diff --git a/crates/proof-of-sql/src/sql/proof_plans/group_by_exec.rs b/crates/proof-of-sql/src/sql/proof_plans/group_by_exec.rs
index 2d8fb8d90..e2a2d9163 100644
--- a/crates/proof-of-sql/src/sql/proof_plans/group_by_exec.rs
+++ b/crates/proof-of-sql/src/sql/proof_plans/group_by_exec.rs
@@ -23,7 +23,7 @@ use crate::{
 use alloc::{boxed::Box, vec, vec::Vec};
 use bumpalo::Bump;
 use core::{iter, iter::repeat_with};
-use num_traits::One;
+use num_traits::{One, Zero};
 use proof_of_sql_parser::Identifier;
 use serde::{Deserialize, Serialize};
 
@@ -128,12 +128,14 @@ impl ProofPlan for GroupByExec {
 
         let alpha = builder.consume_post_result_challenge();
         let beta = builder.consume_post_result_challenge();
+        let output_one_eval = builder.consume_one_evaluation();
 
         verify_group_by(
             builder,
             alpha,
             beta,
             input_one_eval,
+            output_one_eval,
             (group_by_evals, aggregate_evals, where_eval),
             (
                 group_by_result_columns_evals.clone(),
@@ -171,7 +173,6 @@ impl ProofPlan for GroupByExec {
             .chain(sum_result_columns_evals)
             .chain(iter::once(count_column_eval))
             .collect::<Vec<_>>();
-        let output_one_eval = builder.consume_one_evaluation();
         Ok(TableEvaluation::new(column_evals, output_one_eval))
     }
 
@@ -343,38 +344,39 @@ fn verify_group_by<S: Scalar>(
     builder: &mut VerificationBuilder<S>,
     alpha: S,
     beta: S,
-    one_eval: S,
+    input_one_eval: S,
+    output_one_eval: S,
     (g_in_evals, sum_in_evals, sel_in_eval): (Vec<S>, Vec<S>, S),
     (g_out_evals, sum_out_evals, count_out_eval): (Vec<S>, Vec<S>, S),
 ) -> Result<(), ProofError> {
-    // g_in_fold = alpha + sum beta^j * g_in[j]
-    let g_in_fold_eval = alpha * one_eval + fold_vals(beta, &g_in_evals);
-    // g_out_bar_fold = alpha + sum beta^j * g_out_bar[j]
-    let g_out_bar_fold_eval = alpha * one_eval + fold_vals(beta, &g_out_evals);
-    // sum_in_fold = 1 + sum beta^(j+1) * sum_in[j]
-    let sum_in_fold_eval = one_eval + beta * fold_vals(beta, &sum_in_evals);
-    // sum_out_bar_fold = count_out_bar + sum beta^(j+1) * sum_out_bar[j]
-    let sum_out_bar_fold_eval = count_out_eval + beta * fold_vals(beta, &sum_out_evals);
+    // g_in_fold = alpha * sum beta^j * g_in[j]
+    let g_in_fold_eval = alpha * fold_vals(beta, &g_in_evals);
+    // g_out_fold = alpha * sum beta^j * g_out[j]
+    let g_out_fold_eval = alpha * fold_vals(beta, &g_out_evals);
+    // sum_in_fold = input_ones + sum beta^(j+1) * sum_in[j]
+    let sum_in_fold_eval = input_one_eval + beta * fold_vals(beta, &sum_in_evals);
+    // sum_out_fold = count_out + sum beta^(j+1) * sum_out[j]
+    let sum_out_fold_eval = count_out_eval + beta * fold_vals(beta, &sum_out_evals);
 
     let g_in_star_eval = builder.consume_intermediate_mle();
     let g_out_star_eval = builder.consume_intermediate_mle();
 
-    // sum g_in_star * sel_in * sum_in_fold - g_out_star * sum_out_bar_fold = 0
+    // sum g_in_star * sel_in * sum_in_fold - g_out_star * sum_out_fold = 0
     builder.produce_sumcheck_subpolynomial_evaluation(
         SumcheckSubpolynomialType::ZeroSum,
-        g_in_star_eval * sel_in_eval * sum_in_fold_eval - g_out_star_eval * sum_out_bar_fold_eval,
+        g_in_star_eval * sel_in_eval * sum_in_fold_eval - g_out_star_eval * sum_out_fold_eval,
     );
 
-    // g_in_star * g_in_fold - input_ones = 0
+    // g_in_star + g_in_star * g_in_fold - input_ones = 0
     builder.produce_sumcheck_subpolynomial_evaluation(
         SumcheckSubpolynomialType::Identity,
-        g_in_star_eval * g_in_fold_eval - one_eval,
+        g_in_star_eval + g_in_star_eval * g_in_fold_eval - input_one_eval,
     );
 
-    // g_out_star * g_out_bar_fold - input_ones = 0
+    // g_out_star + g_out_star * g_out_fold - output_ones = 0
     builder.produce_sumcheck_subpolynomial_evaluation(
         SumcheckSubpolynomialType::Identity,
-        g_out_star_eval * g_out_bar_fold_eval - one_eval,
+        g_out_star_eval + g_out_star_eval * g_out_fold_eval - output_one_eval,
     );
 
