Additional assistant audit for release preparation
Audit date: 3 October 2026. Scope: the supplied PROOF.md, APPENDIX.md, and exact verifier sources. This is an additional AI review, not independent human peer review or proof-assistant formalization.
Conclusion
No fatal mathematical error was found in the stated order-seven, rank-at-most-three theorem or in the supplied higher-order extension. The proof is a written analytic derivation plus finite exact rational checks. It is suitable for release as a research manuscript and reproducibility package with its present restricted scope and review-status disclosures.
The source arguments for diagonal normalization, permutation-monomial Cauchy–Schwarz, the fractional Bregman row bound, the complex Gaussian permanent identity, spherical log means, the top-direction measure tilt, eigenvalue elimination, the three-endpoint envelope reduction, and the infinite-order tail were checked directly. In particular, the conditional expectation and tilted logarithm constants yield exactly t - 12 at order seven and t - 1 - 11(n-1)/6 at general order. The piecewise convexity argument does not require convexity across interpolation knots.
Executed verification
Fresh replay used copied scripts in an isolated temporary directory, leaving the source package unchanged.
| Script | Result |
|---|---|
verify_q7.py |
PASS: 49/49 exact checks, including the optional analytic-tail finite checks |
verify_q8_q9.py |
PASS: all 21 closed intervals for n=8 and all 12 for n=9; complete domain coverage and no unresolved intervals |
audit_symbolic.py |
PASS: 21/21 symbolic and exact-fixture checks, using SymPy 1.14.0 and mpmath 1.3.0 installed into an isolated scratch dependency directory |
The initial optional audit attempts found SymPy absent; installing it into an isolated scratch dependency directory enabled a successful fresh replay. The 49 checks, 33 intervals, and 21 symbolic checks are different kinds of verification and should not be presented as a single proof-completeness percentage.
Source verification
- The author-maintained Yair Lavi page was freshly read and explicitly identifies the complex rank-at-most-three case as covered in every order except seven. It supports describing this as a documented gap. It does not establish priority.
- Ma's arXiv:2609.37461v1 was freshly read. Its Sections 2 and 3 supply the attributed row-interpolation, Gaussian-sphere, and spectral-tilt ingredients. These should remain credited rather than characterized as entirely new methods.
- The Li–Zhao–Luo Preprints.org page exists. Its through-order-six result is an external preprint dependency for the all-order rank-three corollary. That paper was not independently audited here.
Defensible release scope
The principal theorem is:
per(A ∘ B) ≤ (24/25) per(A) per(B)
for complex Hermitian positive semidefinite 7×7 matrices A and B, each of rank at most three. The package also provides separate higher-order constants for n≥7. The coefficient is certified and convenient, not claimed optimal. The inequality is strict when both matrices have positive diagonal; the non-strict wording correctly includes zero-permanent cases.
The all-order rank-at-most-three Chollet statement is conditional on accepting the cited through-order-six theorem. The unrestricted arbitrary-rank conjecture is outside the manuscript's result. Publication priority is unconfirmed. No experimental benchmarking, human peer review, or machine formalization is established by these verifiers.
Minor presentation suggestion
For maximum self-containment, add one sentence in the infinite-tail appendix: increasingness of (k!)^(1/k) follows because (k+1)^k > k!, so b_{k+1}^{k+1}=(k+1)k! > b_k^{k+1}. This closes the only background property invoked there without a local argument. It is a routine clarification, not a proof blocker.
Blockers
None for a scoped, unreviewed research release. The optional symbolic audit requires SymPy, while the main exact verifiers need only the Python standard library. No claim of first proof or unrestricted resolution is justified by this audit.