diff --git a/gaussian_observations_localization_v2.txt b/gaussian_observations_localization_v2.txt
index 30eff88..e075a78 100644
--- a/gaussian_observations_localization_v2.txt
+++ b/gaussian_observations_localization_v2.txt
@@ -58,7 +58,7 @@ For the replica experiment the likelihood of the tuple has a closed form, becaus
 
   rho_A(X, y) = rho_{0,k}(X, y) * int f_A(s) mu_{X,y}(ds)
 
-when rho_{0,k}(X, y) > 0 and the integral is finite, and rho_A(X, y) = 0 on rank failures or on unused fibers where the integral is infinite. The product 0 * infinity never occurs. Let pi_A be the law of (s_1, ..., s_p, X) under the replica construction, and let lambda be the probability measure mu^{otimes p}(ds) gamma_{k, B_s}(dX), defined arbitrarily on the null set where J = 0. Then pi_A is absolutely continuous with respect to lambda and
+when rho_{0,k}(X, y) > 0 and the integral is finite, and rho_A(X, y) = 0 on rank failures or on unused fibers where the integral is infinite. The product 0 * infinity never occurs. Let pi_A be the law of (s_1, ..., s_p, X) under the replica construction, and let lambda be the probability measure mu^{⊗ p}(ds) gamma_{k, B_s}(dX), defined arbitrarily on the null set where J = 0. Then pi_A is absolutely continuous with respect to lambda and
 
   d pi_A / d lambda (s, X) = (2 pi)^{-k (p-1)/2} J(s)^{-k} * prod_{i=1}^p f_A(s_i) / rho_A(X, X s_1)^{p-1}.
 
@@ -70,11 +70,11 @@ 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^{otimes 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 inequality 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^{otimes 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
+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
 
-  E D(L(s | A, w) || mu^{otimes p}) <= log W + p D(A) + (p-1) E[-log rho_A(X, Y)] - (k(p-1)/2) log(2 pi) - k E log J,
+  E D(L(s | A, w) || mu^{⊗ p}) <= log W + p D(A) + (p-1) E[-log rho_A(X, Y)] - (k(p-1)/2) log(2 pi) - k E log J,
 
 which is finite by the likelihood integrability. The label entropy is bounded by the cross entropy with the Gaussian density of N(X c_A, r_A^2 I_k), because the conditional signal law is supported in B(c_A, r_A) and freshness gives E(||Y - X c_A||^2 / r_A^2 | A) <= k. Hence
 
@@ -82,9 +82,9 @@ which is finite by the likelihood integrability. The label entropy is bounded by
 
 Substituting this and log J = sum_i log delta_i, and using log(r_A / delta_i) <= Delta_i so that k E log r_A - k E log delta_i <= k E Delta_i, gives
 
-  E D(L(s | A, w) || mu^{otimes p}) <= log W + p D(A) + (k(p-1)/2) + k E sum_i Delta_i.
+  E D(L(s | A, w) || mu^{⊗ p}) <= log W + p D(A) + (k(p-1)/2) + k E sum_i Delta_i.
 
-Now draw I uniform on {1, ..., p}, independent of everything, use s_I as the signal for the next block, and set A_* = (A, w, I, s_1, ..., s_{I-1}). The triple (s_I, A, w) has the original (S, A, w) law, because each draw has that marginal and I is independent. This finite relative entropy also makes each conditional coordinate law, given its predecessors, absolutely continuous with respect to mu. The product-reference chain rule and uniformity of I give the exact identity D(A_*) = (1/p) E D(L(s | A, w) || mu^{otimes p}). Dividing the last display by p and using (p-1)/p <= 1,
+Now draw I uniform on {1, ..., p}, independent of everything, use s_I as the signal for the next block, and set A_* = (A, w, I, s_1, ..., s_{I-1}). The triple (s_I, A, w) has the original (S, A, w) law, because each draw has that marginal and I is independent. This finite relative entropy also makes each conditional coordinate law, given its predecessors, absolutely continuous with respect to mu. The product-reference chain rule and uniformity of I give the exact identity D(A_*) = (1/p) E D(L(s | A, w) || mu^{⊗ p}). Dividing the last display by p and using (p-1)/p <= 1,
 
   D(A_*) <= D(A) + (log W)/p + k/2 + k E Delta_I.