C engine for the six-variable Dumont–Foata certificates: desktop kit and second-machine reproduction (k = 21–51)

This record contains the C engine used as a computational cross-check for the article "A monotonicity property of the six-variable Dumont–Foata polynomials" (10.5281/zenodo.23827461), together with a complete rerun of its large-k certificates on a second machine. WHAT THE ENGINE COMPUTES For each k it builds, in exact integer arithmetic (multi-modular with Chinese-remainder reconstruction), the polynomials Q_k and 2P_k of the article in the variables X, Y, Z, δ, and certifies their signs on the box 1 ≤ X, Y, Z ≤ 2, 3/4 ≤ δ ≤ 3 by bounding the positive and negative parts separately. The computation is organised around the δ-shift identity ∂m_k/∂δ = k m_(k−1). Under the current proof of the article, these certificates are an independent cross-check of the analytic argument for large k, not a part the theorems depend on. REPRODUCTION ON A SECOND MACHINE Hardware and software: AMD Ryzen 9 5950X (no AVX-512), Windows Subsystem for Linux (Ubuntu), gcc 15. Because this processor lacks AVX-512, the kit builds kernel5_portable.c, which performs the same arithmetic as the original AVX-512 kernel; before release it passed the v4 gate with AVX-512 disabled and gave results identical to the AVX-512 build at k = 30. Gates: gate.py 10 of 10 checks passed; gate4.py 3 of 3 passed. Results: for k = 40 to 51, identical to the recorded v4 results in all 13 non-timing fields (term counts, both certificates, the four certificate margins, the negative-term count, exactness). For k = 21 to 24, identical to the recorded v4.1 reference run in all 5 fields it holds (term counts, both certificates, negative terms). For k = 25 to 39, identical to the recorded v2 run in all 9 fields it holds (the same plus the four certificate margins). This is a reproduction of the same program on different hardware, compiler and kernel path; it is not an independent implementation. FILES c_engine_desktop_kit.zip: the engine sources, the three C kernels (including kernel5_portable.c), build.sh, gate.py, gate4.py, run4.py, compare.py, compare_21_39.py, the recorded reference results for k = 21–51, and README_DESKTOP.txt with step-by-step instructions. c_engine_reproduction.zip: the results of the rerun (all_results.jsonl, r21_39.jsonl and the per-k result files), BUILD_INFO.txt, the gate outputs and the per-k logs. TO REPRODUCE On Linux, macOS, or Windows with WSL: unpack the kit, run sh build.sh, python3 gate.py and python3 gate4.py, then python3 run4.py --from K --to K --out results.jsonl for any K, and python3 compare.py results.jsonl. AI ASSISTANCE The work was directed by the author. Most of the computational work was done with Claude (Anthropic) and Gemini (Google), with support from Grok (xAI). The author ran the reproduction on his own machine and takes responsibility for the contents.