELATION_LENGTH + 1;
     static constexpr size_t NUM_RELATIONS = std::tuple_size<Relations>::value;
 
     // For instances of this flavour, used in folding, we need a unique sumcheck batching challenges for each
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/primitives/curves/bn254.hpp b/barretenberg/cpp/src/barretenberg/stdlib/primitives/curves/bn254.hpp
index 5dfde40f293..a28ed6c87a1 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/primitives/curves/bn254.hpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/primitives/curves/bn254.hpp
@@ -18,6 +18,8 @@ template <typename CircuitBuilder> struct bn254 {
     using ScalarFieldNative = curve::BN254::ScalarField;
     using BaseFieldNative = curve::BN254::BaseField;
     using GroupNative = curve::BN254::Group;
+    using ElementNative = GroupNative::element;
+    using AffineElementNative = GroupNative::affine_element;
 
     // Stdlib types corresponding to those defined in the native description of the curve.
     // Note: its useful to have these type names match the native analog exactly so that components that digest a Curve
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.hpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.hpp
index 67a8d8cc783..44a6520aa16 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.hpp
@@ -14,7 +14,7 @@ template <typename Flavor> class DeciderRecursiveVerifier_ {
     using VerifierCommitmentKey = typename Flavor::VerifierCommitmentKey;
     using NativeVerificationKey = typename Flavor::NativeVerificationKey;
     using Builder = typename Flavor::CircuitBuilder;
-    using AlphaType = typename Flavor::AlphaType;
+    using RelationSeparator = typename Flavor::RelationSeparator;
     using PairingPoints = std::array<GroupElement, 2>;
 
   public:
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
index f776755e4d4..02fb5ae1ef1 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
@@ -6,10 +6,12 @@ template <class VerifierInstances>
 void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::receive_accumulator(const std::shared_ptr<Instance>& inst,
                                                                            const std::string& domain_separator)
 {
-    inst->instance_size = transcript->template receive_from_prover<uint32_t>(domain_separator + "_instance_size");
-    inst->log_instance_size = static_cast<size_t>(numeric::get_msb(inst->instance_size));
-    inst->public_input_size =
+    const auto instance_size = transcript->template receive_from_prover<uint32_t>(domain_separator + "_instance_size");
+    const auto public_input_size =
         transcript->template receive_from_prover<uint32_t>(domain_separator + "_public_input_size");
+    inst->instance_size = static_cast<size_t>(instance_size.get_value());
+    inst->log_instance_size = static_cast<size_t>(numeric::get_msb(inst->instance_size));
+    inst->public_input_size = static_cast<size_t>(public_input_size.get_value());
 
     for (size_t i = 0; i < inst->public_input_size; ++i) {
         auto public_input_i =
@@ -57,10 +59,12 @@ template <class VerifierInstances>
 void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::receive_and_finalise_instance(
     const std::shared_ptr<Instance>& inst, const std::string& domain_separator)
 {
-    inst->instance_size = transcript->template receive_from_prover<uint32_t>(domain_separator + "_instance_size");
-    inst->log_instance_size = static_cast<size_t>(numeric::get_msb(inst->instance_size));
-    inst->public_input_size =
+    const auto instance_size = transcript->template receive_from_prover<uint32_t>(domain_separator + "_instance_size");
+    const auto public_input_size =
         transcript->template receive_from_prover<uint32_t>(domain_separator + "_public_input_size");
+    inst->instance_size = static_cast<size_t>(instance_size.get_value());
+    inst->log_instance_size = static_cast<size_t>(numeric::get_msb(inst->instance_size));
+    inst->public_input_size = static_cast<size_t>(public_input_size.get_value());
 
     for (size_t i = 0; i < inst->public_input_size; ++i) {
         auto public_input_i =
@@ -68,9 +72,11 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::receive_and_finalise_inst
         inst->public_inputs.emplace_back(public_input_i);
     }
 
-    inst->pub_inputs_offset =
+    const auto pub_inputs_offset =
         transcript->template receive_from_prover<uint32_t>(domain_separator + "_pub_inputs_offset");
 
+    inst->pub_inputs_offset = static_cast<size_t>(pub_inputs_offset.get_value());
+
     auto labels = inst->commitment_labels;
     auto& witness_commitments = inst->witness_commitments;
     witness_commitments.w_l = transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.w_l);
@@ -88,9 +94,10 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::receive_and_finalise_inst
     witness_commitments.z_lookup =
         transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.z_lookup);
 
-    const FF public_input_delta = compute_public_input_delta<Flavor>(
+    const FF public_input_delta = proof_system::honk::compute_public_input_delta<Flavor>(
         inst->public_inputs, beta, gamma, inst->instance_size, inst->pub_inputs_offset);
-    const FF lookup_grand_product_delta = compute_lookup_grand_product_delta<FF>(beta, gamma, inst->instance_size);
+    const FF lookup_grand_product_delta =
+        proof_system::honk::compute_lookup_grand_product_delta<FF>(beta, gamma, inst->instance_size);
     inst->relation_parameters =
         RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
 
@@ -108,14 +115,13 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::receive_and_finalise_inst
 
 // TODO(https://github.com/AztecProtocol/barretenberg/issues/795): The rounds prior to actual verifying are common
 // between decider and folding verifier and could be somehow shared so we do not duplicate code so much.
-template <class VerifierInstances>
-void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::prepare_for_folding(const std::vector<uint8_t>& fold_data)
+template <class VerifierInstances> void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::prepare_for_folding()
 {
-    transcript = std::make_shared<Transcript>(fold_data);
     auto index = 0;
     auto inst = instances[0];
     auto domain_separator = std::to_string(index);
-    inst->is_accumulator = transcript->template receive_from_prover<bool>(domain_separator + "is_accumulator");
+    const auto is_accumulator = transcript->template receive_from_prover<bool>(domain_separator + "is_accumulator");
+    inst->is_accumulator = static_cast<bool>(is_accumulator.get_value());
     if (inst->is_accumulator) {
         receive_accumulator(inst, domain_separator);
     } else {
@@ -142,9 +148,14 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::prepare_for_folding(const
 }
 
 template <class VerifierInstances>
-bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std::vector<uint8_t> fold_data)
+bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(const plonk::proof& proof)
 {
-    prepare_for_folding(fold_data);
+    using Transcript = typename Flavor::Transcript;
+    using ElementNative = typename Flavor::Curve::ElementNative;
+    using AffineElementNative = typename Flavor::Curve::AffineElementNative;
+
+    transcript = std::make_shared<Transcript>(builder, proof.proof_data);
+    prepare_for_folding();
 
     auto delta = transcript->get_challenge("delta");
     auto accumulator = get_accumulator();
@@ -155,9 +166,7 @@ bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
         perturbator_coeffs[idx] = transcript->template receive_from_prover<FF>("perturbator_" + std::to_string(idx));
     }
 
-    if (perturbator_coeffs[0] != accumulator->target_sum) {
-        return false;
-    }
+    perturbator_coeffs[0].assert_equal(accumulator->target_sum);
 
     auto perturbator = Polynomial<FF>(perturbator_coeffs);
     FF perturbator_challenge = transcript->get_challenge("perturbator_challenge");
@@ -171,7 +180,7 @@ bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
     }
     Univariate<FF, VerifierInstances::BATCHED_EXTENDED_LENGTH, VerifierInstances::NUM> combiner_quotient(
         combiner_quotient_evals);
-    FF combiner_challenge = transcript->get_challenge("combiner_quotient_challenge");
+    FF combiner_challenge = transcript->get_challenge("combiner_quotient_challenge").get_value();
     auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge);
 
     auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
@@ -181,26 +190,30 @@ bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
     auto expected_next_target_sum =
         perturbator_at_challenge * lagranges[0] + vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
     auto next_target_sum = transcript->template receive_from_prover<FF>("next_target_sum");
-    bool verified = (expected_next_target_sum == next_target_sum);
+    expected_next_target_sum.assert_equal(next_target_sum);
+
     auto expected_betas_star = update_gate_challenges(perturbator_challenge, accumulator->gate_challenges, deltas);
     for (size_t idx = 0; idx < accumulator->log_instance_size; idx++) {
         auto beta_star = transcript->template receive_from_prover<FF>("next_gate_challenge_" + std::to_string(idx));
-        verified = verified & (expected_betas_star[idx] == beta_star);
+        verified = verified & (expected_betas_star[idx] == beta_star.get_value());
     }
 
     // Compute ϕ and verify against the data received from the prover
     WitnessCommitments acc_witness_commitments;
     auto witness_labels = commitment_labels.get_witness();
     size_t comm_idx = 0;
+    auto random_generator = AffineElementNative(ElementNative::random_element());
+    auto sth = Commitment::from_witness(builder, random_generator);
     for (auto& expected_comm : acc_witness_commitments.get_all()) {
-        expected_comm = Commitment::infinity();
+        expected_comm = sth;
         size_t inst = 0;
         for (auto& instance : instances) {
             expected_comm = expected_comm + instance->witness_commitments.get_all()[comm_idx] * lagranges[inst];
             inst++;
         }
+        expected_comm -= sth;
         auto comm = transcript->template receive_from_prover<Commitment>("next_" + witness_labels[comm_idx]);
-        verified = verified & (comm == expected_comm);
+        verified = verified & (comm.get_value() == expected_comm);
         comm_idx++;
     }
 
@@ -213,23 +226,21 @@ bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
             inst++;
         }
         auto el = transcript->template receive_from_prover<FF>("next_public_input" + std::to_string(el_idx));
-        verified = verified & (el == expected_el);
+        verified = verified & (el.get_value() == expected_el);
         el_idx++;
     }
 
-    AlphaType expected_alphas;
-    size_t alpha_idx = 0;
-    for (auto& alpha : expected_alphas) {
-        alpha = FF(0);
+    for (size_t alpha_idx = 0; alpha_idx < NUM_SUBRELATIONS - 1; alpha_idx++) {
+        FF alpha(0);
         size_t instance_idx = 0;
         for (auto& instance : instances) {
             alpha += instance->alphas[alpha_idx] * lagranges[instance_idx];
             instance_idx++;
         }
         auto next_alpha = transcript->template receive_from_prover<FF>("next_alpha_" + std::to_string(alpha_idx));
-        verified = verified & (alpha == next_alpha);
-        alpha_idx++;
+        verified = verified & (alpha == next_alpha.get_value());
     }
+
     auto expected_parameters = proof_system::RelationParameters<FF>{};
     for (size_t inst_idx = 0; inst_idx < VerifierInstances::NUM; inst_idx++) {
         auto instance = instances[inst_idx];
@@ -243,20 +254,21 @@ bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
     }
 
     auto next_eta = transcript->template receive_from_prover<FF>("next_eta");
-    verified = verified & (next_eta == expected_parameters.eta);
+    verified = verified & (next_eta.get_value() == expected_parameters.eta);
 
     auto next_beta = transcript->template receive_from_prover<FF>("next_beta");
-    verified = verified & (next_beta == expected_parameters.beta);
+    verified = verified & (next_beta.get_value() == expected_parameters.beta);
 
     auto next_gamma = transcript->template receive_from_prover<FF>("next_gamma");
-    verified = verified & (next_gamma == expected_parameters.gamma);
+    verified = verified & (next_gamma.get_value() == expected_parameters.gamma);
 
     auto next_public_input_delta = transcript->template receive_from_prover<FF>("next_public_input_delta");
-    verified = verified & (next_public_input_delta == expected_parameters.public_input_delta);
+    verified = verified & (next_public_input_delta.get_value() == expected_parameters.public_input_delta);
 
     auto next_lookup_grand_product_delta =
         transcript->template receive_from_prover<FF>("next_lookup_grand_product_delta");
-    verified = verified & (next_lookup_grand_product_delta == expected_parameters.lookup_grand_product_delta);
+    verified =
+        verified & (next_lookup_grand_product_delta.get_value() == expected_parameters.lookup_grand_product_delta);
 
     auto acc_vk = std::make_shared<VerificationKey>(instances[0]->instance_size, instances[0]->public_input_size);
     auto vk_labels = commitment_labels.get_precomputed();
@@ -269,7 +281,7 @@ bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
             inst++;
         }
         auto vk = transcript->template receive_from_prover<Commitment>("next_" + vk_labels[vk_idx]);
-        verified = verified & (vk == expected_vk);
+        verified = verified & (vk.get_value() == expected_vk);
         vk_idx++;
     }
 
@@ -277,8 +289,9 @@ bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
 }
 
 template class ProtoGalaxyRecursiveVerifier_<
-    VerifierInstances_<proof_system::honk::flavor::UltraRecursive_<UltraCircuitBuilder>, 2>>;
+    proof_system::honk::VerifierInstances_<proof_system::honk::flavor::UltraRecursive_<UltraCircuitBuilder>, 2>>;
+template class ProtoGalaxyRecursiveVerifier_<
+    proof_system::honk::VerifierInstances_<proof_system::honk::flavor::UltraRecursive_<GoblinUltraCircuitBuilder>, 2>>;
 template class ProtoGalaxyRecursiveVerifier_<
-    VerifierInstances_<proof_system::honk::flavor::UltraRecursive_<GoblinUltraCircuitBuilder>, 2>>;
-template class ProtoGalaxyRecursiveVerifier_<proof_system::honk::flavor::GoblinUltraRecursive, 2>;
+    proof_system::honk::VerifierInstances_<proof_system::honk::flavor::GoblinUltraRecursive, 2>>;
 } // namespace proof_system::plonk::stdlib::recursion::honk
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp
index b67ba4a5a8f..5f559393831 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp
@@ -1,17 +1,16 @@
 #pragma once
 #include "barretenberg/flavor/flavor.hpp"
-#include "barretenberg/flavor/goblin_ultra.hpp"
-#include "barretenberg/flavor/ultra.hpp"
+#include "barretenberg/flavor/goblin_ultra_recursive.hpp"
+#include "barretenberg/flavor/ultra_recursive.hpp"
 #include "barretenberg/plonk/proof_system/types/proof.hpp"
 #include "barretenberg/protogalaxy/folding_result.hpp"
+#include "barretenberg/stdlib/recursion/honk/transcript/transcript.hpp"
 #include "barretenberg/sumcheck/instance/instances.hpp"
-#include "barretenberg/transcript/transcript.hpp"
 
 namespace proof_system::plonk::stdlib::recursion::honk {
 template <class VerifierInstances> class ProtoGalaxyRecursiveVerifier_ {
   public:
     using Flavor = typename VerifierInstances::Flavor;
-    using Transcript = typename Flavor::Transcript;
     using FF = typename Flavor::FF;
     using Commitment = typename Flavor::Commitment;
     using GroupElement = typename Flavor::GroupElement;
@@ -20,17 +19,18 @@ template <class VerifierInstances> class ProtoGalaxyRecursiveVerifier_ {
     using WitnessCommitments = typename Flavor::WitnessCommitments;
     using CommitmentLabels = typename Flavor::CommitmentLabels;
     using Builder = typename Flavor::CircuitBuilder;
-    using AlphaType = typename Flavor::AlphaType;
+    using RelationSeparator = typename Flavor::RelationSeparator;
     using PairingPoints = std::array<GroupElement, 2>;
 
     static constexpr size_t NUM_SUBRELATIONS = Flavor::NUM_SUBRELATIONS;
 
     VerifierInstances instances;
 
-    std::shared_ptr<Transcript> transcript = std::make_shared<Transcript>();
-
     CommitmentLabels commitment_labels;
 
+    Builder* builder;
+    std::shared_ptr<Transcript<Builder>> transcript;
+
     ProtoGalaxyRecursiveVerifier_(VerifierInstances insts)
         : instances(insts){};
     ~ProtoGalaxyRecursiveVerifier_() = default;
@@ -68,7 +68,7 @@ template <class VerifierInstances> class ProtoGalaxyRecursiveVerifier_ {
      *
      * @param fold_data The data transmitted via the transcript by the prover.
      */
-    void prepare_for_folding(const std::vector<uint8_t>&);
+    void prepare_for_folding();
 
     /**
      * @brief Instantiatied the accumulator (i.e. the relaxed instance) from the transcript.
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/utility/utility.hpp b/barretenberg/cpp/src/barretenberg/stdlib/utility/utility.hpp
index 451ef2c8e97..e6d2f36424c 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/utility/utility.hpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/utility/utility.hpp
@@ -115,6 +115,10 @@ template <typename Builder> class StdlibTypesUtility {
         using type = uint32_t;
     };
 
+    template <size_t LENGTH> struct NativeType<bool, LENGTH> {
+        using type = bool;
+    };
+
     template <size_t LENGTH> struct NativeType<field_ct, LENGTH> {
         using type = FF;
     };

From c5fdb82ae4145645809c729b8e662f7ed1398162 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Thu, 11 Jan 2024 15:45:29 +0000
Subject: [PATCH 77/86] compiles

---
 .../protogalaxy_recursive_verifier.cpp        | 34 ++++++++++---------
 1 file changed, 18 insertions(+), 16 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
index 02fb5ae1ef1..e70ccd3de6c 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
@@ -166,7 +166,7 @@ bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(cons
         perturbator_coeffs[idx] = transcript->template receive_from_prover<FF>("perturbator_" + std::to_string(idx));
     }
 
-    perturbator_coeffs[0].assert_equal(accumulator->target_sum);
+    perturbator_coeffs[0].assert_equal(accumulator->target_sum, "F(0) != e");
 
     auto perturbator = Polynomial<FF>(perturbator_coeffs);
     FF perturbator_challenge = transcript->get_challenge("perturbator_challenge");
@@ -190,12 +190,12 @@ bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(cons
     auto expected_next_target_sum =
         perturbator_at_challenge * lagranges[0] + vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
     auto next_target_sum = transcript->template receive_from_prover<FF>("next_target_sum");
-    expected_next_target_sum.assert_equal(next_target_sum);
+    expected_next_target_sum.assert_equal(next_target_sum, "next_target_sum mismatch");
 
     auto expected_betas_star = update_gate_challenges(perturbator_challenge, accumulator->gate_challenges, deltas);
     for (size_t idx = 0; idx < accumulator->log_instance_size; idx++) {
         auto beta_star = transcript->template receive_from_prover<FF>("next_gate_challenge_" + std::to_string(idx));
-        verified = verified & (expected_betas_star[idx] == beta_star.get_value());
+        expected_betas_star[idx].assert_equal(beta_star, "beta mismatch at: " + std::to_string(idx));
     }
 