     Ok(())
@@ -393,39 +395,41 @@ pub fn prove_group_by<'a, S: Scalar>(
     (g_out, sum_out, count_out): (&[Column<S>], &[&'a [S]], &'a [i64]),
     n: usize,
 ) {
-    let m_out = count_out.len();
+    let m = count_out.len();
     let input_ones = alloc.alloc_slice_fill_copy(n, true);
+    let output_ones = alloc.alloc_slice_fill_copy(m, true);
 
-    // g_in_fold = alpha + sum beta^j * g_in[j]
-    let g_in_fold = alloc.alloc_slice_fill_copy(n, alpha);
-    fold_columns(g_in_fold, One::one(), beta, g_in);
+    // g_in_fold = alpha * sum beta^j * g_in[j]
+    let g_in_fold = alloc.alloc_slice_fill_copy(n, Zero::zero());
+    fold_columns(g_in_fold, alpha, beta, g_in);
 
-    // g_out_bar_fold = alpha + sum beta^j * g_out_bar[j]
-    let g_out_bar_fold = alloc.alloc_slice_fill_copy(n, alpha);
-    fold_columns(g_out_bar_fold, One::one(), beta, g_out);
+    // g_out_fold = alpha * sum beta^j * g_out[j]
+    let g_out_fold = alloc.alloc_slice_fill_copy(m, Zero::zero());
+    fold_columns(g_out_fold, alpha, beta, g_out);
 
     // sum_in_fold = 1 + sum beta^(j+1) * sum_in[j]
     let sum_in_fold = alloc.alloc_slice_fill_copy(n, One::one());
     fold_columns(sum_in_fold, beta, beta, sum_in);
 
-    // sum_out_bar_fold = count_out_bar + sum beta^(j+1) * sum_out_bar[j]
-    let sum_out_bar_fold = alloc.alloc_slice_fill_default(n);
-    slice_ops::slice_cast_mut(count_out, sum_out_bar_fold);
-    fold_columns(sum_out_bar_fold, beta, beta, sum_out);
+    // sum_out_fold = count_out + sum beta^(j+1) * sum_out[j]
+    let sum_out_fold = alloc.alloc_slice_fill_default(m);
+    slice_ops::slice_cast_mut(count_out, sum_out_fold);
+    fold_columns(sum_out_fold, beta, beta, sum_out);
 
-    // g_in_star = g_in_fold^(-1)
+    // g_in_star = (1 + g_in_fold)^(-1)
     let g_in_star = alloc.alloc_slice_copy(g_in_fold);
+    slice_ops::add_const::<S, S>(g_in_star, One::one());
     slice_ops::batch_inversion(g_in_star);
 
-    // g_out_star = g_out_bar_fold^(-1), which is simply alpha^(-1) when beyond the output length
-    let g_out_star = alloc.alloc_slice_copy(g_out_bar_fold);
-    g_out_star[m_out..].fill(alpha.inv().expect("alpha should never be 0"));
-    slice_ops::batch_inversion(&mut g_out_star[..m_out]);
+    // g_out_star = (1 + g_out_fold)^(-1)
+    let g_out_star = alloc.alloc_slice_copy(g_out_fold);
+    slice_ops::add_const::<S, S>(g_out_star, One::one());
+    slice_ops::batch_inversion(g_out_star);
 
     builder.produce_intermediate_mle(g_in_star as &[_]);
     builder.produce_intermediate_mle(g_out_star as &[_]);
 
-    // sum g_in_star * sel_in * sum_in_fold - g_out_star * sum_out_bar_fold = 0
+    // sum g_in_star * sel_in * sum_in_fold - g_out_star * sum_out_fold = 0
     builder.produce_sumcheck_subpolynomial(
         SumcheckSubpolynomialType::ZeroSum,
         vec![
@@ -439,18 +443,16 @@ pub fn prove_group_by<'a, S: Scalar>(
             ),
             (
                 -S::one(),
-                vec![
-                    Box::new(g_out_star as &[_]),
-                    Box::new(sum_out_bar_fold as &[_]),
-                ],
+                vec![Box::new(g_out_star as &[_]), Box::new(sum_out_fold as &[_])],
             ),
         ],
     );
 
-    // g_in_star * g_in_fold - input_ones = 0
+    // g_in_star + g_in_star * g_in_fold - input_ones = 0
     builder.produce_sumcheck_subpolynomial(
         SumcheckSubpolynomialType::Identity,
         vec![
+            (S::one(), vec![Box::new(g_in_star as &[_])]),
             (
                 S::one(),
                 vec![Box::new(g_in_star as &[_]), Box::new(g_in_fold as &[_])],
@@ -459,18 +461,16 @@ pub fn prove_group_by<'a, S: Scalar>(
         ],
     );
 
-    // g_out_star * g_out_bar_fold - input_ones = 0
+    // g_out_star + g_out_star * g_out_fold - output_ones = 0
     builder.produce_sumcheck_subpolynomial(
         SumcheckSubpolynomialType::Identity,
         vec![
+            (S::one(), vec![Box::new(g_out_star as &[_])]),
             (
                 S::one(),
-                vec![
-                    Box::new(g_out_star as &[_]),
-                    Box::new(g_out_bar_fold as &[_]),
-                ],
+                vec![Box::new(g_out_star as &[_]), Box::new(g_out_fold as &[_])],
             ),
-            (-S::one(), vec![Box::new(input_ones as &[_])]),
+            (-S::one(), vec![Box::new(output_ones as &[_])]),
         ],
     );
 }