diff --git a/gaussian_observations_localization_v2.txt b/gaussian_observations_localization_v2.txt
index 4cfb57f..70ca928 100644
--- a/gaussian_observations_localization_v2.txt
+++ b/gaussian_observations_localization_v2.txt
@@ -24,8 +24,8 @@ The weight has a plain meaning. When the differences are independent, the points
 
 Let mu = f P with f bounded, nonnegative, and Borel, so mu is a finite measure that need not be a probability. For a full-rank matrix A let p_A(y) be the density of A u at the label y in R^k for u ~ P, and let P_{A,y} be the conditional law of u given Au = y; on an unused label the latter may be chosen arbitrarily. Put h_A(y) = integral of f with respect to P_{A,y}, set p_A(y) = p_A^0(y) h_A(y) where p_A^0 is the untilded density for P, and define the tilted fiber mu_{A,y}(du) = (f(u)/h_A(y)) P_{A,y}(du) when h_A(y) > 0. Then for every nonnegative Borel test H(A, y, u),
 
-  integral Gamma_{k,d}(dA) integral_{R^k} p_A(y)^t integral H(A, y, u) mu_{A,y}^{⊗t}(du) dy
-    = (2 pi)^{-k m/2} integral J(u)^{-k} mu^{⊗t}(du) integral H(A, A u_1, u) Gamma_{k,d}^{F_u}(dA).
+  integral Gamma_{k,d}(dA) integral_{R^k} p_A(y)^t integral H(A, y, u) mu_{A,y}^{\otimes t}(du) dy
+    = (2 pi)^{-k m/2} integral J(u)^{-k} mu^{\otimes t}(du) integral H(A, A u_1, u) Gamma_{k,d}^{F_u}(dA).
 
 Both sides are finite measures on (A, y, u) before H is inserted, and both are supported on the incidence A u_i = y for all i, on full-column-rank D_u, and on full-row-rank A; the right side places the label at y = A u_1. In the spherical case the shared fiber has a positive radius.
 
@@ -33,15 +33,15 @@ The coarea formula for the map (A, u) -> (A, A u_1), together with the standard
 
 The proof fixes a bounded continuous test H and first treats P itself. Let phi_eta be a standard approximate identity on R^k, either uniform on the radius-eta ball or the density of N(0, eta^2 I_k). The mollified functional
 
-  I_eta = integral (P)^{⊗t}(du) Gamma_{k,d}(dA) H(A, A u_1, u) prod_{i=2}^{t} phi_eta(A u_i - A u_1)
+  I_eta = integral (P)^{\otimes t}(du) Gamma_{k,d}(dA) H(A, A u_1, u) prod_{i=2}^{t} phi_eta(A u_i - A u_1)
 
-converges as eta decreases to zero to the left side above. Indeed, p_A(y)^t P_{A,y}^{⊗t}(du) dy is precisely the law of (y, u) under which the t independent signal points are conditioned to share the single label y; the product of the phi_eta differences tests the constraints A u_i = A u_1 in the limit. On the other hand, for a fixed tuple u with rank D_u = m, the image Psi_u(A) = (A u_2 - A u_1, ..., A u_t - A u_1) in (R^k)^m is a centered Gaussian whose per-row covariance is the Gram matrix of the differences, so det(D_u^T D_u) appears with exponent k/2; the density of Psi_u(A) at the origin is exactly (2 pi)^{-k m/2} J(u)^{-k}, and the conditional law of A given Psi_u(A) = 0 is Gamma_{k,d}^{F_u}. The product of the phi_eta is an approximate identity on (R^k)^m at the origin, so I_eta also converges to the right side. Comparing the two limits proves the identity for mu = P. For a general bounded density f, the two t-fold factors change in the same way: p_A(y)^t mu_{A,y}^{⊗t}(du) equals p_A^0(y)^t (prod_i f(u_i)) P_{A,y}^{⊗t}(du) because the tilted density h_A(y)^t cancels against the t denominators of mu_{A,y}^{⊗t}. Thus inserting prod_i f(u_i) into the test, which the right side does automatically through mu^{⊗t}, reduces the claim to the base case. The differentiation and conditional-density versions of p_A agree for Lebesgue-almost every y and at the constrained label for the finite reference measure; taking lower limits of ball averages yields a jointly measurable choice.
+converges as eta decreases to zero to the left side above. Indeed, p_A(y)^t P_{A,y}^{\otimes t}(du) dy is precisely the law of (y, u) under which the t independent signal points are conditioned to share the single label y; the product of the phi_eta differences tests the constraints A u_i = A u_1 in the limit. On the other hand, for a fixed tuple u with rank D_u = m, the image Psi_u(A) = (A u_2 - A u_1, ..., A u_t - A u_1) in (R^k)^m is a centered Gaussian whose per-row covariance is the Gram matrix of the differences, so det(D_u^T D_u) appears with exponent k/2; the density of Psi_u(A) at the origin is exactly (2 pi)^{-k m/2} J(u)^{-k}, and the conditional law of A given Psi_u(A) = 0 is Gamma_{k,d}^{F_u}. The product of the phi_eta is an approximate identity on (R^k)^m at the origin, so I_eta also converges to the right side. Comparing the two limits proves the identity for mu = P. For a general bounded density f, the two t-fold factors change in the same way: p_A(y)^t mu_{A,y}^{\otimes t}(du) equals p_A^0(y)^t (prod_i f(u_i)) P_{A,y}^{\otimes t}(du) because the tilted density h_A(y)^t cancels against the t denominators of mu_{A,y}^{\otimes t}. Thus inserting prod_i f(u_i) into the test, which the right side does automatically through mu^{\otimes t}, reduces the claim to the base case. The differentiation and conditional-density versions of p_A agree for Lebesgue-almost every y and at the constrained label for the finite reference measure; taking lower limits of ball averages yields a jointly measurable choice.
 The identity pays for itself as a finiteness statement about inverse simplex volumes, and this is the quantitative engine of every block estimate below. Fix integers k >= 1 and p >= 2 with k + p + 1 < d, and draw u_1, ..., u_p independently from P. For almost every tuple the p - 1 difference vectors u_2 - u_1, ..., u_p - u_1 are linearly independent, so J(u) > 0. Then
 