     // Compute ϕ and verify against the data received from the prover
@@ -213,7 +213,8 @@ bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(cons
         }
         expected_comm -= sth;
         auto comm = transcript->template receive_from_prover<Commitment>("next_" + witness_labels[comm_idx]);
-        verified = verified & (comm.get_value() == expected_comm);
+        comm.x.assert_equal(expected_comm.x);
+        comm.y.assert_equal(expected_comm.y);
         comm_idx++;
     }
 
@@ -225,8 +226,8 @@ bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(cons
             expected_el += instance->public_inputs[el_idx] * lagranges[inst];
             inst++;
         }
-        auto el = transcript->template receive_from_prover<FF>("next_public_input" + std::to_string(el_idx));
-        verified = verified & (el.get_value() == expected_el);
+        auto next_el = transcript->template receive_from_prover<FF>("next_public_input" + std::to_string(el_idx));
+        next_el.assert_equal(expected_el);
         el_idx++;
     }
 
@@ -238,7 +239,7 @@ bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(cons
             instance_idx++;
         }
         auto next_alpha = transcript->template receive_from_prover<FF>("next_alpha_" + std::to_string(alpha_idx));
-        verified = verified & (alpha == next_alpha.get_value());
+        next_alpha.assert_equal(alpha);
     }
 
     auto expected_parameters = proof_system::RelationParameters<FF>{};
@@ -254,38 +255,39 @@ bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(cons
     }
 
     auto next_eta = transcript->template receive_from_prover<FF>("next_eta");
-    verified = verified & (next_eta.get_value() == expected_parameters.eta);
+    next_eta.assert_equal(expected_parameters.eta);
 
     auto next_beta = transcript->template receive_from_prover<FF>("next_beta");
-    verified = verified & (next_beta.get_value() == expected_parameters.beta);
+    next_beta.assert_equal(expected_parameters.beta);
 
     auto next_gamma = transcript->template receive_from_prover<FF>("next_gamma");
-    verified = verified & (next_gamma.get_value() == expected_parameters.gamma);
+    next_gamma.assert_equal(expected_parameters.gamma);
 
     auto next_public_input_delta = transcript->template receive_from_prover<FF>("next_public_input_delta");
-    verified = verified & (next_public_input_delta.get_value() == expected_parameters.public_input_delta);
+    next_public_input_delta.assert_equal(expected_parameters.public_input_delta);
 
     auto next_lookup_grand_product_delta =
         transcript->template receive_from_prover<FF>("next_lookup_grand_product_delta");
-    verified =
-        verified & (next_lookup_grand_product_delta.get_value() == expected_parameters.lookup_grand_product_delta);
+    next_lookup_grand_product_delta.assert_equal(expected_parameters.lookup_grand_product_delta);
 
     auto acc_vk = std::make_shared<VerificationKey>(instances[0]->instance_size, instances[0]->public_input_size);
     auto vk_labels = commitment_labels.get_precomputed();
     size_t vk_idx = 0;
     for (auto& expected_vk : acc_vk->get_all()) {
         size_t inst = 0;
-        expected_vk = Commitment::infinity();
+        expected_vk = sth;
         for (auto& instance : instances) {
             expected_vk = expected_vk + instance->verification_key->get_all()[vk_idx] * lagranges[inst];
             inst++;
         }
         auto vk = transcript->template receive_from_prover<Commitment>("next_" + vk_labels[vk_idx]);
-        verified = verified & (vk.get_value() == expected_vk);
+        expected_vk -= sth;
+        vk.x.assert_equal(expected_vk.x);
+        vk.y.assert_equal(expected_vk.y);
         vk_idx++;
     }
 
-    return verified;
+    return true;
 }
 
 template class ProtoGalaxyRecursiveVerifier_<

From d0f760db823358d05adebb582ed9f9025941debd Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Fri, 12 Jan 2024 15:56:58 +0000
Subject: [PATCH 78/86] add tests

---
 .../protogalaxy/protogalaxy_prover.cpp        |   4 -
 .../protogalaxy_recursive_verifier.test.cpp   | 104 ++++++++++++++++++
 2 files changed, 104 insertions(+), 4 deletions(-)
 create mode 100644 barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
index e735ff8cab9..afd3b70396f 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_prover.cpp
@@ -269,14 +269,10 @@ std::shared_ptr<typename ProverInstances::Instance> ProtoGalaxyProver_<ProverIns
     return next_accumulator;
 }
 
-// TODO(#https://github.com/AztecProtocol/barretenberg/issues/689): finalise implementation this function
 template <class ProverInstances>
 FoldingResult<typename ProverInstances::Flavor> ProtoGalaxyProver_<ProverInstances>::fold_instances()
 {
     prepare_for_folding();
-
-    // TODO(#https://github.com/AztecProtocol/barretenberg/issues/740): Handle the case where we are folding for the
-    // first time and accumulator is 0
     FF delta = transcript->get_challenge("delta");
 
     auto accumulator = get_accumulator();
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
new file mode 100644
index 00000000000..89b7cd5699b
--- /dev/null
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
@@ -0,0 +1,104 @@
+#include "barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp"
+#include "barretenberg/common/test.hpp"
+#include "barretenberg/flavor/ultra_recursive.hpp"
+#include "barretenberg/stdlib/hash/blake3s/blake3s.hpp"
+#include "barretenberg/stdlib/hash/pedersen/pedersen.hpp"
+#include "barretenberg/stdlib/primitives/curves/bn254.hpp"
+#include "barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.hpp"
+#include "barretenberg/ultra_honk/ultra_composer.hpp"
+
+namespace proof_system::plonk::stdlib::recursion::honk {
+template <typename BuilderType> class ProtogalaxyRecursiveVerifierTest : public testing::Test {
+    // Define types relevant for testing
+    using UltraFlavor = ::proof_system::honk::flavor::Ultra;
+    using GoblinUltraFlavor = ::proof_system::honk::flavor::GoblinUltra;
+    using UltraComposer = ::proof_system::honk::UltraComposer_<UltraFlavor>;
+    using GoblinUltraComposer = ::proof_system::honk::UltraComposer_<GoblinUltraFlavor>;
+
+    using InnerFlavor = UltraFlavor;
+    using InnerComposer = UltraComposer;
+    using InnerBuilder = typename InnerComposer::CircuitBuilder;
+    using InnerCurve = bn254<InnerBuilder>;
+    using Commitment = InnerFlavor::Commitment;
+    using FF = InnerFlavor::FF;
+
+    // Types for recursive verifier circuit
+    using RecursiveFlavor = ::proof_system::honk::flavor::UltraRecursive_<BuilderType>;
+    using VerifierInstances = ::proof_system::honk::VerifierInstances_<RecursiveFlavor, 2>;
+    using FoldingRecursiveVerifier = ProtoGalaxyRecursiveVerifier_<VerifierInstances>;
+    using OuterBuilder = BuilderType;
+    using DeciderRecursiveVerifier = DeciderRecursiveVerifier_<RecursiveFlavor>;
+
+    /**
+     * @brief Create a non-trivial arbitrary inner circuit, the proof of which will be recursively verified
+     *
+     * @param builder
+     * @param public_inputs
+     * @param log_num_gates
+     */
+    static void create_inner_circuit(InnerBuilder& builder, size_t log_num_gates = 10)
+    {
+        using fr_ct = InnerCurve::ScalarField;
+        using fq_ct = InnerCurve::BaseField;
+        using public_witness_ct = InnerCurve::public_witness_ct;
+        using witness_ct = InnerCurve::witness_ct;
+        using byte_array_ct = InnerCurve::byte_array_ct;
+        using fr = typename InnerCurve::ScalarFieldNative;
+
+        // Create 2^log_n many add gates based on input log num gates
+        const size_t num_gates = 1 << log_num_gates;
+        for (size_t i = 0; i < num_gates; ++i) {
+            fr a = fr::random_element();
+            uint32_t a_idx = builder.add_variable(a);
+
+            fr b = fr::random_element();
+            fr c = fr::random_element();
+            fr d = a + b + c;
+            uint32_t b_idx = builder.add_variable(b);
+            uint32_t c_idx = builder.add_variable(c);
+            uint32_t d_idx = builder.add_variable(d);
+
+            builder.create_big_add_gate({ a_idx, b_idx, c_idx, d_idx, fr(1), fr(1), fr(1), fr(-1), fr(0) });
+        }
+
+        // Define some additional non-trivial but arbitrary circuit logic
+        fr_ct a(public_witness_ct(&builder, fr::random_element()));
+        fr_ct b(public_witness_ct(&builder, fr::random_element()));
+        fr_ct c(public_witness_ct(&builder, fr::random_element()));
+
+        for (size_t i = 0; i < 32; ++i) {
+            a = (a * b) + b + a;
+            a = a.madd(b, c);
+        }
+        pedersen_hash<InnerBuilder>::hash({ a, b });
+        byte_array_ct to_hash(&builder, "nonsense test data");
+        blake3s(to_hash);
+
+        fr bigfield_data = fr::random_element();
+        fr bigfield_data_a{ bigfield_data.data[0], bigfield_data.data[1], 0, 0 };
+        fr bigfield_data_b{ bigfield_data.data[2], bigfield_data.data[3], 0, 0 };
+
+        fq_ct big_a(fr_ct(witness_ct(&builder, bigfield_data_a.to_montgomery_form())), fr_ct(witness_ct(&builder, 0)));
+        fq_ct big_b(fr_ct(witness_ct(&builder, bigfield_data_b.to_montgomery_form())), fr_ct(witness_ct(&builder, 0)));
+
+        big_a* big_b;
+    };
+
+  public:
+    static void SetUpTestSuite() { barretenberg::srs::init_crs_factory("../srs_db/ignition"); }
+
+    /**
+     * @brief Create inner circuit and call check_circuit on it
+     *
+     */
+    static void test_inner_circuit()
+    {
+        InnerBuilder builder;
+
+        create_inner_circuit(builder);
+
+        bool result = builder.check_circuit();
+        EXPECT_EQ(result, true);
+    }
+};
+} // namespace proof_system::plonk::stdlib::recursion::honk
\ No newline at end of file

From edda2b93aee8747458f17b5b87e8a705723230ed Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Tue, 16 Jan 2024 14:16:33 +0000
Subject: [PATCH 79/86] not working

---
 .../flavor/goblin_ultra_recursive.hpp         |  2 +
 .../barretenberg/flavor/ultra_recursive.hpp   |  2 +
 .../barretenberg/polynomials/polynomial.hpp   |  1 +
 .../circuit_builder/ultra_circuit_builder.hpp |  6 +--
 .../protogalaxy_recursive_verifier.cpp        | 13 ++---
 .../protogalaxy_recursive_verifier.hpp        |  9 ++--
 .../protogalaxy_recursive_verifier.test.cpp   | 52 +++++++++++++++++--
 .../verifier/ultra_recursive_verifier.hpp     |  5 --
 8 files changed, 68 insertions(+), 22 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
index d9f275bf21b..59e94e36ebb 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
@@ -47,6 +47,8 @@ class GoblinUltraRecursive {
     using Commitment = Curve::Element;
     using CommitmentHandle = Curve::Element;
     using NativeVerificationKey = flavor::GoblinUltra::VerificationKey;
+    using Polynomial = barretenberg::Polynomial<FF>;
+    using PolynomialHandle = std::span<FF>;
 
     // Note(luke): Eventually this may not be needed at all
     using VerifierCommitmentKey = pcs::VerifierCommitmentKey<Curve>;
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
index a8ad134fc4e..12522a65fc0 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
@@ -55,6 +55,8 @@ template <typename BuilderType> class UltraRecursive_ {
     using CommitmentHandle = typename Curve::Element;
     using FF = typename Curve::ScalarField;
     using NativeVerificationKey = flavor::Ultra::VerificationKey;
+    using Polynomial = barretenberg::Polynomial<FF>;
+    using PolynomialHandle = std::span<FF>;
 
     // Note(luke): Eventually this may not be needed at all
     using VerifierCommitmentKey = pcs::VerifierCommitmentKey<Curve>;
diff --git a/barretenberg/cpp/src/barretenberg/polynomials/polynomial.hpp b/barretenberg/cpp/src/barretenberg/polynomials/polynomial.hpp
index d8424e445fd..7b31aa25843 100644
--- a/barretenberg/cpp/src/barretenberg/polynomials/polynomial.hpp
+++ b/barretenberg/cpp/src/barretenberg/polynomials/polynomial.hpp
@@ -276,6 +276,7 @@ template <typename Fr> inline std::ostream& operator<<(std::ostream& os, Polynom
 extern template class Polynomial<barretenberg::fr>;
 // N.B. grumpkin polynomials don't support fast fourier transforms using roots of unity!
 extern template class Polynomial<grumpkin::fr>;
+// I think these are necessary
 
 using polynomial = Polynomial<barretenberg::fr>;
 
diff --git a/barretenberg/cpp/src/barretenberg/proof_system/circuit_builder/ultra_circuit_builder.hpp b/barretenberg/cpp/src/barretenberg/proof_system/circuit_builder/ultra_circuit_builder.hpp
index 14c7f9cc2b9..397fbddb0aa 100644
--- a/barretenberg/cpp/src/barretenberg/proof_system/circuit_builder/ultra_circuit_builder.hpp
+++ b/barretenberg/cpp/src/barretenberg/proof_system/circuit_builder/ultra_circuit_builder.hpp
@@ -1,7 +1,7 @@
 #pragma once
-// #include "barretenberg/plonk/proof_system/constants.hpp"
-// #include "barretenberg/plonk/proof_system/types/polynomial_manifest.hpp"
-// #include "barretenberg/plonk/proof_system/types/prover_settings.hpp"
+#include "barretenberg/plonk/proof_system/constants.hpp"
+#include "barretenberg/plonk/proof_system/types/polynomial_manifest.hpp"
+#include "barretenberg/plonk/proof_system/types/prover_settings.hpp"
 #include "barretenberg/polynomials/polynomial.hpp"
 #include "barretenberg/proof_system/op_queue/ecc_op_queue.hpp"
 #include "barretenberg/proof_system/plookup_tables/plookup_tables.hpp"
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
index e70ccd3de6c..277837673e5 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
@@ -1,4 +1,5 @@
 #include "protogalaxy_recursive_verifier.hpp"
+#include "barretenberg/polynomials/polynomial.hpp"
 #include "barretenberg/proof_system/library/grand_product_delta.hpp"
 namespace proof_system::plonk::stdlib::recursion::honk {
 
@@ -148,13 +149,13 @@ template <class VerifierInstances> void ProtoGalaxyRecursiveVerifier_<VerifierIn
 }
 
 template <class VerifierInstances>
-bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(const plonk::proof& proof)
+bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std::vector<uint8_t> proof)
 {
     using Transcript = typename Flavor::Transcript;
     using ElementNative = typename Flavor::Curve::ElementNative;
     using AffineElementNative = typename Flavor::Curve::AffineElementNative;
 
-    transcript = std::make_shared<Transcript>(builder, proof.proof_data);
+    transcript = std::make_shared<Transcript>(builder, proof);
     prepare_for_folding();
 
     auto delta = transcript->get_challenge("delta");
@@ -168,9 +169,9 @@ bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(cons
 
     perturbator_coeffs[0].assert_equal(accumulator->target_sum, "F(0) != e");
 
-    auto perturbator = Polynomial<FF>(perturbator_coeffs);
+    auto perturbator = Polynomial(perturbator_coeffs);
     FF perturbator_challenge = transcript->get_challenge("perturbator_challenge");
-    auto perturbator_at_challenge = perturbator.evaluate(perturbator_challenge);
+    auto perturbator_at_challenge = perturbator.evaluate(perturbator_challenge.get_value());
 
     // The degree of K(X) is dk - k - 1 = k(d - 1) - 1. Hence we need  k(d - 1) evaluations to represent it.
     std::array<FF, VerifierInstances::BATCHED_EXTENDED_LENGTH - VerifierInstances::NUM> combiner_quotient_evals;
@@ -180,8 +181,8 @@ bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(cons
     }
     Univariate<FF, VerifierInstances::BATCHED_EXTENDED_LENGTH, VerifierInstances::NUM> combiner_quotient(
         combiner_quotient_evals);
-    FF combiner_challenge = transcript->get_challenge("combiner_quotient_challenge").get_value();
-    auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge);
+    FF combiner_challenge = transcript->get_challenge("combiner_quotient_challenge");
+    auto combiner_quotient_at_challenge = combiner_quotient.evaluate(combiner_challenge); // fine recursive i think
 
     auto vanishing_polynomial_at_challenge = combiner_challenge * (combiner_challenge - FF(1));
     auto lagranges = std::vector<FF>{ FF(1) - combiner_challenge, combiner_challenge };
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp
index 5f559393831..dd744e32dc1 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp
@@ -21,6 +21,7 @@ template <class VerifierInstances> class ProtoGalaxyRecursiveVerifier_ {
     using Builder = typename Flavor::CircuitBuilder;
     using RelationSeparator = typename Flavor::RelationSeparator;
     using PairingPoints = std::array<GroupElement, 2>;
+    using Polynomial = typename Flavor::Polynomial;
 
     static constexpr size_t NUM_SUBRELATIONS = Flavor::NUM_SUBRELATIONS;
 
@@ -31,9 +32,9 @@ template <class VerifierInstances> class ProtoGalaxyRecursiveVerifier_ {
     Builder* builder;
     std::shared_ptr<Transcript<Builder>> transcript;
 
