C engine results: all large-k certificates for the six-variable Dumont–Foata monotonicity theorem reproduced exactly on a second machine (k = 21–51)

This record documents a reproduction, on a second machine, of the computer certificates for large k that accompany the article "A monotonicity property of the six-variable Dumont–Foata polynomials" (10.5281/zenodo.23827461).WHAT WAS REPRODUCEDFor every k from 21 to 51, the exact polynomials Q_k and 2P_k of the article were recomputed and their signs certified on the box 1 ≤ X, Y, Z ≤ 2, 3/4 ≤ δ ≤ 3: Q_k > 0 and 2P_k < 0. All arithmetic is exact integer arithmetic.RESULTSFor every k = 21 to 51 the rerun matches the recorded reference results in every field the reference holds: all certificate fields for k = 40 to 51; term counts, both certificates and the negative term for k = 21 to 24; the same plus the four certificate margins for k = 25 to 39. All built-in checks passed. Machine: AMD Ryzen 9 5950X, Windows Subsystem for Linux, gcc 15.These certificates are a computational cross-check of the article's analytic argument for large k; the theorems do not depend on them.FILESc_engine_results_k21_51.zip contains RESULTS.md (the verdict for every k), results_k21_51.csv (the results table), exact_certificate_sums.json (the exact certificate values for every k), FINGERPRINTS.txt, and verify_published_results.py, which lets anyone check the results from this data alone, without the engine.SOURCE CODEThe software is private and not published. FINGERPRINTS.txt lists the SHA-256 fingerprint of each source file, so the exact code that produced these results can be identified and verified on request.COGNITIVE ARCHITECTURE This work was conceived and directed by the author, Joshua W. Davis, as cognitive architect. The author originated the research question; conceived the overall approach; designed the verification protocol under which every result is judged (exact arithmetic only, built-in checks that must pass, negative controls that must fail, fingerprinted records, and reproduction on independent hardware); selected and arranged the results reported here; and made every determination of what is established and what is not. AI systems were used as tools within this architecture, under the author's direction and review. This record is one instance of the protocol's final gate: reproduction on independent hardware. AI ASSISTANCE Computational work and drafting were carried out with AI tools: mainly Claude (Anthropic) and Gemini (Google), with support from Grok (xAI), at the author's direction. The author reviewed, selected and approved all content, ran the reproduction on his own machine, and takes responsibility for the contents. RIGHTS The source code that produced these results is proprietary to Joshua W. Davis and is not licensed or published. The data and text of this record are released under Creative Commons Attribution 4.0.