diff --git a/src/fri/verifier.rs b/src/fri/verifier.rs
index a135f5f0..45c94173 100644
--- a/src/fri/verifier.rs
+++ b/src/fri/verifier.rs
@@ -196,6 +196,11 @@ fn fri_combine_initial<F: Field + Extendable<D>, const D: usize>(
     let wire_composition_eval = reduce_with_iter(wire_evals, alpha_powers.clone());
     let zeta_frob = zeta.frobenius();
     let wire_eval = reduce_with_iter(&os.wires, alpha_powers.clone());
+    // We want to compute `sum a^i*phi(w_i)`, where `phi` denotes the Frobenius automorphism.
+    // Since `phi^D=id` and `phi` is a field automorphism, we have the following equalities:
+    // `sum a^i*phi(w_i) = sum phi(phi^(D-1)(a^i)*w_i) = phi(sum phi^(D-1)(a)^i*w_i)`
+    // So we can compute the original sum using only one call to the `D-1`-repeated Frobenius of alpha,
+    // and one call at the end of the sum.
     let alpha_powers_frob = alpha_powers.repeated_frobenius(D - 1);
     let wire_eval_frob = reduce_with_iter(&os.wires, alpha_powers_frob).frobenius();
     let wires_interpol = interpolant(&[(zeta, wire_eval), (zeta_frob, wire_eval_frob)]);
diff --git a/src/gadgets/split_base.rs b/src/gadgets/split_base.rs
index 50a5e579..10189841 100644
--- a/src/gadgets/split_base.rs
+++ b/src/gadgets/split_base.rs
@@ -18,7 +18,7 @@ impl<F: Extendable<D>, const D: usize> CircuitBuilder<F, D> {
 
         Target::wires_from_range(
             gate,
-            BaseSumGate::<B>::WIRE_LIMBS_START..BaseSumGate::<B>::WIRE_LIMBS_START + num_limbs,
+            BaseSumGate::<B>::START_LIMBS..BaseSumGate::<B>::START_LIMBS + num_limbs,
         )
     }
 
diff --git a/src/gadgets/split_join.rs b/src/gadgets/split_join.rs
index 647b0ef5..2a388e35 100644
--- a/src/gadgets/split_join.rs
+++ b/src/gadgets/split_join.rs
@@ -29,7 +29,7 @@ impl<F: Extendable<D>, const D: usize> CircuitBuilder<F, D> {
     /// Verifies that the decomposition is correct by using `k` `BaseSum<2>` gates
     /// with `k` such that `k*num_routed_wires>=num_bits`.
     pub(crate) fn split_le(&mut self, integer: Target, num_bits: usize) -> Vec<Target> {
-        let num_limbs = self.config.num_routed_wires - BaseSumGate::<2>::WIRE_LIMBS_START;
+        let num_limbs = self.config.num_routed_wires - BaseSumGate::<2>::START_LIMBS;
         let k = ceil_div_usize(num_bits, num_limbs);
         let gates = (0..k)
             .map(|_| self.add_gate_no_constants(BaseSumGate::<2>::new(num_limbs)))
@@ -39,7 +39,7 @@ impl<F: Extendable<D>, const D: usize> CircuitBuilder<F, D> {
         for &gate in &gates {
             bits.extend(Target::wires_from_range(
                 gate,
-                BaseSumGate::<2>::WIRE_LIMBS_START..BaseSumGate::<2>::WIRE_LIMBS_START + num_limbs,
+                BaseSumGate::<2>::START_LIMBS..BaseSumGate::<2>::START_LIMBS + num_limbs,
             ));
         }
         bits.drain(num_bits..);
diff --git a/src/gates/base_sum.rs b/src/gates/base_sum.rs
index 150a492e..0792ea7a 100644
--- a/src/gates/base_sum.rs
+++ b/src/gates/base_sum.rs
@@ -1,3 +1,5 @@
+use std::ops::Range;
+
 use crate::circuit_builder::CircuitBuilder;
 use crate::field::extension_field::target::ExtensionTarget;
 use crate::field::extension_field::Extendable;
@@ -8,9 +10,8 @@ use crate::plonk_common::{reduce_with_powers, reduce_with_powers_recursive};
 use crate::target::Target;
 use crate::vars::{EvaluationTargets, EvaluationVars};
 use crate::witness::PartialWitness;
-use std::ops::Range;
 
-/// A gate which can sum base W limbs and the reversed limbs.
+/// A gate which can sum base B limbs and the reversed limbs.
 #[derive(Debug)]
 pub struct BaseSumGate<const B: usize> {
     num_limbs: usize,
@@ -23,11 +24,11 @@ impl<const B: usize> BaseSumGate<B> {
 
     pub const WIRE_SUM: usize = 0;
     pub const WIRE_REVERSED_SUM: usize = 1;
-    pub const WIRE_LIMBS_START: usize = 2;
+    pub const START_LIMBS: usize = 2;
 
     /// Returns the index of the `i`th limb wire.
     pub fn limbs(&self) -> Range<usize> {
-        Self::WIRE_LIMBS_START..Self::WIRE_LIMBS_START + self.num_limbs
+        Self::START_LIMBS..Self::START_LIMBS + self.num_limbs
     }
 }
 
@@ -66,10 +67,10 @@ impl<F: Extendable<D>, const D: usize, const B: usize> Gate<F, D> for BaseSumGat
         let mut limbs = vars.local_wires[self.limbs()].to_vec();
         let computed_sum = reduce_with_powers_recursive(builder, &limbs, base);
         limbs.reverse();
-        let reversed_computed_sum = reduce_with_powers_recursive(builder, &limbs, base);
+        let computed_reversed_sum = reduce_with_powers_recursive(builder, &limbs, base);
         let mut constraints = vec![
             builder.sub_extension(computed_sum, sum),
-            builder.sub_extension(reversed_computed_sum, reversed_sum),
+            builder.sub_extension(computed_reversed_sum, reversed_sum),
         ];
         for limb in limbs {
             constraints.push({
@@ -138,8 +139,7 @@ impl<F: Field, const B: usize> SimpleGenerator<F> for BaseSplitGenerator<B> {
             "Integer too large to fit in given number of limbs"
         );
 
-        let limbs = (BaseSumGate::<B>::WIRE_LIMBS_START
-            ..BaseSumGate::<B>::WIRE_LIMBS_START + self.num_limbs)
+        let limbs = (BaseSumGate::<B>::START_LIMBS..BaseSumGate::<B>::START_LIMBS + self.num_limbs)
             .map(|i| Target::wire(self.gate_index, i));
         let limbs_value = (0..self.num_limbs)
             .scan(sum_value, |acc, _| {