-    ProtoGalaxyRecursiveVerifier_(VerifierInstances insts)
-        : instances(insts){};
-    ~ProtoGalaxyRecursiveVerifier_() = default;
+    explicit ProtoGalaxyRecursiveVerifier_(Builder* builder)
+        : instances(VerifierInstances())
+        , builder(builder){};
     /**
      * @brief Given a new round challenge δ for each iteration of the full ProtoGalaxy protocol, compute the vector
      * [δ, δ^2,..., δ^t] where t = logn and n is the size of the instance.
@@ -87,7 +88,7 @@ template <class VerifierInstances> class ProtoGalaxyRecursiveVerifier_ {
      * accumulator, received from the prover is the same as that produced by the verifier.
      *
      */
-    bool verify_folding_proof(const plonk::proof& proof);
+    bool verify_folding_proof(std::vector<uint8_t> proof);
 };
 
 extern template class ProtoGalaxyRecursiveVerifier_<
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
index 89b7cd5699b..944f2fc0403 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
@@ -8,15 +8,16 @@
 #include "barretenberg/ultra_honk/ultra_composer.hpp"
 
 namespace proof_system::plonk::stdlib::recursion::honk {
-template <typename BuilderType> class ProtogalaxyRecursiveVerifierTest : public testing::Test {
+template <typename BuilderType> class ProtogalaxyRecursiveTest : public testing::Test {
     // Define types relevant for testing
     using UltraFlavor = ::proof_system::honk::flavor::Ultra;
     using GoblinUltraFlavor = ::proof_system::honk::flavor::GoblinUltra;
     using UltraComposer = ::proof_system::honk::UltraComposer_<UltraFlavor>;
     using GoblinUltraComposer = ::proof_system::honk::UltraComposer_<GoblinUltraFlavor>;
 
-    using InnerFlavor = UltraFlavor;
+    using InnerFlavor = UltraFlavor; // type this
     using InnerComposer = UltraComposer;
+    using Instance = ::proof_system::honk::ProverInstance_<InnerFlavor>;
     using InnerBuilder = typename InnerComposer::CircuitBuilder;
     using InnerCurve = bn254<InnerBuilder>;
     using Commitment = InnerFlavor::Commitment;
@@ -24,10 +25,12 @@ template <typename BuilderType> class ProtogalaxyRecursiveVerifierTest : public
 
     // Types for recursive verifier circuit
     using RecursiveFlavor = ::proof_system::honk::flavor::UltraRecursive_<BuilderType>;
-    using VerifierInstances = ::proof_system::honk::VerifierInstances_<RecursiveFlavor, 2>;
-    using FoldingRecursiveVerifier = ProtoGalaxyRecursiveVerifier_<VerifierInstances>;
+    using RecursiveVerifierInstances = ::proof_system::honk::VerifierInstances_<RecursiveFlavor, 2>;
+    using FoldingRecursiveVerifier = ProtoGalaxyRecursiveVerifier_<RecursiveVerifierInstances>;
     using OuterBuilder = BuilderType;
     using DeciderRecursiveVerifier = DeciderRecursiveVerifier_<RecursiveFlavor>;
+    using NativeVerifierInstances = ::proof_system::honk::VerifierInstances_<InnerFlavor, 2>;
+    using NativeFoldingVerifier = proof_system::honk::ProtoGalaxyVerifier_<NativeVerifierInstances>;
 
     /**
      * @brief Create a non-trivial arbitrary inner circuit, the proof of which will be recursively verified
@@ -100,5 +103,46 @@ template <typename BuilderType> class ProtogalaxyRecursiveVerifierTest : public
         bool result = builder.check_circuit();
         EXPECT_EQ(result, true);
     }
+
+    static void test_recursive_folding()
+    {
+        InnerBuilder builder1;
+        create_inner_circuit(builder1);
+        InnerBuilder builder2;
+        create_inner_circuit(builder2);
+        InnerComposer inner_composer = InnerComposer();
+        auto instance1 = inner_composer.create_instance(builder1);
+        auto instance2 = inner_composer.create_instance(builder2);
+        auto instances = std::vector<std::shared_ptr<Instance>>{ instance1, instance2 };
+        // commitment key accessible
+        auto inner_folding_prover = inner_composer.create_folding_prover(instances, inner_composer.commitment_key);
+        auto inner_folding_proof = inner_folding_prover.fold_instances();
+        // auto native_folding_verifier = inner_composer.create_folding_verifier();
+        // auto res = native_folding_verifier.verify_folding_proof(inner_folding_proof.folding_data);
+        // EXPECT_EQ(res, true);
+
+        OuterBuilder outer_circuit;
+        FoldingRecursiveVerifier verifier{ &outer_circuit };
+        auto res = verifier.verify_folding_proof(inner_folding_proof.folding_data);
+        EXPECT_EQ(res, true);
+        info("Recursive Verifier Ultra: num gates = ", outer_circuit.num_gates);
+
+        // Check for a failure flag in the recursive verifier circuit
+        EXPECT_EQ(outer_circuit.failed(), false) << outer_circuit.err();
+    }
 };
+
+using BuilderTypes = testing::Types<UltraCircuitBuilder, GoblinUltraCircuitBuilder>;
+TYPED_TEST_SUITE(ProtogalaxyRecursiveTest, BuilderTypes);
+
+TYPED_TEST(ProtogalaxyRecursiveTest, InnerCircuit)
+{
+    TestFixture::test_inner_circuit();
+}
+
+TYPED_TEST(ProtogalaxyRecursiveTest, FirstRound)
+{
+    TestFixture::test_recursive_folding();
+}
+
 } // namespace proof_system::plonk::stdlib::recursion::honk
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.hpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.hpp
index 6077fe241ef..5522f0f72e5 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/ultra_recursive_verifier.hpp
@@ -20,11 +20,6 @@ template <typename Flavor> class UltraRecursiveVerifier_ {
 
     explicit UltraRecursiveVerifier_(Builder* builder,
                                      const std::shared_ptr<NativeVerificationKey>& native_verifier_key);
-    UltraRecursiveVerifier_(UltraRecursiveVerifier_&& other) = delete;
-    UltraRecursiveVerifier_(const UltraRecursiveVerifier_& other) = delete;
-    UltraRecursiveVerifier_& operator=(const UltraRecursiveVerifier_& other) = delete;
-    UltraRecursiveVerifier_& operator=(UltraRecursiveVerifier_&& other) = delete;
-    ~UltraRecursiveVerifier_() = default;
 
     // TODO(luke): Eventually this will return something like aggregation_state but I'm simplifying for now until we
     // determine the exact interface. Simply returns the two pairing points.

From 08942a37b56c00035b4e85612aeffb058fe020fb Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Wed, 17 Jan 2024 16:55:55 +0000
Subject: [PATCH 80/86] first working thingy

---
 .../flavor/goblin_ultra_recursive.hpp         |  2 -
 .../barretenberg/flavor/ultra_recursive.hpp   |  2 -
 .../verifier/decider_recursive_verifier.cpp   |  6 +-
 .../verifier/decider_recursive_verifier.hpp   |  5 +-
 .../protogalaxy_recursive_verifier.cpp        |  5 +-
 .../protogalaxy_recursive_verifier.hpp        | 12 +++-
 .../protogalaxy_recursive_verifier.test.cpp   | 67 +++++++++++++++----
 7 files changed, 71 insertions(+), 28 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
index 59e94e36ebb..d9f275bf21b 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra_recursive.hpp
@@ -47,8 +47,6 @@ class GoblinUltraRecursive {
     using Commitment = Curve::Element;
     using CommitmentHandle = Curve::Element;
     using NativeVerificationKey = flavor::GoblinUltra::VerificationKey;
-    using Polynomial = barretenberg::Polynomial<FF>;
-    using PolynomialHandle = std::span<FF>;
 
     // Note(luke): Eventually this may not be needed at all
     using VerifierCommitmentKey = pcs::VerifierCommitmentKey<Curve>;
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
index 12522a65fc0..a8ad134fc4e 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
@@ -55,8 +55,6 @@ template <typename BuilderType> class UltraRecursive_ {
     using CommitmentHandle = typename Curve::Element;
     using FF = typename Curve::ScalarField;
     using NativeVerificationKey = flavor::Ultra::VerificationKey;
-    using Polynomial = barretenberg::Polynomial<FF>;
-    using PolynomialHandle = std::span<FF>;
 
     // Note(luke): Eventually this may not be needed at all
     using VerifierCommitmentKey = pcs::VerifierCommitmentKey<Curve>;
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.cpp
index 847d1a087eb..140c35a178b 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.cpp
@@ -7,10 +7,8 @@
 namespace proof_system::plonk::stdlib::recursion::honk {
 
 template <typename Flavor>
-DeciderRecursiveVerifier_<Flavor>::DeciderRecursiveVerifier_(
-    Builder* builder, const std::shared_ptr<NativeVerificationKey>& native_verifier_key)
-    : key(std::make_shared<VerificationKey>(builder, native_verifier_key))
-    , builder(builder)
+DeciderRecursiveVerifier_<Flavor>::DeciderRecursiveVerifier_(Builder* builder)
+    : builder(builder)
 {}
 
 /**
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.hpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.hpp
index 44a6520aa16..253965dd1b5 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.hpp
@@ -12,18 +12,15 @@ template <typename Flavor> class DeciderRecursiveVerifier_ {
     using GroupElement = typename Flavor::GroupElement;
     using VerificationKey = typename Flavor::VerificationKey;
     using VerifierCommitmentKey = typename Flavor::VerifierCommitmentKey;
-    using NativeVerificationKey = typename Flavor::NativeVerificationKey;
     using Builder = typename Flavor::CircuitBuilder;
     using RelationSeparator = typename Flavor::RelationSeparator;
     using PairingPoints = std::array<GroupElement, 2>;
 
   public:
-    explicit DeciderRecursiveVerifier_(Builder* builder,
-                                       const std::shared_ptr<NativeVerificationKey>& native_verifier_key);
+    explicit DeciderRecursiveVerifier_(Builder* builder);
 
     PairingPoints verify_proof(const plonk::proof& proof);
 
-    std::shared_ptr<VerificationKey> key;
     std::map<std::string, Commitment> commitments;
     std::shared_ptr<VerifierCommitmentKey> pcs_verification_key;
     Builder* builder;
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
index 277837673e5..83724f3232d 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
@@ -169,10 +169,9 @@ bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
 
     perturbator_coeffs[0].assert_equal(accumulator->target_sum, "F(0) != e");
 
-    auto perturbator = Polynomial(perturbator_coeffs);
+    // auto perturbator = Polynomial<FF>(perturbator_coeffs);
     FF perturbator_challenge = transcript->get_challenge("perturbator_challenge");
-    auto perturbator_at_challenge = perturbator.evaluate(perturbator_challenge.get_value());
-
+    auto perturbator_at_challenge = evaluate_perturbator(perturbator_coeffs, perturbator_challenge);
     // The degree of K(X) is dk - k - 1 = k(d - 1) - 1. Hence we need  k(d - 1) evaluations to represent it.
     std::array<FF, VerifierInstances::BATCHED_EXTENDED_LENGTH - VerifierInstances::NUM> combiner_quotient_evals;
     for (size_t idx = 0; idx < VerifierInstances::BATCHED_EXTENDED_LENGTH - VerifierInstances::NUM; idx++) {
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp
index dd744e32dc1..35f03eac7e5 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp
@@ -21,7 +21,6 @@ template <class VerifierInstances> class ProtoGalaxyRecursiveVerifier_ {
     using Builder = typename Flavor::CircuitBuilder;
     using RelationSeparator = typename Flavor::RelationSeparator;
     using PairingPoints = std::array<GroupElement, 2>;
-    using Polynomial = typename Flavor::Polynomial;
 
     static constexpr size_t NUM_SUBRELATIONS = Flavor::NUM_SUBRELATIONS;
 
@@ -89,6 +88,17 @@ template <class VerifierInstances> class ProtoGalaxyRecursiveVerifier_ {
      *
      */
     bool verify_folding_proof(std::vector<uint8_t> proof);
+
+    static FF evaluate_perturbator(std::vector<FF> coeffs, FF z)
+    {
+        FF z_acc = FF(1);
+        FF result = FF(0);
+        for (size_t i = 0; i < coeffs.size(); i++) {
+            result += coeffs[i] * z_acc;
+            z_acc *= z;
+        }
+        return result;
+    };
 };
 
 extern template class ProtoGalaxyRecursiveVerifier_<
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
index 722353cc7c8..781c085c296 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
@@ -104,12 +104,36 @@ template <typename BuilderType> class ProtogalaxyRecursiveTest : public testing:
         EXPECT_EQ(result, true);
     }
 
+    static void test_evaluate_perturbator()
+    {
+        OuterBuilder builder;
+        using fr_ct = bn254<OuterBuilder>::ScalarField;
+        using fr = bn254<OuterBuilder>::ScalarFieldNative;
+
+        std::vector<fr> coeffs;
+        std::vector<fr_ct> coeffs_ct;
+        for (size_t idx = 0; idx < 8; idx++) {
+            auto el = fr::random_element();
+            coeffs.emplace_back(el);
+            coeffs_ct.emplace_back(fr_ct(&builder, el));
+        }
+        Polynomial<fr> poly(coeffs);
+        fr point = fr::random_element();
+        fr_ct point_ct(fr_ct(&builder, point));
+        auto res1 = poly.evaluate(point);
+
+        auto res2 = FoldingRecursiveVerifier::evaluate_perturbator(coeffs_ct, point_ct);
+        EXPECT_EQ(res1, res2.get_value());
+    }
+
     static void test_recursive_folding()
     {
         InnerBuilder builder1;
-        create_inner_circuit(builder1);
+        // create_inner_circuit(builder1);
+        builder1.add_public_variable(FF(1));
         InnerBuilder builder2;
-        create_inner_circuit(builder2);
+        builder2.add_public_variable(FF(1));
+        // create_inner_circuit(builder2);
         InnerComposer inner_composer = InnerComposer();
         auto instance1 = inner_composer.create_instance(builder1);
         auto instance2 = inner_composer.create_instance(builder2);
@@ -117,18 +141,32 @@ template <typename BuilderType> class ProtogalaxyRecursiveTest : public testing:
         // commitment key accessible
         auto inner_folding_prover = inner_composer.create_folding_prover(instances, inner_composer.commitment_key);
         auto inner_folding_proof = inner_folding_prover.fold_instances();
-        auto native_folding_verifier = inner_composer.create_folding_verifier();
-        auto res = native_folding_verifier.verify_folding_proof(inner_folding_proof.folding_data);
-        EXPECT_EQ(res, true);
-
-        // OuterBuilder outer_circuit;
-        // FoldingRecursiveVerifier verifier{ &outer_circuit };
-        // auto res = verifier.verify_folding_proof(inner_folding_proof.folding_data);
+        // auto native_folding_verifier = inner_composer.create_folding_verifier();
+        // auto res = native_folding_verifier.verify_folding_proof(inner_folding_proof.folding_data);
         // EXPECT_EQ(res, true);
-        // info("Recursive Verifier Ultra: num gates = ", outer_circuit.num_gates);
 
-        // // Check for a failure flag in the recursive verifier circuit
-        // EXPECT_EQ(outer_circuit.failed(), false) << outer_circuit.err();
+        OuterBuilder outer_folding_circuit;
+        FoldingRecursiveVerifier verifier{ &outer_folding_circuit };
+        auto res = verifier.verify_folding_proof(inner_folding_proof.folding_data);
+        EXPECT_EQ(res, true);
+        info("Recursive Verifier with Ultra instances: num gates = ", outer_folding_circuit.num_gates);
+
+        // Check for a failure flag in the recursive verifier circuit
+        EXPECT_EQ(outer_folding_circuit.failed(), false) << outer_folding_circuit.err();
+
+        // TODO: construct and verify a proof for the recursive folding verifier
+
+        auto inner_decider_prover =
+            inner_composer.create_decider_prover(inner_folding_proof.accumulator, inner_composer.commitment_key);
+        auto inner_decider_proof = inner_decider_prover.construct_proof();
+        OuterBuilder outer_decider_circuit;
+        DeciderRecursiveVerifier decider_verifier{ &outer_decider_circuit };
+        auto pairing_points = decider_verifier.verify_proof(inner_decider_proof);
+        static_cast<void>(pairing_points);
+        // need to check these pairing points
+        info("Decider Recursive Verifier: num gates = ", outer_decider_circuit.num_gates);
+        // Check for a failure flag in the recursive verifier circuit
+        EXPECT_EQ(outer_decider_circuit.failed(), false) << outer_decider_circuit.err();
     }
 };
 
@@ -140,6 +178,11 @@ TYPED_TEST(ProtogalaxyRecursiveTest, InnerCircuit)
     TestFixture::test_inner_circuit();
 }
 
+TYPED_TEST(ProtogalaxyRecursiveTest, NewEvaluate)
+{
+    TestFixture::test_evaluate_perturbator();
+}
+
 TYPED_TEST(ProtogalaxyRecursiveTest, FirstRound)
 {
     TestFixture::test_recursive_folding();

From 52d31b96a0d01b2a66e37ecf0e201da91eb561e8 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Thu, 18 Jan 2024 14:39:26 +0000
Subject: [PATCH 81/86] cleanups

---
 .../protogalaxy_bench/protogalaxy.bench.cpp   |  2 +-
 .../protogalaxy_recursive_verifier.cpp        | 31 ++++---
 .../protogalaxy_recursive_verifier.hpp        | 22 +++--
 .../protogalaxy_recursive_verifier.test.cpp   | 80 ++++++++++++++-----
 .../ultra_honk/protogalaxy.test.cpp           |  4 +-
 .../ultra_honk/ultra_composer.hpp             |  3 +-
 6 files changed, 96 insertions(+), 46 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/benchmark/protogalaxy_bench/protogalaxy.bench.cpp b/barretenberg/cpp/src/barretenberg/benchmark/protogalaxy_bench/protogalaxy.bench.cpp
index cecb40bbf9f..348b05d1756 100644
--- a/barretenberg/cpp/src/barretenberg/benchmark/protogalaxy_bench/protogalaxy.bench.cpp
+++ b/barretenberg/cpp/src/barretenberg/benchmark/protogalaxy_bench/protogalaxy.bench.cpp
@@ -30,7 +30,7 @@ void fold_one(State& state) noexcept
     std::shared_ptr<Instance> instance_1 = construct_instance();
     std::shared_ptr<Instance> instance_2 = construct_instance();
 