-  integral J(u)^{-(k+1)} P^{⊗p}(du) < infinity.
+  integral J(u)^{-(k+1)} P^{\otimes p}(du) < infinity.
 
 Apply the identity with row count r = k + 1 and tuple size t = p, so m = p - 1 and the dimension condition r + m = k + p <= d - 2 is the hypothesis. Take the test identically one and base density f = 1, so each fiber mu_{A,y} = P_{A,y} is a probability and the left side collapses to
 
   L = integral Gamma_{k+1,d}(dA) integral_{R^{k+1}} p_A(y)^p dy.
 
-The projected density of a uniform sphere point under a full-rank (k+1)-by-d matrix A is supported on the ellipsoid {y : y^T (A A^T)^{-1} y <= 1} and equals, up to the normalising constant c_{d,k+1} of that projection, the function det(A A^T)^{-1/2} (1 - y^T (A A^T)^{-1} y)^{(d - k - 3)/2}. The exponent is nonnegative because k + 3 <= d; changing to z = (A A^T)^{-1/2} y turns the inner integral into sqrt(det(A A^T)) times a bounded function of z on the unit ball, so it is at most c_{d,k+1}^{p-1} det(A A^T)^{-(p-1)/2} for a finite normalizing constant. Gram-Schmidt writes sqrt(det(A A^T)) as the product of successive perpendicular lengths of the k + 1 independent Gaussian rows; those lengths live in dimensions d, d - 1, ..., d - k, every one of which exceeds p - 1 because k + p + 1 < d. A Gaussian perpendicular length in dimension at least p has a finite inverse moment of order p - 1, so iterating conditional expectation makes the double integral L finite. On the right side of the identity the number L equals (2 pi)^{-(k+1)(p-1)/2} times the total mass of Gamma_{k+1,d}^{F_u}, which is one, times integral J(u)^{-(k+1)} P^{⊗p}(du); hence this inverse moment is finite.
+The projected density of a uniform sphere point under a full-rank (k+1)-by-d matrix A is supported on the ellipsoid {y : y^T (A A^T)^{-1} y <= 1} and equals, up to the normalising constant c_{d,k+1} of that projection, the function det(A A^T)^{-1/2} (1 - y^T (A A^T)^{-1} y)^{(d - k - 3)/2}. The exponent is nonnegative because k + 3 <= d; changing to z = (A A^T)^{-1/2} y turns the inner integral into sqrt(det(A A^T)) times a bounded function of z on the unit ball, so it is at most c_{d,k+1}^{p-1} det(A A^T)^{-(p-1)/2} for a finite normalizing constant. Gram-Schmidt writes sqrt(det(A A^T)) as the product of successive perpendicular lengths of the k + 1 independent Gaussian rows; those lengths live in dimensions d, d - 1, ..., d - k, every one of which exceeds p - 1 because k + p + 1 < d. A Gaussian perpendicular length in dimension at least p has a finite inverse moment of order p - 1, so iterating conditional expectation makes the double integral L finite. On the right side of the identity the number L equals (2 pi)^{-(k+1)(p-1)/2} times the total mass of Gamma_{k+1,d}^{F_u}, which is one, times integral J(u)^{-(k+1)} P^{\otimes p}(du); hence this inverse moment is finite.