diff --git a/gaussian_observations_localization_v2.txt b/gaussian_observations_localization_v2.txt
index e075a78..db39495 100644
--- a/gaussian_observations_localization_v2.txt
+++ b/gaussian_observations_localization_v2.txt
@@ -70,7 +70,7 @@ Logarithmic integrability is checked before any entropy difference is taken. Bec
 
   int f_A (log f_A)_- dP <= 1/e,
 
-and this with finiteness of the conditional relative entropy gives averaged absolute integrability of log f_A(s_i). For almost every (A, X), the positive part of log(g(y) / rho_A(X, y)) against rho_A(X, y) dy is at most 1/e for standard Gaussian density g on R^r, and the negative part of log rho_A(X, Y) is integrable because -log g(Y) = (r/2) log(2 pi) + ||Y||^2 / 2 with E||Y||^2 = r. Setting Z = (2 pi)^{-k(p-1)/2} int J^{-k} d mu^{⊗ p}, which is finite and positive, and defining Q by density (2 pi)^{-k(p-1)/2} J^{-k} / Z relative to lambda, gives pi_A << Q with the same null sets. The log likelihood log Z + sum_i log f_A(s_i) - (p-1) log rho_A(X, Y) then has an integrable positive part and a negative part of integral at most 1/e, so full integrability of log rho_A(X, Y) follows without using log J. The extra-moment bound allows exponential tilting in (log J)_-, and Hadamard's inequality gives (log J)_+ <= (p-1) log 2 because every chord has length at most two.
+and this with finiteness of the conditional relative entropy gives averaged absolute integrability of log f_A(s_i). For almost every (A, X), the positive part of log(g(y) / rho_A(X, y)) against rho_A(X, y) dy is at most 1/e for standard Gaussian density g on R^r, and the negative part of log rho_A(X, Y) is integrable because -log g(Y) = (r/2) log(2 pi) + ||Y||^2 / 2 with E||Y||^2 = r. Setting Z = (2 pi)^{-k(p-1)/2} int J^{-k} d mu^{⊗ p}, which is finite and positive, and defining Q by density (2 pi)^{-k(p-1)/2} J^{-k} / Z relative to lambda, gives pi_A << Q with the same null sets. The log likelihood log Z + sum_i log f_A(s_i) - (p-1) log rho_A(X, Y) then has an integrable positive part and a negative part of integral at most 1/e, so full integrability of log rho_A(X, Y) follows without using log J. The extra-moment bound allows exponential tilting in (log J)_-, and Hadamard's ineuality gives (log J)_+ <= (p-1) log 2 because every chord has length at most two.
 
 The likelihood formula yields the information that one uniformly selected posterior draw carries. For i >= 2 put delta_i = dist(s_i, aff(s_1, ..., s_{i-1})), the distance from s_i to the affine span of its predecessors, and Delta_i = max(0, log(r_A / delta_i)); set Delta_1 = 0. Gram-Schmidt gives J = prod_{i=2}^p delta_i. The likelihood formula makes the tuple marginal absolutely continuous with respect to mu^{⊗ p}, so every delta_i is positive almost surely, and delta_i <= 2. Taking logarithms of the likelihood, discarding X by data processing, and conditioning on the block destination w adds at most log W, where W bounds the destination alphabet. The product-reference chain rule for a p-fold product gives