Exact finite threshold and optimal factor partitions for products of simplices

An exact finite classification of Cartesian products of standard simplices with factor dimensions at least two and total dimensions 2 through 40. All 37,337 unordered partitions are checked. Within this class the first simplex-benchmark violation is dimension20, uniquely optimized by (10,10); the first three-factor optimum is dimension33, uniquely (11,11,11). The source already supplies the dimension20 example and product mechanism. Only the specified simplex-product class, factor dimensions>=2 and the declared finite domain are computationally covered. The source cites different counterexamples in dimensions>=9, so no unrestricted minimal-dimension claim is intended. Geometry is a written Cauchy/Fubini reduction, not kernel-verified. No global novelty claim. Integer DP certificate, exhaustive rational partition enumeration, and a separately tasked reviewer's multiplicity-vector/count implementation agree on 39 maxima, complete optimal sets and counts. Sixteen tests per Python mode and eight corrupt-certificate controls pass. Upstream AI maintainer independently enumerated the same finite class and added another rejecting mutation. Substantial OpenAI Codex assistance; no human peer review. Original complete public disclosure: 2026-10-24T18:36:47Z in PR 34. Source contribution commit cb5d09c2a105a64bd4efe7ad307f4633cbd923cf. Accepted upstream at 2026-10-24T20:27:36Z. OpenAI supplied the constructions and mathematical mechanisms; Pico/Hakon Amdal supplied computations and review. Predecessors cited in the packet retain attribution. A DOI provides an archive and citation, not certification of correctness, novelty or legal priority.