-    auto folding_prover = composer.create_folding_prover({ instance_1, instance_2 }, composer.commitment_key);
+    auto folding_prover = composer.create_folding_prover({ instance_1, instance_2 });
 
     for (auto _ : state) {
         auto proof = folding_prover.fold_instances();
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
index 83724f3232d..bef58642e77 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
@@ -149,7 +149,7 @@ template <class VerifierInstances> void ProtoGalaxyRecursiveVerifier_<VerifierIn
 }
 
 template <class VerifierInstances>
-bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std::vector<uint8_t> proof)
+void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std::vector<uint8_t> proof)
 {
     using Transcript = typename Flavor::Transcript;
     using ElementNative = typename Flavor::Curve::ElementNative;
@@ -168,9 +168,8 @@ bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
     }
 
     perturbator_coeffs[0].assert_equal(accumulator->target_sum, "F(0) != e");
-
-    // auto perturbator = Polynomial<FF>(perturbator_coeffs);
     FF perturbator_challenge = transcript->get_challenge("perturbator_challenge");
+
     auto perturbator_at_challenge = evaluate_perturbator(perturbator_coeffs, perturbator_challenge);
     // The degree of K(X) is dk - k - 1 = k(d - 1) - 1. Hence we need  k(d - 1) evaluations to represent it.
     std::array<FF, VerifierInstances::BATCHED_EXTENDED_LENGTH - VerifierInstances::NUM> combiner_quotient_evals;
@@ -190,12 +189,12 @@ bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
     auto expected_next_target_sum =
         perturbator_at_challenge * lagranges[0] + vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
     auto next_target_sum = transcript->template receive_from_prover<FF>("next_target_sum");
-    expected_next_target_sum.assert_equal(next_target_sum, "next_target_sum mismatch");
+    expected_next_target_sum.assert_equal(next_target_sum, "next target sum mismatch");
 
     auto expected_betas_star = update_gate_challenges(perturbator_challenge, accumulator->gate_challenges, deltas);
     for (size_t idx = 0; idx < accumulator->log_instance_size; idx++) {
         auto beta_star = transcript->template receive_from_prover<FF>("next_gate_challenge_" + std::to_string(idx));
-        expected_betas_star[idx].assert_equal(beta_star, "beta mismatch at: " + std::to_string(idx));
+        expected_betas_star[idx].assert_equal(beta_star, " next gate challenge mismatch at: " + std::to_string(idx));
     }
 
     // Compute ϕ and verify against the data received from the prover
@@ -227,19 +226,19 @@ bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
             inst++;
         }
         auto next_el = transcript->template receive_from_prover<FF>("next_public_input" + std::to_string(el_idx));
-        next_el.assert_equal(expected_el);
+        next_el.assert_equal(expected_el, "folded public input mismatch at: " + std::to_string(el_idx));
         el_idx++;
     }
 
     for (size_t alpha_idx = 0; alpha_idx < NUM_SUBRELATIONS - 1; alpha_idx++) {
-        FF alpha(0);
+        FF expected_alpha(0);
         size_t instance_idx = 0;
         for (auto& instance : instances) {
-            alpha += instance->alphas[alpha_idx] * lagranges[instance_idx];
+            expected_alpha += instance->alphas[alpha_idx] * lagranges[instance_idx];
             instance_idx++;
         }
         auto next_alpha = transcript->template receive_from_prover<FF>("next_alpha_" + std::to_string(alpha_idx));
-        next_alpha.assert_equal(alpha);
+        next_alpha.assert_equal(expected_alpha, "folded relation separator mismatch at: " + std::to_string(alpha_idx));
     }
 
     auto expected_parameters = proof_system::RelationParameters<FF>{};
@@ -255,20 +254,22 @@ bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
     }
 
     auto next_eta = transcript->template receive_from_prover<FF>("next_eta");
-    next_eta.assert_equal(expected_parameters.eta);
+    next_eta.assert_equal(expected_parameters.eta, "relation parameter eta mismatch");
 
     auto next_beta = transcript->template receive_from_prover<FF>("next_beta");
-    next_beta.assert_equal(expected_parameters.beta);
+    next_beta.assert_equal(expected_parameters.beta, "relation parameter beta mismatch");
 
     auto next_gamma = transcript->template receive_from_prover<FF>("next_gamma");
-    next_gamma.assert_equal(expected_parameters.gamma);
+    next_gamma.assert_equal(expected_parameters.gamma, "relation parameter gamma mismatch");
 
     auto next_public_input_delta = transcript->template receive_from_prover<FF>("next_public_input_delta");
-    next_public_input_delta.assert_equal(expected_parameters.public_input_delta);
+    next_public_input_delta.assert_equal(expected_parameters.public_input_delta,
+                                         "relation parameter public input delta mismatch");
 
     auto next_lookup_grand_product_delta =
         transcript->template receive_from_prover<FF>("next_lookup_grand_product_delta");
-    next_lookup_grand_product_delta.assert_equal(expected_parameters.lookup_grand_product_delta);
+    next_lookup_grand_product_delta.assert_equal(expected_parameters.lookup_grand_product_delta,
+                                                 "relation parameter lookup grand product delta mismatch");
 
     auto acc_vk = std::make_shared<VerificationKey>(instances[0]->instance_size, instances[0]->public_input_size);
     auto vk_labels = commitment_labels.get_precomputed();
@@ -286,8 +287,6 @@ bool ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
         vk.y.assert_equal(expected_vk.y);
         vk_idx++;
     }
-
-    return true;
 }
 
 template class ProtoGalaxyRecursiveVerifier_<
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp
index 35f03eac7e5..1b88894d399 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp
@@ -86,16 +86,28 @@ template <class VerifierInstances> class ProtoGalaxyRecursiveVerifier_ {
      * @brief Run the folding protocol on the verifier side to establish whether the public data ϕ of the new
      * accumulator, received from the prover is the same as that produced by the verifier.
      *
+     * @details In the recursive setting this function doesn't return anything because the equality checks performed by
+     * the recursive verifier, ensuring the folded ϕ*, e* and β* on the verifier side correspond to what has been sent
+     * by the prover, are expressed as constraints.
      */
-    bool verify_folding_proof(std::vector<uint8_t> proof);
+    void verify_folding_proof(std::vector<uint8_t> proof);
 
-    static FF evaluate_perturbator(std::vector<FF> coeffs, FF z)
+    /**
+     * @brief Evaluates the perturbator at a  given scalar, in a sequential manner for the recursive setting.
+     *
+     * @details This method is equivalent to the ones in the Polynomial class that evaluate a polynomial in coefficient
+     * form, class used in the native verifier for constructing and computing the perturbator. We implement this
+     * functionality here in the recursive folding verifier to avoid instantiating the entire Polynomial class on
+     * stdlib::bn254. The evaluation needs to be done sequentially asw e don't support a parallel_for in circuits.
+     *
+     */
+    static FF evaluate_perturbator(std::vector<FF> coeffs, FF point)
     {
-        FF z_acc = FF(1);
+        FF point_acc = FF(1);
         FF result = FF(0);
         for (size_t i = 0; i < coeffs.size(); i++) {
-            result += coeffs[i] * z_acc;
-            z_acc *= z;
+            result += coeffs[i] * point_acc;
+            point_acc *= point;
         }
         return result;
     };
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
index 781c085c296..500fb4c5426 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
@@ -15,7 +15,7 @@ template <typename BuilderType> class ProtogalaxyRecursiveTest : public testing:
     using UltraComposer = ::proof_system::honk::UltraComposer_<UltraFlavor>;
     using GoblinUltraComposer = ::proof_system::honk::UltraComposer_<GoblinUltraFlavor>;
 
-    using InnerFlavor = UltraFlavor; // type this
+    using InnerFlavor = UltraFlavor;
     using InnerComposer = UltraComposer;
     using Instance = ::proof_system::honk::ProverInstance_<InnerFlavor>;
     using InnerBuilder = typename InnerComposer::CircuitBuilder;
@@ -24,14 +24,26 @@ template <typename BuilderType> class ProtogalaxyRecursiveTest : public testing:
     using FF = InnerFlavor::FF;
 
     // Types for recursive verifier circuit
+    // cannot do on Goblin
     using RecursiveFlavor = ::proof_system::honk::flavor::UltraRecursive_<BuilderType>;
     using RecursiveVerifierInstances = ::proof_system::honk::VerifierInstances_<RecursiveFlavor, 2>;
     using FoldingRecursiveVerifier = ProtoGalaxyRecursiveVerifier_<RecursiveVerifierInstances>;
     using OuterBuilder = BuilderType;
     using DeciderRecursiveVerifier = DeciderRecursiveVerifier_<RecursiveFlavor>;
+    using DeciderVerifier = ::proof_system::honk::DeciderVerifier_<InnerFlavor>;
     using NativeVerifierInstances = ::proof_system::honk::VerifierInstances_<InnerFlavor, 2>;
     using NativeFoldingVerifier = proof_system::honk::ProtoGalaxyVerifier_<NativeVerifierInstances>;
 
+    // Helper for getting composer for prover/verifier of recursive (outer) circuit
+    template <typename BuilderT> static auto get_outer_composer()
+    {
+        if constexpr (IsGoblinBuilder<BuilderT>) {
+            return GoblinUltraComposer();
+        } else {
+            return UltraComposer();
+        }
+    }
+
     /**
      * @brief Create a non-trivial arbitrary inner circuit, the proof of which will be recursively verified
      *
@@ -88,7 +100,7 @@ template <typename BuilderType> class ProtogalaxyRecursiveTest : public testing:
     };
 
   public:
-    static void SetUpTestSuite() { barretenberg::srs::init_crs_factory("../srs_db/ignition"); }
+    static void SetUpTestSuite() { bb::srs::init_crs_factory("../srs_db/ignition"); }
 
     /**
      * @brief Create inner circuit and call check_circuit on it
@@ -104,6 +116,11 @@ template <typename BuilderType> class ProtogalaxyRecursiveTest : public testing:
         EXPECT_EQ(result, true);
     }
 
+    /**
+     * @brief Ensure that evaluating the perturbator in the recursive folding verifier returns the same result as
+     * evaluating in Polynomial class.
+     *
+     */
     static void test_evaluate_perturbator()
     {
         OuterBuilder builder;
@@ -128,49 +145,72 @@ template <typename BuilderType> class ProtogalaxyRecursiveTest : public testing:
 
     static void test_recursive_folding()
     {
+        // Create two arbitrary circuits of size 2^10
         InnerBuilder builder1;
-        // create_inner_circuit(builder1);
-        builder1.add_public_variable(FF(1));
+        create_inner_circuit(builder1, 20);
         InnerBuilder builder2;
-        builder2.add_public_variable(FF(1));
-        // create_inner_circuit(builder2);
+        create_inner_circuit(builder2, 20);
+
         InnerComposer inner_composer = InnerComposer();
         auto instance1 = inner_composer.create_instance(builder1);
         auto instance2 = inner_composer.create_instance(builder2);
         auto instances = std::vector<std::shared_ptr<Instance>>{ instance1, instance2 };
-        // commitment key accessible
-        auto inner_folding_prover = inner_composer.create_folding_prover(instances, inner_composer.commitment_key);
+
+        // Generate a folding proof
+        auto inner_folding_prover = inner_composer.create_folding_prover(instances);
         auto inner_folding_proof = inner_folding_prover.fold_instances();
-        // auto native_folding_verifier = inner_composer.create_folding_verifier();
-        // auto res = native_folding_verifier.verify_folding_proof(inner_folding_proof.folding_data);
-        // EXPECT_EQ(res, true);
 
+        // Create a recursive folding verifier circuit for the folding proof of the two instances
         OuterBuilder outer_folding_circuit;
         FoldingRecursiveVerifier verifier{ &outer_folding_circuit };
-        auto res = verifier.verify_folding_proof(inner_folding_proof.folding_data);
-        EXPECT_EQ(res, true);
+        verifier.verify_folding_proof(inner_folding_proof.folding_data);
         info("Recursive Verifier with Ultra instances: num gates = ", outer_folding_circuit.num_gates);
 
+        // Perform native folding verification and ensure it returns the same result (either true or false) as calling
+        // check_circuit on the recursive folding verifier
+        auto native_folding_verifier = inner_composer.create_folding_verifier();
+        auto native_folding_result = native_folding_verifier.verify_folding_proof(inner_folding_proof.folding_data);
+        EXPECT_EQ(native_folding_result, outer_folding_circuit.check_circuit());
+
         // Check for a failure flag in the recursive verifier circuit
         EXPECT_EQ(outer_folding_circuit.failed(), false) << outer_folding_circuit.err();
 
-        // TODO: construct and verify a proof for the recursive folding verifier
-
-        auto inner_decider_prover =
-            inner_composer.create_decider_prover(inner_folding_proof.accumulator, inner_composer.commitment_key);
+        // Create a decider proof for the relaxed instance obtained through folding
+        auto inner_decider_prover = inner_composer.create_decider_prover(inner_folding_proof.accumulator);
         auto inner_decider_proof = inner_decider_prover.construct_proof();
+
+        // Create a decider verifier circuit for recursively verifying the decider proof
         OuterBuilder outer_decider_circuit;
         DeciderRecursiveVerifier decider_verifier{ &outer_decider_circuit };
         auto pairing_points = decider_verifier.verify_proof(inner_decider_proof);
-        static_cast<void>(pairing_points);
-        // need to check these pairing points
         info("Decider Recursive Verifier: num gates = ", outer_decider_circuit.num_gates);
         // Check for a failure flag in the recursive verifier circuit
         EXPECT_EQ(outer_decider_circuit.failed(), false) << outer_decider_circuit.err();
+
+        // Perform native verification then perform the pairing on the outputs of the recursive
+        //  decider verifier and check that the result agrees.
+        DeciderVerifier native_decider_verifier =
+            inner_composer.create_decider_verifier(inner_folding_proof.accumulator);
+        auto native_result = native_decider_verifier.verify_proof(inner_decider_proof);
+        auto recursive_result = native_decider_verifier.pcs_verification_key->pairing_check(
+            pairing_points[0].get_value(), pairing_points[1].get_value());
+        EXPECT_EQ(native_result, recursive_result);
+
+        // Construct and verify a proof of the recursive decider verifier circuit
+        {
+            auto composer = get_outer_composer<OuterBuilder>();
+            auto instance = composer.create_instance(outer_decider_circuit);
+            auto prover = composer.create_prover(instance);
+            auto verifier = composer.create_verifier(instance);
+            auto proof = prover.construct_proof();
+            bool verified = verifier.verify_proof(proof);
+
+            ASSERT(verified);
+        }
     }
 };
 
-using BuilderTypes = testing::Types<UltraCircuitBuilder, GoblinUltraCircuitBuilder>;
+using BuilderTypes = testing::Types<GoblinUltraCircuitBuilder>;
 TYPED_TEST_SUITE(ProtogalaxyRecursiveTest, BuilderTypes);
 
 TYPED_TEST(ProtogalaxyRecursiveTest, InnerCircuit)
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
index 5eb362767fb..bc4a694a924 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/protogalaxy.test.cpp
@@ -51,7 +51,7 @@ std::shared_ptr<Instance> fold_and_verify(const std::vector<std::shared_ptr<Inst
                                           UltraComposer& composer,
                                           bool expected_result)
 {
-    auto folding_prover = composer.create_folding_prover(instances, composer.commitment_key);
+    auto folding_prover = composer.create_folding_prover(instances);
     auto folding_verifier = composer.create_folding_verifier();
 
     auto proof = folding_prover.fold_instances();
@@ -80,7 +80,7 @@ void check_accumulator_target_sum_manual(std::shared_ptr<Instance>& accumulator,
 }
 void decide_and_verify(std::shared_ptr<Instance>& accumulator, UltraComposer& composer, bool expected_result)
 {
-    auto decider_prover = composer.create_decider_prover(accumulator, composer.commitment_key);
+    auto decider_prover = composer.create_decider_prover(accumulator);
     auto decider_verifier = composer.create_decider_verifier(accumulator);
     auto decision = decider_prover.construct_proof();
     auto verified = decider_verifier.verify_proof(decision);
diff --git a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp
index 37110a3b657..1d2037fc557 100644
--- a/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp
+++ b/barretenberg/cpp/src/barretenberg/ultra_honk/ultra_composer.hpp
@@ -110,8 +110,7 @@ template <UltraFlavor Flavor> class UltraComposer_ {
      */
     MergeVerifier_<Flavor> create_merge_verifier() { return MergeVerifier_<Flavor>(); }
 
-    ProtoGalaxyProver_<ProverInstances> create_folding_prover(const std::vector<std::shared_ptr<Instance>>& instances,
-                                                              const std::shared_ptr<CommitmentKey>& commitment_key)
+    ProtoGalaxyProver_<ProverInstances> create_folding_prover(const std::vector<std::shared_ptr<Instance>>& instances)
     {
         ProtoGalaxyProver_<ProverInstances> output_state(instances, commitment_key);
 

From 167ad393585b339b3f7adcf484d6f4f4c5a79674 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Fri, 19 Jan 2024 13:27:18 +0000
Subject: [PATCH 82/86] stuff that works

---
 .../barretenberg/flavor/ultra_recursive.hpp   |  51 ++++----
 .../protogalaxy/protogalaxy_verifier.cpp      |   1 +
 .../protogalaxy_recursive_verifier.cpp        |  13 +-
 .../protogalaxy_recursive_verifier.test.cpp   | 123 ++++++++++++++----
 .../barretenberg/transcript/transcript.hpp    |   6 +-
 5 files changed, 132 insertions(+), 62 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
index a8ad134fc4e..60cc3e5e2d4 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
@@ -346,32 +346,31 @@ template <typename BuilderType> class UltraRecursive_ {
             this->z_lookup = "Z_LOOKUP";
             this->sorted_accum = "SORTED_ACCUM";
 
