# Verification of the spherical-triangle counterexample

The manuscript proves the whole stated quantitative counterexample, not
only a finite numerical instance. A conditional coupling bounds the L2
norm of the conditional expectation onto every m-dimensional projection
by 2 sqrt(2 log 2) sqrt(m) times (1/sqrt(d+2)+1/sqrt(d-m)). Cauchy-Schwarz
therefore controls every measurable projection test of L2 norm at most one,
with uniformity over every orientation and every linear basis.

Independent Gaussian coordinates and exact Gram-Schmidt identities reduce
the triangle statistic to a three-dimensional Gaussian limit of covariance
1 on the diagonal and 1/4 off the diagonal. Bounded continuous cosine
products permit passage to the limit. Exact centering yields triangle
average 7/16384 and function variance 225/2048. The fixed threshold
7/32768 then contradicts any dimension-free inverse bounds.

The reproducibility program uses Python's standard-library Fraction class.
Its normal and optimized runs agree. It checks the triangle Gram matrix,
the Gaussian covariance and eigenvectors, all eight sign contributions,
exact cosine moments, four finite fourth-moment contractions and a rotated
projection example. It is arithmetic support, not a proof assistant, and
does not pretend finite tests certify every subspace or a limit theorem.

All five compiled PDF pages were visually inspected. Author details are
in the footnote and AI assistance is disclosed in the text. No clipping,
overflow, missing glyphs or unresolved citations were found. A harmless
underfull bibliography line does not affect content or readability.

The complete proof passed originating-model self-audit. No independent
review, formal verification or exhaustive novelty certificate is claimed.
The related zero-sum-free set problem and planar/finite-field variants
are explicitly excluded, as is a purely qualitative finite-dimensional
interpretation of the unquantified word 'correlate'.