# Verification of the coordinate intersection counterexample

The written proof establishes an all-prime family for p>=5 in a countably generated hyperfinite finite von Neumann algebra. It verifies the Lie bracket, class-three multiplication, exact subgroup generation, faithful expected embeddings, permutation exchangeability, minimality, scalar tail, the p²-versus-p intersection defect, and the least enlarged factorization. The group-to-von-Neumann passage is justified by Fourier expectations and a separating regular vector.

The standalone exact checker verifies nine rational polynomial associators in 27 independent variables, 2,744 basis Jacobi triples, three complete two-generator subgroups of order 3,125, the intersection order 25 and singleton order 5, 25 disjoint subgroup pairs, 486 permutation-equivariance checks, 36 ordered commutator formulas and 900 seeded exact associativity/inverse/exponent cases. Normal and optimized modes agree. The whole group of order 5^9 is not exhaustively enumerated; finite computations are regression evidence rather than the general proof.

Thirteen complete relevant source pages were visually inspected. The five manuscript pages were rendered and inspected without clipping, broken mathematics or overflow; the built-in compiler and a local Tectonic build succeeded without final-log warnings. A first macro-grouping error and a missing optional dependency were fixed with their original versions retained in the research archive.

The original researcher's detailed self-audit is complete. No independent review or proof-assistant verification occurred. The bounded primary-source search identified no matching construction but does not certify novelty or absolute priority. The result is limited to retaining the original coordinate ranges: the ambient algebra does admit the enlarged factorization constructed in the paper. The unrestricted wording of the original source conjecture is not declared closed.

The source ZIP is checked separately for a clean portable rebuild and exact checker replay before the package is registered as ready. There is no DOI, external publication or public HF announcement yet.