-            // The ones beginning with "__" are only used for debugging
-            this->q_c = "__Q_C";
-            this->q_l = "__Q_L";
-            this->q_r = "__Q_R";
-            this->q_o = "__Q_O";
-            this->q_4 = "__Q_4";
-            this->q_m = "__Q_M";
-            this->q_arith = "__Q_ARITH";
-            this->q_sort = "__Q_SORT";
-            this->q_elliptic = "__Q_ELLIPTIC";
-            this->q_aux = "__Q_AUX";
-            this->q_lookup = "__Q_LOOKUP";
-            this->sigma_1 = "__SIGMA_1";
-            this->sigma_2 = "__SIGMA_2";
-            this->sigma_3 = "__SIGMA_3";
-            this->sigma_4 = "__SIGMA_4";
-            this->id_1 = "__ID_1";
-            this->id_2 = "__ID_2";
-            this->id_3 = "__ID_3";
-            this->id_4 = "__ID_4";
-            this->table_1 = "__TABLE_1";
-            this->table_2 = "__TABLE_2";
-            this->table_3 = "__TABLE_3";
-            this->table_4 = "__TABLE_4";
-            this->lagrange_first = "__LAGRANGE_FIRST";
-            this->lagrange_last = "__LAGRANGE_LAST";
+            this->q_c = "Q_C";
+            this->q_l = "Q_L";
+            this->q_r = "Q_R";
+            this->q_o = "Q_O";
+            this->q_4 = "Q_4";
+            this->q_m = "Q_M";
+            this->q_arith = "Q_ARITH";
+            this->q_sort = "Q_SORT";
+            this->q_elliptic = "Q_ELLIPTIC";
+            this->q_aux = "Q_AUX";
+            this->q_lookup = "Q_LOOKUP";
+            this->sigma_1 = "SIGMA_1";
+            this->sigma_2 = "SIGMA_2";
+            this->sigma_3 = "SIGMA_3";
+            this->sigma_4 = "SIGMA_4";
+            this->id_1 = "ID_1";
+            this->id_2 = "ID_2";
+            this->id_3 = "ID_3";
+            this->id_4 = "ID_4";
+            this->table_1 = "TABLE_1";
+            this->table_2 = "TABLE_2";
+            this->table_3 = "TABLE_3";
+            this->table_4 = "TABLE_4";
+            this->lagrange_first = "LAGRANGE_FIRST";
+            this->lagrange_last = "LAGRANGE_LAST";
         };
     };
 
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index 871d4ad23b8..c52eb02ce92 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -156,6 +156,7 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
     }
 
     if (perturbator_coeffs[0] != accumulator->target_sum) {
+        info("here");
         return false;
     }
 
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
index bef58642e77..8a89978ce11 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
@@ -167,7 +167,7 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
         perturbator_coeffs[idx] = transcript->template receive_from_prover<FF>("perturbator_" + std::to_string(idx));
     }
 
-    perturbator_coeffs[0].assert_equal(accumulator->target_sum, "F(0) != e");
+    accumulator->target_sum.assert_equal(perturbator_coeffs[0], "F(0) != e");
     FF perturbator_challenge = transcript->get_challenge("perturbator_challenge");
 
     auto perturbator_at_challenge = evaluate_perturbator(perturbator_coeffs, perturbator_challenge);
@@ -201,16 +201,15 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
     WitnessCommitments acc_witness_commitments;
     auto witness_labels = commitment_labels.get_witness();
     size_t comm_idx = 0;
-    auto random_generator = AffineElementNative(ElementNative::random_element());
-    auto sth = Commitment::from_witness(builder, random_generator);
+    auto random_generator = Commitment::from_witness(builder, AffineElementNative(ElementNative::random_element()));
     for (auto& expected_comm : acc_witness_commitments.get_all()) {
-        expected_comm = sth;
+        expected_comm = random_generator;
         size_t inst = 0;
         for (auto& instance : instances) {
             expected_comm = expected_comm + instance->witness_commitments.get_all()[comm_idx] * lagranges[inst];
             inst++;
         }
-        expected_comm -= sth;
+        expected_comm -= random_generator;
         auto comm = transcript->template receive_from_prover<Commitment>("next_" + witness_labels[comm_idx]);
         comm.x.assert_equal(expected_comm.x);
         comm.y.assert_equal(expected_comm.y);
@@ -276,13 +275,13 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
     size_t vk_idx = 0;
     for (auto& expected_vk : acc_vk->get_all()) {
         size_t inst = 0;
-        expected_vk = sth;
+        expected_vk = random_generator;
         for (auto& instance : instances) {
             expected_vk = expected_vk + instance->verification_key->get_all()[vk_idx] * lagranges[inst];
             inst++;
         }
         auto vk = transcript->template receive_from_prover<Commitment>("next_" + vk_labels[vk_idx]);
-        expected_vk -= sth;
+        expected_vk -= random_generator;
         vk.x.assert_equal(expected_vk.x);
         vk.y.assert_equal(expected_vk.y);
         vk_idx++;
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
index 500fb4c5426..e401b09eb49 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
@@ -8,7 +8,8 @@
 #include "barretenberg/ultra_honk/ultra_composer.hpp"
 
 namespace proof_system::plonk::stdlib::recursion::honk {
-template <typename BuilderType> class ProtogalaxyRecursiveTest : public testing::Test {
+class ProtogalaxyRecursiveTest : public testing::Test {
+  public:
     // Define types relevant for testing
     using UltraFlavor = ::proof_system::honk::flavor::Ultra;
     using GoblinUltraFlavor = ::proof_system::honk::flavor::GoblinUltra;
@@ -25,10 +26,10 @@ template <typename BuilderType> class ProtogalaxyRecursiveTest : public testing:
 
     // Types for recursive verifier circuit
     // cannot do on Goblin
-    using RecursiveFlavor = ::proof_system::honk::flavor::UltraRecursive_<BuilderType>;
+    using OuterBuilder = GoblinUltraCircuitBuilder;
+    using RecursiveFlavor = ::proof_system::honk::flavor::UltraRecursive_<OuterBuilder>;
     using RecursiveVerifierInstances = ::proof_system::honk::VerifierInstances_<RecursiveFlavor, 2>;
     using FoldingRecursiveVerifier = ProtoGalaxyRecursiveVerifier_<RecursiveVerifierInstances>;
-    using OuterBuilder = BuilderType;
     using DeciderRecursiveVerifier = DeciderRecursiveVerifier_<RecursiveFlavor>;
     using DeciderVerifier = ::proof_system::honk::DeciderVerifier_<InnerFlavor>;
     using NativeVerifierInstances = ::proof_system::honk::VerifierInstances_<InnerFlavor, 2>;
@@ -143,19 +144,9 @@ template <typename BuilderType> class ProtogalaxyRecursiveTest : public testing:
         EXPECT_EQ(res1, res2.get_value());
     }
 
-    static void test_recursive_folding()
+    static std::shared_ptr<Instance> fold_and_verify(const std::vector<std::shared_ptr<Instance>>& instances,
+                                                     InnerComposer& inner_composer)
     {
-        // Create two arbitrary circuits of size 2^10
-        InnerBuilder builder1;
-        create_inner_circuit(builder1, 20);
-        InnerBuilder builder2;
-        create_inner_circuit(builder2, 20);
-
-        InnerComposer inner_composer = InnerComposer();
-        auto instance1 = inner_composer.create_instance(builder1);
-        auto instance2 = inner_composer.create_instance(builder2);
-        auto instances = std::vector<std::shared_ptr<Instance>>{ instance1, instance2 };
-
         // Generate a folding proof
         auto inner_folding_prover = inner_composer.create_folding_prover(instances);
         auto inner_folding_proof = inner_folding_prover.fold_instances();
@@ -172,11 +163,48 @@ template <typename BuilderType> class ProtogalaxyRecursiveTest : public testing:
         auto native_folding_result = native_folding_verifier.verify_folding_proof(inner_folding_proof.folding_data);
         EXPECT_EQ(native_folding_result, outer_folding_circuit.check_circuit());
 
+        // Ensure that the underlying native and recursive folding verification algorithms agree by ensuring
+        // the manifests produced by each agree.
+        auto recursive_folding_manifest = verifier.transcript->get_manifest();
+        auto native_folding_manifest = native_folding_verifier.transcript->get_manifest();
+
+        for (size_t i = 0; i < recursive_folding_manifest.size(); ++i) {
+            EXPECT_EQ(recursive_folding_manifest[i], native_folding_manifest[i]);
+        }
+
         // Check for a failure flag in the recursive verifier circuit
         EXPECT_EQ(outer_folding_circuit.failed(), false) << outer_folding_circuit.err();
 
+        return inner_folding_proof.accumulator;
+    }
+
+    static void test_recursive_folding()
+    {
+        // Create two arbitrary circuits for the first round of folding
+        InnerBuilder builder1;
+
+        create_inner_circuit(builder1);
+        InnerBuilder builder2;
+        builder2.add_public_variable(FF(1));
+        create_inner_circuit(builder2);
+
+        InnerComposer inner_composer = InnerComposer();
+        auto instance1 = inner_composer.create_instance(builder1);
+        auto instance2 = inner_composer.create_instance(builder2);
+        auto instances = std::vector<std::shared_ptr<Instance>>{ instance1, instance2 };
+
+        auto accumulator = fold_and_verify(instances, inner_composer);
+
+        // Create another circuit to do a second round of folding
+        InnerBuilder builder3;
+        create_inner_circuit(builder3);
+        auto instance3 = inner_composer.create_instance(builder3);
+        instances = std::vector<std::shared_ptr<Instance>>{ accumulator, instance3 };
+
+        accumulator = fold_and_verify(instances, inner_composer);
+
         // Create a decider proof for the relaxed instance obtained through folding
-        auto inner_decider_prover = inner_composer.create_decider_prover(inner_folding_proof.accumulator);
+        auto inner_decider_prover = inner_composer.create_decider_prover(accumulator);
         auto inner_decider_proof = inner_decider_prover.construct_proof();
 
         // Create a decider verifier circuit for recursively verifying the decider proof
@@ -189,13 +217,20 @@ template <typename BuilderType> class ProtogalaxyRecursiveTest : public testing:
 
         // Perform native verification then perform the pairing on the outputs of the recursive
         //  decider verifier and check that the result agrees.
-        DeciderVerifier native_decider_verifier =
-            inner_composer.create_decider_verifier(inner_folding_proof.accumulator);
+        DeciderVerifier native_decider_verifier = inner_composer.create_decider_verifier(accumulator);
         auto native_result = native_decider_verifier.verify_proof(inner_decider_proof);
         auto recursive_result = native_decider_verifier.pcs_verification_key->pairing_check(
             pairing_points[0].get_value(), pairing_points[1].get_value());
         EXPECT_EQ(native_result, recursive_result);
 
+        // Ensure that the underlying native and recursive decider verification algorithms agree by ensuring
+        // the manifests produced are the same.
+        auto recursive_decider_manifest = decider_verifier.transcript->get_manifest();
+        auto native_decider_manifest = native_decider_verifier.transcript->get_manifest();
+        for (size_t i = 0; i < recursive_decider_manifest.size(); ++i) {
+            EXPECT_EQ(recursive_decider_manifest[i], native_decider_manifest[i]);
+        }
+
         // Construct and verify a proof of the recursive decider verifier circuit
         {
             auto composer = get_outer_composer<OuterBuilder>();
@@ -208,24 +243,60 @@ template <typename BuilderType> class ProtogalaxyRecursiveTest : public testing:
             ASSERT(verified);
         }
     }
+
+    static void test_failing_recursive_deciding()
+    {
+        // Create two arbitrary circuits for the first round of folding
+        InnerBuilder builder1;
+
+        create_inner_circuit(builder1);
+        InnerBuilder builder2;
+        builder2.add_public_variable(FF(1));
+        create_inner_circuit(builder2);
+
+        InnerComposer inner_composer = InnerComposer();
+        auto instance1 = inner_composer.create_instance(builder1);
+        auto instance2 = inner_composer.create_instance(builder2);
+        auto instances = std::vector<std::shared_ptr<Instance>>{ instance1, instance2 };
+
+        auto accumulator = fold_and_verify(instances, inner_composer);
+
+        // Tamper with the accumulator by changing the target sum
+        accumulator->target_sum = FF::random_element();
+
+        // Create a decider proof for the relaxed instance obtained through folding
+        auto inner_decider_prover = inner_composer.create_decider_prover(accumulator);
+        auto inner_decider_proof = inner_decider_prover.construct_proof();
+
+        // Create a decider verifier circuit for recursively verifying the decider proof
+        OuterBuilder outer_decider_circuit;
+        DeciderRecursiveVerifier decider_verifier{ &outer_decider_circuit };
+        decider_verifier.verify_proof(inner_decider_proof);
+        info("Decider Recursive Verifier: num gates = ", outer_decider_circuit.num_gates);
+
+        // We expect the decider circuit check to fail due to the bad proof
+        EXPECT_FALSE(outer_decider_circuit.check_circuit());
+    }
 };
 
-using BuilderTypes = testing::Types<GoblinUltraCircuitBuilder>;
-TYPED_TEST_SUITE(ProtogalaxyRecursiveTest, BuilderTypes);
+TEST_F(ProtogalaxyRecursiveTest, InnerCircuit)
+{
+    test_inner_circuit();
+}
 
-TYPED_TEST(ProtogalaxyRecursiveTest, InnerCircuit)
+TEST_F(ProtogalaxyRecursiveTest, NewEvaluate)
 {
-    TestFixture::test_inner_circuit();
+    test_evaluate_perturbator();
 }
 
-TYPED_TEST(ProtogalaxyRecursiveTest, NewEvaluate)
+TEST_F(ProtogalaxyRecursiveTest, FullProtogalaxyRecursiveTest)
 {
-    TestFixture::test_evaluate_perturbator();
+    test_recursive_folding();
 }
 
-TYPED_TEST(ProtogalaxyRecursiveTest, FirstRound)
+TEST_F(ProtogalaxyRecursiveTest, TamperedDeciderProof)
 {
-    TestFixture::test_recursive_folding();
+    test_failing_recursive_deciding();
 }
 
 } // namespace proof_system::plonk::stdlib::recursion::honk
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/transcript/transcript.hpp b/barretenberg/cpp/src/barretenberg/transcript/transcript.hpp
index a809c4b801f..4a85e02b74f 100644
--- a/barretenberg/cpp/src/barretenberg/transcript/transcript.hpp
+++ b/barretenberg/cpp/src/barretenberg/transcript/transcript.hpp
@@ -10,9 +10,9 @@
 namespace proof_system::honk {
 
 template <typename T, typename... U>
-concept Loggable = (std::same_as<T, bb::fr> || std::same_as<T, grumpkin::fr> ||
-                    std::same_as<T, bb::g1::affine_element> || std::same_as<T, grumpkin::g1::affine_element> ||
-                    std::same_as<T, uint32_t>);
+concept Loggable =
+    (std::same_as<T, bb::fr> || std::same_as<T, grumpkin::fr> || std::same_as<T, bb::g1::affine_element> ||
+     std::same_as<T, grumpkin::g1::affine_element> || std::same_as<T, uint32_t>);
 
 // class TranscriptManifest;
 class TranscriptManifest {

From 2a772e0f887e67d3bdaa17bea683a6190cceff65 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Fri, 19 Jan 2024 13:35:47 +0000
Subject: [PATCH 83/86] fix formatting

---
 .../protogalaxy_recursive_verifier.test.cpp   | 37 +++++++++++++++++++
 .../barretenberg/transcript/transcript.hpp    |  6 +--
 2 files changed, 40 insertions(+), 3 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
index e401b09eb49..e59134db211 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
@@ -299,4 +299,41 @@ TEST_F(ProtogalaxyRecursiveTest, TamperedDeciderProof)
     test_failing_recursive_deciding();
 }
 
+TEST_F(ProtogalaxyRecursiveTest, TamperedAccumulator)
+{
+    // Create two arbitrary circuits for the first round of folding
+    InnerBuilder builder1;
+
+    create_inner_circuit(builder1);
+    InnerBuilder builder2;
+    builder2.add_public_variable(FF(1));
+    create_inner_circuit(builder2);
+
+    InnerComposer inner_composer = InnerComposer();
+    auto instance1 = inner_composer.create_instance(builder1);
+    auto instance2 = inner_composer.create_instance(builder2);
+    auto instances = std::vector<std::shared_ptr<Instance>>{ instance1, instance2 };
+
+    auto accumulator = fold_and_verify(instances, inner_composer);
+
+    // Create another circuit to do a second round of folding
+    InnerBuilder builder3;
+    create_inner_circuit(builder3);
+    auto instance3 = inner_composer.create_instance(builder3);
+
+    // Tamper with the accumulator
+    instances = std::vector<std::shared_ptr<Instance>>{ accumulator, instance3 };
+    accumulator->prover_polynomials.w_l[1] = FF::random_element();
+
+    // Generate a folding proof
+    auto inner_folding_prover = inner_composer.create_folding_prover(instances);
+    auto inner_folding_proof = inner_folding_prover.fold_instances();
+
+    // Create a recursive folding verifier circuit for the folding proof of the two instances
+    OuterBuilder outer_folding_circuit;
+    FoldingRecursiveVerifier verifier{ &outer_folding_circuit };
+    verifier.verify_folding_proof(inner_folding_proof.folding_data);
+    EXPECT_EQ(outer_folding_circuit.check_circuit(), false);
+}
+
 } // namespace proof_system::plonk::stdlib::recursion::honk
\ No newline at end of file
diff --git a/barretenberg/cpp/src/barretenberg/transcript/transcript.hpp b/barretenberg/cpp/src/barretenberg/transcript/transcript.hpp
index 4a85e02b74f..a809c4b801f 100644
--- a/barretenberg/cpp/src/barretenberg/transcript/transcript.hpp
+++ b/barretenberg/cpp/src/barretenberg/transcript/transcript.hpp
@@ -10,9 +10,9 @@
 namespace proof_system::honk {
 
 template <typename T, typename... U>
-concept Loggable =
-    (std::same_as<T, bb::fr> || std::same_as<T, grumpkin::fr> || std::same_as<T, bb::g1::affine_element> ||
-     std::same_as<T, grumpkin::g1::affine_element> || std::same_as<T, uint32_t>);
+concept Loggable = (std::same_as<T, bb::fr> || std::same_as<T, grumpkin::fr> ||
+                    std::same_as<T, bb::g1::affine_element> || std::same_as<T, grumpkin::g1::affine_element> ||
+                    std::same_as<T, uint32_t>);
 
 // class TranscriptManifest;
 class TranscriptManifest {

From 06089f72d9330eb32017e165005ef86e408256e8 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Fri, 19 Jan 2024 17:00:34 +0000
Subject: [PATCH 84/86] more stuff

