diff --git a/book/src/design/proving-system.md b/book/src/design/proving-system.md
index 04e1eba1..8cc801f6 100644
--- a/book/src/design/proving-system.md
+++ b/book/src/design/proving-system.md
@@ -45,10 +45,10 @@ now serves as a summary of the following sub-sections.
 |                                                                             | $\larr$ | $F = [F_0, F_1, \dots, F_{m - 1}]$ |
 | $\mathbf{A} = [A_0, A_1, \dots, A_{m - 1}]$                                 | $\rarr$ |                                    |
 |                                                                             | $\larr$ | $\theta$                           |
-| $\mathbf{L}_p = [(A'_0, S'_0), \dots, (A'_{m - 1}, S'_{m - 1})]$            | $\rarr$ |                                    |
+| $\mathbf{L} = [(A'_0, S'_0), \dots, (A'_{m - 1}, S'_{m - 1})]$              | $\rarr$ |                                    |
 |                                                                             | $\larr$ | $\beta, \gamma$                    |
 | $\mathbf{P} = [P_0, P_1, \dots, P_{m - 1}]$                                 | $\rarr$ |                                    |
-| $\mathbf{L}_c = [L_0, L_1, \dots, L_{m - 1}]$                               | $\rarr$ |                                    |
+| $\mathbf{Z} = [Z_0, Z_1, \dots, Z_{m - 1}]$                                 | $\rarr$ |                                    |
 |                                                                             | $\larr$ | $y$                                |
 | $h(X) = \frac{\text{gate}_0(X) + \dots + y^i \cdot \text{gate}_i(X)}{t(X)}$ |         |                                    |
 | $h(X) = h_0(X) + \dots + X^{n(d-1)} h_{d-1}(X)$                             |         |                                    |
diff --git a/book/src/design/proving-system/circuit-commitments.md b/book/src/design/proving-system/circuit-commitments.md
index e2910f53..60b07c00 100644
--- a/book/src/design/proving-system/circuit-commitments.md
+++ b/book/src/design/proving-system/circuit-commitments.md
@@ -45,7 +45,7 @@ follows:
 
 Finally, the prover creates blinding commitments for all of the lookups
 
-$$\mathbf{L}_p = \left[ (\text{Commit}(A'(X))), \text{Commit}(S'(X))), \dots \right]$$
+$$\mathbf{L} = \left[ (\text{Commit}(A'(X))), \text{Commit}(S'(X))), \dots \right]$$
 
 and sends them to the verifier.
 
@@ -60,21 +60,21 @@ and sends them to the verifier.
 
   and sends them to the verifier.
 
-## Committing to the lookup permutations
+## Committing to the lookup permutation product columns
 
 In addition to committing to the individual permuted lookups, the prover needs to commit
-to constraint polynomials that enforce the correct relation between the permuted lookups
-and their un-permuted forms:
+to the permutation product column
 
-$$L(X) = \frac{(A_\text{compressed}(X) + \beta)(S_\text{compressed}(X) + \gamma)}{(A'(X) + \beta)(S'(X) + \gamma)}$$
+$$Z(X) = \frac{(A_\text{compressed}(X) + \beta)(S_\text{compressed}(X) + \gamma)}{(A'(X) + \beta)(S'(X) + \gamma)}$$
 
 $\beta$ and $\gamma$ are used to combine the permutation arguments for $A'(X)$ and $S'(X)$
 while keeping them independent. We can reuse $\beta$ and $\gamma$ from the equality
 constraint permutation here because they serve the same purpose in both places, and we
 aren't trying to combine the lookup and equality constraint permutation arguments.
 
-As before, the prover creates blinding commitments for every constraint polynomial
+As before, the prover creates blinding commitments to the permutation product column for
+every lookup
 
-$$\mathbf{L}_c = \left[\text{Commit}(L(X))), \dots \right]$$
+$$\mathbf{Z} = \left[\text{Commit}(Z(X))), \dots \right]$$
 
 and sends them to the verifier.