---
 .../protogalaxy/protogalaxy_verifier.cpp      |   1 -
 .../verifier/decider_recursive_verifier.cpp   |  18 +-
 .../honk/verifier/decider_verifier.test.cpp   |   0
 .../protogalaxy_recursive_verifier.cpp        |  54 ++--
 .../protogalaxy_recursive_verifier.hpp        |   1 +
 .../protogalaxy_recursive_verifier.test.cpp   | 264 ++++++++----------
 6 files changed, 162 insertions(+), 176 deletions(-)
 delete mode 100644 barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_verifier.test.cpp

diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index c52eb02ce92..871d4ad23b8 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -156,7 +156,6 @@ bool ProtoGalaxyVerifier_<VerifierInstances>::verify_folding_proof(std::vector<u
     }
 
     if (perturbator_coeffs[0] != accumulator->target_sum) {
-        info("here");
         return false;
     }
 
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.cpp
index 140c35a178b..dafa1839e49 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.cpp
@@ -30,13 +30,12 @@ std::array<typename Flavor::GroupElement, 2> DeciderRecursiveVerifier_<Flavor>::
     transcript = std::make_shared<Transcript>(builder, proof.proof_data);
     auto inst = std::make_unique<Instance>();
 
-    auto instance_size = transcript->template receive_from_prover<uint32_t>("instance_size");
-    auto public_input_size = transcript->template receive_from_prover<uint32_t>("public_input_size");
-    inst->instance_size = static_cast<size_t>(instance_size.get_value());
-    inst->public_input_size = static_cast<size_t>(public_input_size.get_value());
-    inst->log_instance_size = static_cast<size_t>(numeric::get_msb(inst->instance_size));
+    const auto instance_size = transcript->template receive_from_prover<uint32_t>("instance_size");
+    const auto public_input_size = transcript->template receive_from_prover<uint32_t>("public_input_size");
+    const auto log_instance_size =
+        static_cast<size_t>(numeric::get_msb(static_cast<size_t>(instance_size.get_value())));
 
-    for (size_t i = 0; i < inst->public_input_size; ++i) {
+    for (size_t i = 0; i < static_cast<size_t>(public_input_size.get_value()); ++i) {
         auto public_input_i = transcript->template receive_from_prover<FF>("public_input_" + std::to_string(i));
         inst->public_inputs.emplace_back(public_input_i);
     }
@@ -50,14 +49,13 @@ std::array<typename Flavor::GroupElement, 2> DeciderRecursiveVerifier_<Flavor>::
         RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
 
     for (size_t idx = 0; idx < NUM_SUBRELATIONS - 1; idx++) {
-
         inst->alphas[idx] = transcript->template receive_from_prover<FF>("alpha" + std::to_string(idx));
     }
 
     inst->target_sum = transcript->template receive_from_prover<FF>("target_sum");
 
-    inst->gate_challenges = std::vector<FF>(inst->log_instance_size);
-    for (size_t idx = 0; idx < inst->log_instance_size; idx++) {
+    inst->gate_challenges = std::vector<FF>(log_instance_size);
+    for (size_t idx = 0; idx < log_instance_size; idx++) {
         inst->gate_challenges[idx] =
             transcript->template receive_from_prover<FF>("gate_challenge_" + std::to_string(idx));
     }
@@ -76,7 +74,7 @@ std::array<typename Flavor::GroupElement, 2> DeciderRecursiveVerifier_<Flavor>::
 
     VerifierCommitments commitments{ inst->verification_key, inst->witness_commitments };
 
-    auto sumcheck = Sumcheck(inst->log_instance_size, transcript, inst->target_sum);
+    auto sumcheck = Sumcheck(log_instance_size, transcript, inst->target_sum);
 
     auto [multivariate_challenge, claimed_evaluations, sumcheck_verified] =
         sumcheck.verify(inst->relation_parameters, inst->alphas, inst->gate_challenges);
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_verifier.test.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_verifier.test.cpp
deleted file mode 100644
index e69de29bb2d..00000000000
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
index 8a89978ce11..ceb6fa8aadc 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
@@ -10,9 +10,9 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::receive_accumulator(const
     const auto instance_size = transcript->template receive_from_prover<uint32_t>(domain_separator + "_instance_size");
     const auto public_input_size =
         transcript->template receive_from_prover<uint32_t>(domain_separator + "_public_input_size");
-    inst->instance_size = static_cast<size_t>(instance_size.get_value());
-    inst->log_instance_size = static_cast<size_t>(numeric::get_msb(inst->instance_size));
-    inst->public_input_size = static_cast<size_t>(public_input_size.get_value());
+    inst->instance_size = uint32_t(instance_size.get_value());
+    inst->log_instance_size = uint32_t(numeric::get_msb(inst->instance_size));
+    inst->public_input_size = uint32_t(public_input_size.get_value());
 
     for (size_t i = 0; i < inst->public_input_size; ++i) {
         auto public_input_i =
@@ -63,9 +63,9 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::receive_and_finalise_inst
     const auto instance_size = transcript->template receive_from_prover<uint32_t>(domain_separator + "_instance_size");
     const auto public_input_size =
         transcript->template receive_from_prover<uint32_t>(domain_separator + "_public_input_size");
-    inst->instance_size = static_cast<size_t>(instance_size.get_value());
+    inst->instance_size = static_cast<size_t>(uint256_t(instance_size.get_value()));
     inst->log_instance_size = static_cast<size_t>(numeric::get_msb(inst->instance_size));
-    inst->public_input_size = static_cast<size_t>(public_input_size.get_value());
+    inst->public_input_size = static_cast<size_t>(uint256_t(public_input_size.get_value()));
 
     for (size_t i = 0; i < inst->public_input_size; ++i) {
         auto public_input_i =
@@ -154,6 +154,7 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
     using Transcript = typename Flavor::Transcript;
     using ElementNative = typename Flavor::Curve::ElementNative;
     using AffineElementNative = typename Flavor::Curve::AffineElementNative;
+    using ScalarNative = typename Flavor::Curve::ScalarFieldNative;
 
     transcript = std::make_shared<Transcript>(builder, proof);
     prepare_for_folding();
@@ -167,7 +168,12 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
         perturbator_coeffs[idx] = transcript->template receive_from_prover<FF>("perturbator_" + std::to_string(idx));
     }
 
-    accumulator->target_sum.assert_equal(perturbator_coeffs[0], "F(0) != e");
+    // TODO(): As currently the stdlib transcript is not creating proper constraints linked to Fiat-Shamir we add an
+    // additonal gate to ensure assert_equal is correct. This comparison to 0 can be removed here and below once we have
+    // merged the transcript.
+    auto zero = FF::from_witness(builder, ScalarNative(0));
+    zero.assert_equal(accumulator->target_sum - perturbator_coeffs[0], "F(0) != e");
+
     FF perturbator_challenge = transcript->get_challenge("perturbator_challenge");
 
     auto perturbator_at_challenge = evaluate_perturbator(perturbator_coeffs, perturbator_challenge);
@@ -189,12 +195,13 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
     auto expected_next_target_sum =
         perturbator_at_challenge * lagranges[0] + vanishing_polynomial_at_challenge * combiner_quotient_at_challenge;
     auto next_target_sum = transcript->template receive_from_prover<FF>("next_target_sum");
-    expected_next_target_sum.assert_equal(next_target_sum, "next target sum mismatch");
+    zero.assert_equal(expected_next_target_sum - next_target_sum, "next target sum mismatch");
 
     auto expected_betas_star = update_gate_challenges(perturbator_challenge, accumulator->gate_challenges, deltas);
     for (size_t idx = 0; idx < accumulator->log_instance_size; idx++) {
         auto beta_star = transcript->template receive_from_prover<FF>("next_gate_challenge_" + std::to_string(idx));
-        expected_betas_star[idx].assert_equal(beta_star, " next gate challenge mismatch at: " + std::to_string(idx));
+        zero.assert_equal(beta_star - expected_betas_star[idx],
+                          " next gate challenge mismatch at: " + std::to_string(idx));
     }
 
     // Compute ϕ and verify against the data received from the prover
@@ -209,10 +216,10 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
             expected_comm = expected_comm + instance->witness_commitments.get_all()[comm_idx] * lagranges[inst];
             inst++;
         }
-        expected_comm -= random_generator;
         auto comm = transcript->template receive_from_prover<Commitment>("next_" + witness_labels[comm_idx]);
-        comm.x.assert_equal(expected_comm.x);
-        comm.y.assert_equal(expected_comm.y);
+        auto res = expected_comm - comm;
+        random_generator.x.assert_equal(res.x);
+        random_generator.y.assert_equal(res.y);
         comm_idx++;
     }
 
@@ -225,7 +232,7 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
             inst++;
         }
         auto next_el = transcript->template receive_from_prover<FF>("next_public_input" + std::to_string(el_idx));
-        next_el.assert_equal(expected_el, "folded public input mismatch at: " + std::to_string(el_idx));
+        zero.assert_equal(expected_el - next_el, "folded public input mismatch at: " + std::to_string(el_idx));
         el_idx++;
     }
 
@@ -237,7 +244,8 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
             instance_idx++;
         }
         auto next_alpha = transcript->template receive_from_prover<FF>("next_alpha_" + std::to_string(alpha_idx));
-        next_alpha.assert_equal(expected_alpha, "folded relation separator mismatch at: " + std::to_string(alpha_idx));
+        zero.assert_equal(expected_alpha - next_alpha,
+                          "folded relation separator mismatch at: " + std::to_string(alpha_idx));
     }
 
     auto expected_parameters = proof_system::RelationParameters<FF>{};
@@ -253,22 +261,22 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
     }
 
     auto next_eta = transcript->template receive_from_prover<FF>("next_eta");
-    next_eta.assert_equal(expected_parameters.eta, "relation parameter eta mismatch");
+    zero.assert_equal(expected_parameters.eta - next_eta, "relation parameter eta mismatch");
 
     auto next_beta = transcript->template receive_from_prover<FF>("next_beta");
-    next_beta.assert_equal(expected_parameters.beta, "relation parameter beta mismatch");
+    zero.assert_equal(expected_parameters.beta - next_beta, "relation parameter beta mismatch");
 
     auto next_gamma = transcript->template receive_from_prover<FF>("next_gamma");
-    next_gamma.assert_equal(expected_parameters.gamma, "relation parameter gamma mismatch");
+    zero.assert_equal(expected_parameters.gamma - next_gamma, "relation parameter gamma mismatch");
 
     auto next_public_input_delta = transcript->template receive_from_prover<FF>("next_public_input_delta");
-    next_public_input_delta.assert_equal(expected_parameters.public_input_delta,
-                                         "relation parameter public input delta mismatch");
+    zero.assert_equal(expected_parameters.public_input_delta - next_public_input_delta,
+                      "relation parameter public input delta mismatch");
 
     auto next_lookup_grand_product_delta =
         transcript->template receive_from_prover<FF>("next_lookup_grand_product_delta");
-    next_lookup_grand_product_delta.assert_equal(expected_parameters.lookup_grand_product_delta,
-                                                 "relation parameter lookup grand product delta mismatch");
+    zero.assert_equal(expected_parameters.lookup_grand_product_delta - next_lookup_grand_product_delta,
+                      "relation parameter lookup grand product delta mismatch");
 
     auto acc_vk = std::make_shared<VerificationKey>(instances[0]->instance_size, instances[0]->public_input_size);
     auto vk_labels = commitment_labels.get_precomputed();
@@ -281,9 +289,9 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
             inst++;
         }
         auto vk = transcript->template receive_from_prover<Commitment>("next_" + vk_labels[vk_idx]);
-        expected_vk -= random_generator;
-        vk.x.assert_equal(expected_vk.x);
-        vk.y.assert_equal(expected_vk.y);
+        auto res = expected_vk - vk;
+        random_generator.x.assert_equal(res.x);
+        random_generator.y.assert_equal(res.y);
         vk_idx++;
     }
 }
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp
index 1b88894d399..7cc967a3129 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp
@@ -89,6 +89,7 @@ template <class VerifierInstances> class ProtoGalaxyRecursiveVerifier_ {
      * @details In the recursive setting this function doesn't return anything because the equality checks performed by
      * the recursive verifier, ensuring the folded ϕ*, e* and β* on the verifier side correspond to what has been sent
      * by the prover, are expressed as constraints.
+
      */
     void verify_folding_proof(std::vector<uint8_t> proof);
 
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
index e59134db211..a0d166c5ec3 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
@@ -103,47 +103,6 @@ class ProtogalaxyRecursiveTest : public testing::Test {
   public:
     static void SetUpTestSuite() { bb::srs::init_crs_factory("../srs_db/ignition"); }
 
-    /**
-     * @brief Create inner circuit and call check_circuit on it
-     *
-     */
-    static void test_inner_circuit()
-    {
-        InnerBuilder builder;
-
-        create_inner_circuit(builder);
-
-        bool result = builder.check_circuit();
-        EXPECT_EQ(result, true);
-    }
-
-    /**
-     * @brief Ensure that evaluating the perturbator in the recursive folding verifier returns the same result as
-     * evaluating in Polynomial class.
-     *
-     */
-    static void test_evaluate_perturbator()
-    {
-        OuterBuilder builder;
-        using fr_ct = bn254<OuterBuilder>::ScalarField;
-        using fr = bn254<OuterBuilder>::ScalarFieldNative;
-
-        std::vector<fr> coeffs;
-        std::vector<fr_ct> coeffs_ct;
-        for (size_t idx = 0; idx < 8; idx++) {
-            auto el = fr::random_element();
-            coeffs.emplace_back(el);
-            coeffs_ct.emplace_back(fr_ct(&builder, el));
-        }
-        Polynomial<fr> poly(coeffs);
-        fr point = fr::random_element();
-        fr_ct point_ct(fr_ct(&builder, point));
-        auto res1 = poly.evaluate(point);
-
-        auto res2 = FoldingRecursiveVerifier::evaluate_perturbator(coeffs_ct, point_ct);
-        EXPECT_EQ(res1, res2.get_value());
-    }
-
     static std::shared_ptr<Instance> fold_and_verify(const std::vector<std::shared_ptr<Instance>>& instances,
                                                      InnerComposer& inner_composer)
     {
@@ -177,126 +136,147 @@ class ProtogalaxyRecursiveTest : public testing::Test {
 
         return inner_folding_proof.accumulator;
     }
+};
+/**
+ * @brief Create inner circuit and call check_circuit on it
+ *
+ */
+TEST_F(ProtogalaxyRecursiveTest, InnerCircuit)
+{
+    InnerBuilder builder;
 
-    static void test_recursive_folding()
-    {
-        // Create two arbitrary circuits for the first round of folding
-        InnerBuilder builder1;
-
-        create_inner_circuit(builder1);
-        InnerBuilder builder2;
-        builder2.add_public_variable(FF(1));
-        create_inner_circuit(builder2);
-
-        InnerComposer inner_composer = InnerComposer();
-        auto instance1 = inner_composer.create_instance(builder1);
-        auto instance2 = inner_composer.create_instance(builder2);
-        auto instances = std::vector<std::shared_ptr<Instance>>{ instance1, instance2 };
-
-        auto accumulator = fold_and_verify(instances, inner_composer);
-
-        // Create another circuit to do a second round of folding
-        InnerBuilder builder3;
-        create_inner_circuit(builder3);
-        auto instance3 = inner_composer.create_instance(builder3);
-        instances = std::vector<std::shared_ptr<Instance>>{ accumulator, instance3 };
-
-        accumulator = fold_and_verify(instances, inner_composer);
-
-        // Create a decider proof for the relaxed instance obtained through folding
-        auto inner_decider_prover = inner_composer.create_decider_prover(accumulator);
-        auto inner_decider_proof = inner_decider_prover.construct_proof();
-
-        // Create a decider verifier circuit for recursively verifying the decider proof
-        OuterBuilder outer_decider_circuit;
-        DeciderRecursiveVerifier decider_verifier{ &outer_decider_circuit };
-        auto pairing_points = decider_verifier.verify_proof(inner_decider_proof);
-        info("Decider Recursive Verifier: num gates = ", outer_decider_circuit.num_gates);
-        // Check for a failure flag in the recursive verifier circuit
-        EXPECT_EQ(outer_decider_circuit.failed(), false) << outer_decider_circuit.err();
-
-        // Perform native verification then perform the pairing on the outputs of the recursive
-        //  decider verifier and check that the result agrees.
-        DeciderVerifier native_decider_verifier = inner_composer.create_decider_verifier(accumulator);
-        auto native_result = native_decider_verifier.verify_proof(inner_decider_proof);
-        auto recursive_result = native_decider_verifier.pcs_verification_key->pairing_check(
-            pairing_points[0].get_value(), pairing_points[1].get_value());
-        EXPECT_EQ(native_result, recursive_result);
-
-        // Ensure that the underlying native and recursive decider verification algorithms agree by ensuring
-        // the manifests produced are the same.
-        auto recursive_decider_manifest = decider_verifier.transcript->get_manifest();
-        auto native_decider_manifest = native_decider_verifier.transcript->get_manifest();
-        for (size_t i = 0; i < recursive_decider_manifest.size(); ++i) {
-            EXPECT_EQ(recursive_decider_manifest[i], native_decider_manifest[i]);
-        }
+    create_inner_circuit(builder);
 
-        // Construct and verify a proof of the recursive decider verifier circuit
-        {
-            auto composer = get_outer_composer<OuterBuilder>();
-            auto instance = composer.create_instance(outer_decider_circuit);
-            auto prover = composer.create_prover(instance);
-            auto verifier = composer.create_verifier(instance);
-            auto proof = prover.construct_proof();
-            bool verified = verifier.verify_proof(proof);
+    bool result = builder.check_circuit();
+    EXPECT_EQ(result, true);
+}
 
-            ASSERT(verified);
-        }
+/**
+ * @brief Ensure that evaluating the perturbator in the recursive folding verifier returns the same result as
+ * evaluating in Polynomial class.
+ *
+ */
+TEST_F(ProtogalaxyRecursiveTest, NewEvaluate)
+{
+    OuterBuilder builder;
+    using fr_ct = bn254<OuterBuilder>::ScalarField;
+    using fr = bn254<OuterBuilder>::ScalarFieldNative;
+
+    std::vector<fr> coeffs;
+    std::vector<fr_ct> coeffs_ct;
+    for (size_t idx = 0; idx < 8; idx++) {
+        auto el = fr::random_element();
+        coeffs.emplace_back(el);
+        coeffs_ct.emplace_back(fr_ct(&builder, el));
     }
+    Polynomial<fr> poly(coeffs);
+    fr point = fr::random_element();
+    fr_ct point_ct(fr_ct(&builder, point));
+    auto res1 = poly.evaluate(point);
 
-    static void test_failing_recursive_deciding()
-    {
-        // Create two arbitrary circuits for the first round of folding
-        InnerBuilder builder1;
+    auto res2 = FoldingRecursiveVerifier::evaluate_perturbator(coeffs_ct, point_ct);
+    EXPECT_EQ(res1, res2.get_value());
+}
 
-        create_inner_circuit(builder1);
-        InnerBuilder builder2;
-        builder2.add_public_variable(FF(1));
-        create_inner_circuit(builder2);
+TEST_F(ProtogalaxyRecursiveTest, FullProtogalaxyRecursiveTest)
+{
 
-        InnerComposer inner_composer = InnerComposer();
-        auto instance1 = inner_composer.create_instance(builder1);
-        auto instance2 = inner_composer.create_instance(builder2);
-        auto instances = std::vector<std::shared_ptr<Instance>>{ instance1, instance2 };
+    // Create two arbitrary circuits for the first round of folding
+    InnerBuilder builder1;
 
-        auto accumulator = fold_and_verify(instances, inner_composer);
+    create_inner_circuit(builder1);
+    InnerBuilder builder2;
+    builder2.add_public_variable(FF(1));
+    create_inner_circuit(builder2);
 
-        // Tamper with the accumulator by changing the target sum
-        accumulator->target_sum = FF::random_element();
+    InnerComposer inner_composer = InnerComposer();
+    auto instance1 = inner_composer.create_instance(builder1);
+    auto instance2 = inner_composer.create_instance(builder2);
+    auto instances = std::vector<std::shared_ptr<Instance>>{ instance1, instance2 };
 
-        // Create a decider proof for the relaxed instance obtained through folding
-        auto inner_decider_prover = inner_composer.create_decider_prover(accumulator);
-        auto inner_decider_proof = inner_decider_prover.construct_proof();
+    auto accumulator = fold_and_verify(instances, inner_composer);
 
-        // Create a decider verifier circuit for recursively verifying the decider proof
-        OuterBuilder outer_decider_circuit;
-        DeciderRecursiveVerifier decider_verifier{ &outer_decider_circuit };
-        decider_verifier.verify_proof(inner_decider_proof);
-        info("Decider Recursive Verifier: num gates = ", outer_decider_circuit.num_gates);
+    // Create another circuit to do a second round of folding
+    InnerBuilder builder3;
+    create_inner_circuit(builder3);
+    auto instance3 = inner_composer.create_instance(builder3);
+    instances = std::vector<std::shared_ptr<Instance>>{ accumulator, instance3 };
 
-        // We expect the decider circuit check to fail due to the bad proof
-        EXPECT_FALSE(outer_decider_circuit.check_circuit());
+    accumulator = fold_and_verify(instances, inner_composer);
+
+    // Create a decider proof for the relaxed instance obtained through folding
+    auto inner_decider_prover = inner_composer.create_decider_prover(accumulator);
+    auto inner_decider_proof = inner_decider_prover.construct_proof();
+
+    // Create a decider verifier circuit for recursively verifying the decider proof
+    OuterBuilder outer_decider_circuit;
+    DeciderRecursiveVerifier decider_verifier{ &outer_decider_circuit };
+    auto pairing_points = decider_verifier.verify_proof(inner_decider_proof);
+    info("Decider Recursive Verifier: num gates = ", outer_decider_circuit.num_gates);
+    // Check for a failure flag in the recursive verifier circuit
+    EXPECT_EQ(outer_decider_circuit.failed(), false) << outer_decider_circuit.err();
+
+    // Perform native verification then perform the pairing on the outputs of the recursive
+    //  decider verifier and check that the result agrees.
+    DeciderVerifier native_decider_verifier = inner_composer.create_decider_verifier(accumulator);
+    auto native_result = native_decider_verifier.verify_proof(inner_decider_proof);
+    auto recursive_result = native_decider_verifier.pcs_verification_key->pairing_check(pairing_points[0].get_value(),
+                                                                                        pairing_points[1].get_value());
+    EXPECT_EQ(native_result, recursive_result);
+
+    // Ensure that the underlying native and recursive decider verification algorithms agree by ensuring
+    // the manifests produced are the same.
+    auto recursive_decider_manifest = decider_verifier.transcript->get_manifest();
+    auto native_decider_manifest = native_decider_verifier.transcript->get_manifest();
+    for (size_t i = 0; i < recursive_decider_manifest.size(); ++i) {
+        EXPECT_EQ(recursive_decider_manifest[i], native_decider_manifest[i]);
     }
-};
 
-TEST_F(ProtogalaxyRecursiveTest, InnerCircuit)
-{
-    test_inner_circuit();
+    // Construct and verify a proof of the recursive decider verifier circuit
+    {
+        auto composer = get_outer_composer<OuterBuilder>();
+        auto instance = composer.create_instance(outer_decider_circuit);
+        auto prover = composer.create_prover(instance);
+        auto verifier = composer.create_verifier(instance);
+        auto proof = prover.construct_proof();
+        bool verified = verifier.verify_proof(proof);
+
+        ASSERT(verified);
+    }
 }
 
-TEST_F(ProtogalaxyRecursiveTest, NewEvaluate)
+TEST_F(ProtogalaxyRecursiveTest, TamperedDeciderProof)
 {
-    test_evaluate_perturbator();
-}
+    // Create two arbitrary circuits for the first round of folding
+    InnerBuilder builder1;
 
-TEST_F(ProtogalaxyRecursiveTest, FullProtogalaxyRecursiveTest)
-{
-    test_recursive_folding();
-}
+    create_inner_circuit(builder1);
+    InnerBuilder builder2;
+    builder2.add_public_variable(FF(1));
+    create_inner_circuit(builder2);
 
-TEST_F(ProtogalaxyRecursiveTest, TamperedDeciderProof)
-{
-    test_failing_recursive_deciding();
+    InnerComposer inner_composer = InnerComposer();
+    auto instance1 = inner_composer.create_instance(builder1);
+    auto instance2 = inner_composer.create_instance(builder2);
+    auto instances = std::vector<std::shared_ptr<Instance>>{ instance1, instance2 };
+
+    auto accumulator = fold_and_verify(instances, inner_composer);
+
+    // Tamper with the accumulator by changing the target sum
+    accumulator->target_sum = FF::random_element();
+
+    // Create a decider proof for the relaxed instance obtained through folding
+    auto inner_decider_prover = inner_composer.create_decider_prover(accumulator);
+    auto inner_decider_proof = inner_decider_prover.construct_proof();
+
+    // Create a decider verifier circuit for recursively verifying the decider proof
+    OuterBuilder outer_decider_circuit;
+    DeciderRecursiveVerifier decider_verifier{ &outer_decider_circuit };
+    decider_verifier.verify_proof(inner_decider_proof);
+    info("Decider Recursive Verifier: num gates = ", outer_decider_circuit.num_gates);
+
+    // We expect the decider circuit check to fail due to the bad proof
+    EXPECT_FALSE(outer_decider_circuit.check_circuit());
 }
 
 TEST_F(ProtogalaxyRecursiveTest, TamperedAccumulator)

From da3f1587c9e1970e39d532bb044d53a44e273238 Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Tue, 23 Jan 2024 15:57:46 +0000
Subject: [PATCH 85/86] pr review changes

---
 .../src/barretenberg/flavor/goblin_ultra.hpp  | 65 +++++--------------
 .../cpp/src/barretenberg/flavor/ultra.hpp     | 48 ++++----------
 .../barretenberg/flavor/ultra_recursive.hpp   | 48 ++++----------
 .../protogalaxy/protogalaxy_verifier.cpp      | 15 +++++
 .../verifier/decider_recursive_verifier.cpp   |  1 -
 .../protogalaxy_recursive_verifier.cpp        | 31 ++++++---
 .../protogalaxy_recursive_verifier.hpp        | 11 ++--
 .../protogalaxy_recursive_verifier.test.cpp   |  9 +++
 8 files changed, 92 insertions(+), 136 deletions(-)

diff --git a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
index 643cbc79204..5f13c7ac66b 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/goblin_ultra.hpp
@@ -418,42 +418,8 @@ class GoblinUltra {
     template <typename Commitment, typename VerificationKey>
     class VerifierCommitments_ : public AllEntities<Commitment> {
       public:
-        VerifierCommitments_(const std::shared_ptr<VerificationKey>& verification_key)
-        {
-            this->q_m = verification_key->q_m;
-            this->q_l = verification_key->q_l;
-            this->q_r = verification_key->q_r;
-            this->q_o = verification_key->q_o;
-            this->q_4 = verification_key->q_4;
-            this->q_c = verification_key->q_c;
-            this->q_arith = verification_key->q_arith;
-            this->q_sort = verification_key->q_sort;
-            this->q_elliptic = verification_key->q_elliptic;
-            this->q_aux = verification_key->q_aux;
-            this->q_lookup = verification_key->q_lookup;
-            this->q_busread = verification_key->q_busread;
-            this->q_poseidon2_external = verification_key->q_poseidon2_external;
-            this->q_poseidon2_internal = verification_key->q_poseidon2_internal;
-            this->sigma_1 = verification_key->sigma_1;
-            this->sigma_2 = verification_key->sigma_2;
-            this->sigma_3 = verification_key->sigma_3;
-            this->sigma_4 = verification_key->sigma_4;
-            this->id_1 = verification_key->id_1;
-            this->id_2 = verification_key->id_2;
-            this->id_3 = verification_key->id_3;
-            this->id_4 = verification_key->id_4;
-            this->table_1 = verification_key->table_1;
-            this->table_2 = verification_key->table_2;
-            this->table_3 = verification_key->table_3;
-            this->table_4 = verification_key->table_4;
-            this->lagrange_first = verification_key->lagrange_first;
-            this->lagrange_last = verification_key->lagrange_last;
-            this->lagrange_ecc_op = verification_key->lagrange_ecc_op;
-            this->databus_id = verification_key->databus_id;
-        }
-
         VerifierCommitments_(const std::shared_ptr<VerificationKey>& verification_key,
-                             const WitnessEntities<Commitment>& witness_commitments)
+                             const std::optional<WitnessEntities<Commitment>>& witness_commitments = std::nullopt)
         {
             this->q_m = verification_key->q_m;
             this->q_l = verification_key->q_l;
@@ -486,19 +452,22 @@ class GoblinUltra {
             this->lagrange_ecc_op = verification_key->lagrange_ecc_op;
             this->databus_id = verification_key->databus_id;
 
-            this->w_l = witness_commitments.w_l;
-            this->w_r = witness_commitments.w_r;
-            this->w_o = witness_commitments.w_o;
-            this->sorted_accum = witness_commitments.sorted_accum;
-            this->w_4 = witness_commitments.w_4;
-            this->z_perm = witness_commitments.z_perm;
-            this->z_lookup = witness_commitments.z_lookup;
-            this->ecc_op_wire_1 = witness_commitments.ecc_op_wire_1;
-            this->ecc_op_wire_2 = witness_commitments.ecc_op_wire_2;
-            this->ecc_op_wire_3 = witness_commitments.ecc_op_wire_3;
-            this->calldata = witness_commitments.calldata;
-            this->calldata = witness_commitments.calldata_read_counts;
-            this->lookup_inverses = witness_commitments.lookup_inverses;
+            if (witness_commitments.has_value()) {
+                auto commitments = witness_commitments.value();
+                this->w_l = commitments.w_l;
+                this->w_r = commitments.w_r;
+                this->w_o = commitments.w_o;
+                this->sorted_accum = commitments.sorted_accum;
+                this->w_4 = commitments.w_4;
+                this->z_perm = commitments.z_perm;
+                this->z_lookup = commitments.z_lookup;
+                this->ecc_op_wire_1 = commitments.ecc_op_wire_1;
+                this->ecc_op_wire_2 = commitments.ecc_op_wire_2;
+                this->ecc_op_wire_3 = commitments.ecc_op_wire_3;
+                this->calldata = commitments.calldata;
+                this->calldata = commitments.calldata_read_counts;
+                this->lookup_inverses = commitments.lookup_inverses;
+            }
         }
     };
     // Specialize for GoblinUltra (general case used in GoblinUltraRecursive).
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
index 3bf08e29b8f..2fd21fab351 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra.hpp
@@ -412,37 +412,8 @@ class Ultra {
      */
     class VerifierCommitments : public AllEntities<Commitment> {
       public:
-        VerifierCommitments(const std::shared_ptr<VerificationKey>& verification_key)
-        {
-            q_m = verification_key->q_m;
-            q_c = verification_key->q_c;
-            q_l = verification_key->q_l;
-            q_r = verification_key->q_r;
-            q_o = verification_key->q_o;
-            q_4 = verification_key->q_4;
-            q_arith = verification_key->q_arith;
-            q_sort = verification_key->q_sort;
-            q_elliptic = verification_key->q_elliptic;
-            q_aux = verification_key->q_aux;
-            q_lookup = verification_key->q_lookup;
-            sigma_1 = verification_key->sigma_1;
-            sigma_2 = verification_key->sigma_2;
-            sigma_3 = verification_key->sigma_3;
-            sigma_4 = verification_key->sigma_4;
-            id_1 = verification_key->id_1;
-            id_2 = verification_key->id_2;
-            id_3 = verification_key->id_3;
-            id_4 = verification_key->id_4;
-            table_1 = verification_key->table_1;
-            table_2 = verification_key->table_2;
-            table_3 = verification_key->table_3;
-            table_4 = verification_key->table_4;
-            lagrange_first = verification_key->lagrange_first;
-            lagrange_last = verification_key->lagrange_last;
-        }
-
         VerifierCommitments(const std::shared_ptr<VerificationKey>& verification_key,
-                            const WitnessCommitments& witness_commitments)
+                            const std::optional<WitnessCommitments>& witness_commitments = std::nullopt)
         {
             q_m = verification_key->q_m;
             q_c = verification_key->q_c;
@@ -470,13 +441,16 @@ class Ultra {
             lagrange_first = verification_key->lagrange_first;
             lagrange_last = verification_key->lagrange_last;
 
-            w_l = witness_commitments.w_l;
-            w_r = witness_commitments.w_r;
-            w_o = witness_commitments.w_o;
-            sorted_accum = witness_commitments.sorted_accum;
-            w_4 = witness_commitments.w_4;
-            z_perm = witness_commitments.z_perm;
-            z_lookup = witness_commitments.z_lookup;
+            if (witness_commitments.has_value()) {
+                auto commitments = witness_commitments.value();
+                this->w_l = commitments.w_l;
+                this->w_r = commitments.w_r;
+                this->w_o = commitments.w_o;
+                this->sorted_accum = commitments.sorted_accum;
+                this->w_4 = commitments.w_4;
+                this->z_perm = commitments.z_perm;
+                this->z_lookup = commitments.z_lookup;
+            }
         }
     };
 
diff --git a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
index c2d4fb2493f..94955d72f75 100644
--- a/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
+++ b/barretenberg/cpp/src/barretenberg/flavor/ultra_recursive.hpp
@@ -376,37 +376,8 @@ template <typename BuilderType> class UltraRecursive_ {
 
     class VerifierCommitments : public AllEntities<Commitment> {
       public:
-        VerifierCommitments(const std::shared_ptr<VerificationKey>& verification_key)
-        {
-            this->q_m = verification_key->q_m;
-            this->q_l = verification_key->q_l;
-            this->q_r = verification_key->q_r;
-            this->q_o = verification_key->q_o;
-            this->q_4 = verification_key->q_4;
-            this->q_c = verification_key->q_c;
-            this->q_arith = verification_key->q_arith;
-            this->q_sort = verification_key->q_sort;
-            this->q_elliptic = verification_key->q_elliptic;
-            this->q_aux = verification_key->q_aux;
-            this->q_lookup = verification_key->q_lookup;
-            this->sigma_1 = verification_key->sigma_1;
-            this->sigma_2 = verification_key->sigma_2;
-            this->sigma_3 = verification_key->sigma_3;
-            this->sigma_4 = verification_key->sigma_4;
-            this->id_1 = verification_key->id_1;
-            this->id_2 = verification_key->id_2;
-            this->id_3 = verification_key->id_3;
-            this->id_4 = verification_key->id_4;
-            this->table_1 = verification_key->table_1;
-            this->table_2 = verification_key->table_2;
-            this->table_3 = verification_key->table_3;
-            this->table_4 = verification_key->table_4;
-            this->lagrange_first = verification_key->lagrange_first;
-            this->lagrange_last = verification_key->lagrange_last;
-        }
-
         VerifierCommitments(const std::shared_ptr<VerificationKey>& verification_key,
-                            const WitnessCommitments& witness_commitments)
+                            const std::optional<WitnessCommitments>& witness_commitments = std::nullopt)
         {
             this->q_m = verification_key->q_m;
             this->q_l = verification_key->q_l;
@@ -434,13 +405,16 @@ template <typename BuilderType> class UltraRecursive_ {
             this->lagrange_first = verification_key->lagrange_first;
             this->lagrange_last = verification_key->lagrange_last;
 
-            this->w_l = witness_commitments.w_l;
-            this->w_r = witness_commitments.w_r;
-            this->w_o = witness_commitments.w_o;
-            this->sorted_accum = witness_commitments.sorted_accum;
-            this->w_4 = witness_commitments.w_4;
-            this->z_perm = witness_commitments.z_perm;
-            this->z_lookup = witness_commitments.z_lookup;
+            if (witness_commitments.has_value()) {
+                auto commitments = witness_commitments.value();
+                this->w_l = commitments.w_l;
+                this->w_r = commitments.w_r;
+                this->w_o = commitments.w_o;
+                this->sorted_accum = commitments.sorted_accum;
+                this->w_4 = commitments.w_4;
+                this->z_perm = commitments.z_perm;
+                this->z_lookup = commitments.z_lookup;
+            }
         }
     };
 
diff --git a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
index 0e33b27cd77..bf7d4c50117 100644
--- a/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/protogalaxy/protogalaxy_verifier.cpp
@@ -6,17 +6,20 @@ template <class VerifierInstances>
 void ProtoGalaxyVerifier_<VerifierInstances>::receive_accumulator(const std::shared_ptr<Instance>& inst,
                                                                   const std::string& domain_separator)
 {
+    // Get circuit parameters
     inst->instance_size = transcript->template receive_from_prover<uint32_t>(domain_separator + "_instance_size");
     inst->log_instance_size = static_cast<size_t>(numeric::get_msb(inst->instance_size));
     inst->public_input_size =
         transcript->template receive_from_prover<uint32_t>(domain_separator + "_public_input_size");
 
+    // Get folded public inputs
     for (size_t i = 0; i < inst->public_input_size; ++i) {
         auto public_input_i =
             transcript->template receive_from_prover<FF>(domain_separator + "_public_input_" + std::to_string(i));
         inst->public_inputs.emplace_back(public_input_i);
     }
 
+    // Get folded relation parameters
     auto eta = transcript->template receive_from_prover<FF>(domain_separator + "_eta");
     auto beta = transcript->template receive_from_prover<FF>(domain_separator + "_beta");
     auto gamma = transcript->template receive_from_prover<FF>(domain_separator + "_gamma");
@@ -26,6 +29,7 @@ void ProtoGalaxyVerifier_<VerifierInstances>::receive_accumulator(const std::sha
     inst->relation_parameters =
         RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
 
+    // Get the folded relation separator challenges \vec{α}
     for (size_t idx = 0; idx < NUM_SUBRELATIONS - 1; idx++) {
         inst->alphas[idx] =
             transcript->template receive_from_prover<FF>(domain_separator + "_alpha_" + std::to_string(idx));
@@ -33,11 +37,14 @@ void ProtoGalaxyVerifier_<VerifierInstances>::receive_accumulator(const std::sha
 
     inst->target_sum = transcript->template receive_from_prover<FF>(domain_separator + "_target_sum");
 
+    // Get the folded gate challenges, \vec{β} in the paper
     inst->gate_challenges = std::vector<FF>(inst->log_instance_size);
     for (size_t idx = 0; idx < inst->log_instance_size; idx++) {
         inst->gate_challenges[idx] =
             transcript->template receive_from_prover<FF>(domain_separator + "_gate_challenge_" + std::to_string(idx));
     }
+
+    // Get the folded commitments to all witness polynomials
     auto comm_view = inst->witness_commitments.get_all();
     auto witness_labels = inst->commitment_labels.get_witness();
     for (size_t idx = 0; idx < witness_labels.size(); idx++) {
@@ -45,6 +52,7 @@ void ProtoGalaxyVerifier_<VerifierInstances>::receive_accumulator(const std::sha
             transcript->template receive_from_prover<Commitment>(domain_separator + "_" + witness_labels[idx]);
     }
 
+    // Get the folded commitments to selector polynomials
     inst->verification_key = std::make_shared<VerificationKey>(inst->instance_size, inst->public_input_size);
     auto vk_view = inst->verification_key->get_all();
     auto vk_labels = inst->commitment_labels.get_precomputed();
@@ -57,6 +65,7 @@ template <class VerifierInstances>
 void ProtoGalaxyVerifier_<VerifierInstances>::receive_and_finalise_instance(const std::shared_ptr<Instance>& inst,
                                                                             const std::string& domain_separator)
 {
+    // Get circuit parameters and the public inputs
     inst->instance_size = transcript->template receive_from_prover<uint32_t>(domain_separator + "_instance_size");
     inst->log_instance_size = static_cast<size_t>(numeric::get_msb(inst->instance_size));
     inst->public_input_size =
@@ -71,33 +80,39 @@ void ProtoGalaxyVerifier_<VerifierInstances>::receive_and_finalise_instance(cons
     inst->pub_inputs_offset =
         transcript->template receive_from_prover<uint32_t>(domain_separator + "_pub_inputs_offset");
 
+    // Get commitments to first three wire polynomials
     auto labels = inst->commitment_labels;
     auto& witness_commitments = inst->witness_commitments;
     witness_commitments.w_l = transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.w_l);
     witness_commitments.w_r = transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.w_r);
     witness_commitments.w_o = transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.w_o);
 
+    // Get challenge for sorted list batching and wire four memory records commitment
     auto eta = transcript->get_challenge(domain_separator + "_eta");
     witness_commitments.sorted_accum =
         transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.sorted_accum);
     witness_commitments.w_4 = transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.w_4);
 
+    // Get permutation challenges and commitment to permutation and lookup grand products
     auto [beta, gamma] = transcript->get_challenges(domain_separator + "_beta", domain_separator + "_gamma");
     witness_commitments.z_perm =
         transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.z_perm);
     witness_commitments.z_lookup =
         transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.z_lookup);
 
+    // Compute correction terms for grand products
     const FF public_input_delta = compute_public_input_delta<Flavor>(
         inst->public_inputs, beta, gamma, inst->instance_size, inst->pub_inputs_offset);
     const FF lookup_grand_product_delta = compute_lookup_grand_product_delta<FF>(beta, gamma, inst->instance_size);
     inst->relation_parameters =
         RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
 
+    // Get the relation separation challenges
     for (size_t idx = 0; idx < NUM_SUBRELATIONS - 1; idx++) {
         inst->alphas[idx] = transcript->get_challenge(domain_separator + "_alpha_" + std::to_string(idx));
     }
 
+    // Get the commitments to the selector polynomials for the given instance
     inst->verification_key = std::make_shared<VerificationKey>(inst->instance_size, inst->public_input_size);
     auto vk_view = inst->verification_key->get_all();
     auto vk_labels = labels.get_precomputed();
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.cpp
index f0a1b5fe6b8..6578f48cc99 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/decider_recursive_verifier.cpp
@@ -91,7 +91,6 @@ std::array<typename Flavor::GroupElement, 2> DeciderRecursiveVerifier_<Flavor>::
     return pairing_points;
 }
 
-template class DeciderRecursiveVerifier_<bb::honk::flavor::UltraRecursive_<UltraCircuitBuilder>>;
 template class DeciderRecursiveVerifier_<bb::honk::flavor::UltraRecursive_<GoblinUltraCircuitBuilder>>;
 template class DeciderRecursiveVerifier_<bb::honk::flavor::GoblinUltraRecursive_<GoblinUltraCircuitBuilder>>;
 } // namespace bb::stdlib::recursion::honk
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
index 9f13f644fc7..68c366de5b9 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.cpp
@@ -7,6 +7,7 @@ template <class VerifierInstances>
 void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::receive_accumulator(const std::shared_ptr<Instance>& inst,
                                                                            const std::string& domain_separator)
 {
+    // Get circuit parameters
     const auto instance_size = transcript->template receive_from_prover<uint32_t>(domain_separator + "_instance_size");
     const auto public_input_size =
         transcript->template receive_from_prover<uint32_t>(domain_separator + "_public_input_size");
@@ -14,12 +15,14 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::receive_accumulator(const
     inst->log_instance_size = uint32_t(numeric::get_msb(inst->instance_size));
     inst->public_input_size = uint32_t(public_input_size.get_value());
 
+    // Get folded public inputs
     for (size_t i = 0; i < inst->public_input_size; ++i) {
         auto public_input_i =
             transcript->template receive_from_prover<FF>(domain_separator + "_public_input_" + std::to_string(i));
         inst->public_inputs.emplace_back(public_input_i);
     }
 
+    // Get folded relation parameters
     auto eta = transcript->template receive_from_prover<FF>(domain_separator + "_eta");
     auto beta = transcript->template receive_from_prover<FF>(domain_separator + "_beta");
     auto gamma = transcript->template receive_from_prover<FF>(domain_separator + "_gamma");
@@ -29,6 +32,7 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::receive_accumulator(const
     inst->relation_parameters =
         RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
 
+    // Get the folded relation separator challenges \vec{α}
     for (size_t idx = 0; idx < NUM_SUBRELATIONS - 1; idx++) {
         inst->alphas[idx] =
             transcript->template receive_from_prover<FF>(domain_separator + "_alpha_" + std::to_string(idx));
@@ -36,11 +40,14 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::receive_accumulator(const
 
     inst->target_sum = transcript->template receive_from_prover<FF>(domain_separator + "_target_sum");
 
+    // Get the folded gate challenges, \vec{β} in the paper
     inst->gate_challenges = std::vector<FF>(inst->log_instance_size);
     for (size_t idx = 0; idx < inst->log_instance_size; idx++) {
         inst->gate_challenges[idx] =
             transcript->template receive_from_prover<FF>(domain_separator + "_gate_challenge_" + std::to_string(idx));
     }
+
+    // Get the folded commitments to all witness polynomials
     auto comm_view = inst->witness_commitments.get_all();
     auto witness_labels = inst->commitment_labels.get_witness();
     for (size_t idx = 0; idx < witness_labels.size(); idx++) {
@@ -48,6 +55,7 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::receive_accumulator(const
             transcript->template receive_from_prover<Commitment>(domain_separator + "_" + witness_labels[idx]);
     }
 
+    // Get the folded commitments to selector polynomials
     inst->verification_key = std::make_shared<VerificationKey>(inst->instance_size, inst->public_input_size);
     auto vk_view = inst->verification_key->get_all();
     auto vk_labels = inst->commitment_labels.get_precomputed();
@@ -60,6 +68,7 @@ template <class VerifierInstances>
 void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::receive_and_finalise_instance(
     const std::shared_ptr<Instance>& inst, const std::string& domain_separator)
 {
+    // Get circuit parameters and the public inputs
     const auto instance_size = transcript->template receive_from_prover<uint32_t>(domain_separator + "_instance_size");
     const auto public_input_size =
         transcript->template receive_from_prover<uint32_t>(domain_separator + "_public_input_size");
@@ -78,23 +87,27 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::receive_and_finalise_inst
 
     inst->pub_inputs_offset = uint32_t(pub_inputs_offset.get_value());
 
+    // Get commitments to first three wire polynomials
     auto labels = inst->commitment_labels;
     auto& witness_commitments = inst->witness_commitments;
     witness_commitments.w_l = transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.w_l);
     witness_commitments.w_r = transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.w_r);
     witness_commitments.w_o = transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.w_o);
 
+    // Get challenge for sorted list batching and wire four memory records commitment
     auto eta = transcript->get_challenge(domain_separator + "_eta");
     witness_commitments.sorted_accum =
         transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.sorted_accum);
     witness_commitments.w_4 = transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.w_4);
 
+    // Get permutation challenges and commitment to permutation and lookup grand products
     auto [beta, gamma] = transcript->get_challenges(domain_separator + "_beta", domain_separator + "_gamma");
     witness_commitments.z_perm =
         transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.z_perm);
     witness_commitments.z_lookup =
         transcript->template receive_from_prover<Commitment>(domain_separator + "_" + labels.z_lookup);
 
+    // Compute correction terms for grand products
     const FF public_input_delta = bb::honk::compute_public_input_delta<Flavor>(
         inst->public_inputs, beta, gamma, inst->instance_size, inst->pub_inputs_offset);
     const FF lookup_grand_product_delta =
@@ -102,10 +115,12 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::receive_and_finalise_inst
     inst->relation_parameters =
         RelationParameters<FF>{ eta, beta, gamma, public_input_delta, lookup_grand_product_delta };
 
+    // Get the relation separation challenges
     for (size_t idx = 0; idx < NUM_SUBRELATIONS - 1; idx++) {
         inst->alphas[idx] = transcript->get_challenge(domain_separator + "_alpha_" + std::to_string(idx));
     }
 
+    // Get the commitments to the selector polynomials for the given instance
     inst->verification_key = std::make_shared<VerificationKey>(inst->instance_size, inst->public_input_size);
     auto vk_view = inst->verification_key->get_all();
     auto vk_labels = labels.get_precomputed();
@@ -224,16 +239,18 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
     }
 
     std::vector<FF> folded_public_inputs(instances[0]->public_inputs.size(), 0);
-    size_t el_idx = 0;
-    for (auto& expected_el : folded_public_inputs) {
+    size_t public_input_idx = 0;
+    for (auto& expected_public_input : folded_public_inputs) {
         size_t inst = 0;
         for (auto& instance : instances) {
-            expected_el += instance->public_inputs[el_idx] * lagranges[inst];
+            expected_public_input += instance->public_inputs[public_input_idx] * lagranges[inst];
             inst++;
         }
-        auto next_el = transcript->template receive_from_prover<FF>("next_public_input" + std::to_string(el_idx));
-        zero.assert_equal(expected_el - next_el, "folded public input mismatch at: " + std::to_string(el_idx));
-        el_idx++;
+        auto next_public_input =
+            transcript->template receive_from_prover<FF>("next_public_input" + std::to_string(public_input_idx));
+        zero.assert_equal(expected_public_input - next_public_input,
+                          "folded public input mismatch at: " + std::to_string(public_input_idx));
+        public_input_idx++;
     }
 
     for (size_t alpha_idx = 0; alpha_idx < NUM_SUBRELATIONS - 1; alpha_idx++) {
@@ -296,8 +313,6 @@ void ProtoGalaxyRecursiveVerifier_<VerifierInstances>::verify_folding_proof(std:
     }
 }
 
-template class ProtoGalaxyRecursiveVerifier_<
-    bb::honk::VerifierInstances_<bb::honk::flavor::UltraRecursive_<UltraCircuitBuilder>, 2>>;
 template class ProtoGalaxyRecursiveVerifier_<
     bb::honk::VerifierInstances_<bb::honk::flavor::UltraRecursive_<GoblinUltraCircuitBuilder>, 2>>;
 template class ProtoGalaxyRecursiveVerifier_<
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp
index 8e36bd1f414..64b75719542 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.hpp
@@ -71,7 +71,7 @@ template <class VerifierInstances> class ProtoGalaxyRecursiveVerifier_ {
     void prepare_for_folding();
 
     /**
-     * @brief Instantiatied the accumulator (i.e. the relaxed instance) from the transcript.
+     * @brief Instantiate the accumulator (i.e. the relaxed instance) from the transcript.
      *
      */
     void receive_accumulator(const std::shared_ptr<Instance>&, const std::string&);
@@ -96,10 +96,11 @@ template <class VerifierInstances> class ProtoGalaxyRecursiveVerifier_ {
     /**
      * @brief Evaluates the perturbator at a  given scalar, in a sequential manner for the recursive setting.
      *
-     * @details This method is equivalent to the ones in the Polynomial class that evaluate a polynomial in coefficient
-     * form, class used in the native verifier for constructing and computing the perturbator. We implement this
-     * functionality here in the recursive folding verifier to avoid instantiating the entire Polynomial class on
-     * stdlib::bn254. The evaluation needs to be done sequentially asw e don't support a parallel_for in circuits.
+     * @details This method is equivalent to the one in the Polynomial class for evaluating a polynomial, represented by
+     * coefficients in monomial basis, at a given point. The Polynomial class is used in the native verifier for
+     * constructing and computing the perturbator. We implement this separate functionality here in the recursive
+     * folding verifier to avoid instantiating the entire Polynomial class on stdlib::bn254. Furthermore, the evaluation
+     * needs to be done sequentially as we don't support a parallel_for in circuits.
      *
      */
     static FF evaluate_perturbator(std::vector<FF> coeffs, FF point)
diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
index f0bac0f0712..fd39808e192 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
@@ -51,6 +51,9 @@ class ProtogalaxyRecursiveTest : public testing::Test {
      * @param builder
      * @param public_inputs
      * @param log_num_gates
+     *
+     * TODO(https://github.com/AztecProtocol/barretenberg/issues/744): make testing utility with functionality shared
+     * amongst test files
      */
     static void create_inner_circuit(InnerBuilder& builder, size_t log_num_gates = 10)
     {
@@ -178,6 +181,12 @@ TEST_F(ProtogalaxyRecursiveTest, NewEvaluate)
     EXPECT_EQ(res1, res2.get_value());
 }
 
+/**
+ * @brief Recursively verify two rounds of folding valid circuits and then recursive verify the final decider proof,
+ * make sure the verifer circuits pass check_circuit(). Ensure that the algorithm of the recursive and native verifiers
+ * are identical by checking the manifests
+
+ */
 TEST_F(ProtogalaxyRecursiveTest, FullProtogalaxyRecursiveTest)
 {
 

From 4455659933330b362148d5b2ea85280467fa73bf Mon Sep 17 00:00:00 2001
From: maramihali <mara@aztecprotocol.com>
Date: Tue, 23 Jan 2024 16:40:34 +0000
Subject: [PATCH 86/86] passing recursive folding test

---
 .../protogalaxy_recursive_verifier.test.cpp   | 22 +++++++++++++++++++
 1 file changed, 22 insertions(+)

diff --git a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
index fd39808e192..61dee74084f 100644
--- a/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
+++ b/barretenberg/cpp/src/barretenberg/stdlib/recursion/honk/verifier/protogalaxy_recursive_verifier.test.cpp
@@ -181,6 +181,28 @@ TEST_F(ProtogalaxyRecursiveTest, NewEvaluate)
     EXPECT_EQ(res1, res2.get_value());
 }
 
+/**
+ * @brief Tests a simple recursive fold that is valid works as expected.
+ *
+ */
+TEST_F(ProtogalaxyRecursiveTest, RecursiveFoldingTest)
+{
+    // Create two arbitrary circuits for the first round of folding
+    InnerBuilder builder1;
+
+    create_inner_circuit(builder1);
+    InnerBuilder builder2;
+    builder2.add_public_variable(FF(1));
+    create_inner_circuit(builder2);
+
+    InnerComposer inner_composer = InnerComposer();
+    auto instance1 = inner_composer.create_instance(builder1);
+    auto instance2 = inner_composer.create_instance(builder2);
+    auto instances = std::vector<std::shared_ptr<Instance>>{ instance1, instance2 };
+
+    fold_and_verify(instances, inner_composer);
+}
+
 /**
  * @brief Recursively verify two rounds of folding valid circuits and then recursive verify the final decider proof,
  * make sure the verifer circuits pass check_circuit(). Ensure that the algorithm of the recursive and native verifiers