{
  "audit_id": "hales-jewett-lower-bounds-6000a76f-2026-08-30",
  "audited_at": "2026-08-30T01:44:00Z",
  "verdict": "independently_reproduced_under_standard_hales_jewett_definitions",
  "peer_reviewed": false,
  "priority_verdict": "not_established_by_this_audit",
  "source": {
    "title": "Lower Bounds for the Hales-Jewett Numbers via Symmetric and One-Weight Colorings",
    "author": "Younes Mouhib",
    "arxiv_id": "2606.22155v2",
    "arxiv_url": "https://arxiv.org/abs/2606.22155v2",
    "submitted_at": "2026-06-20T17:23:39Z",
    "revised_at": "2026-07-24T18:25:46Z",
    "repository": "https://github.com/ysmouhib/hj-certificates",
    "repository_commit": "6000a76f0e2e6021dc813e65cfd938adc26ec2e3",
    "repository_tree": "d5a19372ace5d1e68e21654943d5f4efde018b2f",
    "repository_commit_date": "2026-07-24T19:44:35+02:00",
    "repository_license": "MIT",
    "archive_doi": "10.5281/zenodo.21537624"
  },
  "claims": [
    {
      "statement": "HJ(3,3) >= 22",
      "witness": "A three-coloring of the 253 cells in the discrete simplex T_21^(3)",
      "certificate_url": "https://raw.githubusercontent.com/ysmouhib/hj-certificates/6000a76f0e2e6021dc813e65cfd938adc26ec2e3/certificates/hj33_ge22_T21.cert",
      "certificate_sha256": "876eabf0c351c36f96160a8ef3baf4b131a1e8a7664be21ebacce39ffdbbdbb5",
      "simplex_cells": 253,
      "corner_tuples_checked": 1771,
      "monochromatic_corner_tuples": 0,
      "negative_controls": {
        "incomplete_coloring_rejected": true,
        "monochromatic_diagonal_rejected": true
      },
      "result": "VERIFIED"
    },
    {
      "statement": "HJ(4,2) >= 14",
      "witness": "A two-coloring of the 560 cells in the discrete simplex T_13^(4)",
      "certificate_url": "https://raw.githubusercontent.com/ysmouhib/hj-certificates/6000a76f0e2e6021dc813e65cfd938adc26ec2e3/certificates/hj42_ge14_T13.cert",
      "certificate_sha256": "ed602c972fc2e3556eacd3b2176aff81420e04736ace49174dc651f94aaf7de4",
      "simplex_cells": 560,
      "corner_tuples_checked": 1820,
      "monochromatic_corner_tuples": 0,
      "negative_controls": {
        "incomplete_coloring_rejected": true,
        "monochromatic_diagonal_rejected": true
      },
      "result": "VERIFIED"
    }
  ],
  "independent_method": {
    "verifier_url": "https://frontier.datastreamssolutions.com/artifacts/public-reproductions/hales-jewett-independent-verifier.py",
    "verifier_sha256": "a106402916ecdc30d914ee96ea8591dd551cc128b8bc90ff80fab251ad3e7a1c",
    "trusted_base": "CPython 3.13.7 standard library",
    "container_image": "python:3.13.7-alpine@sha256:9ba6d8cbebf0fb6546ae71f2a1c14f6ffd2fdab83af7fa5669734ef30ad48844",
    "isolation": "Network disabled; source and root filesystems read-only; Linux capabilities dropped; no-new-privileges, PID, CPU, and memory limits enabled.",
    "derivation": "Each word is colored by its letter-count vector. A combinatorial line with k wildcard positions and fixed count vector v maps to the corner tuple v+k*e_j. Exhaustive absence of monochromatic corner tuples gives a line-free coloring in the full grid.",
    "negative_controls": [
      "Remove one simplex cell and require rejection.",
      "Force the pure-letter diagonal cells to one color and require detection of a monochromatic line."
    ]
  },
  "candidate_verifier_replays": [
    {
      "verifier_url": "https://raw.githubusercontent.com/ysmouhib/hj-certificates/6000a76f0e2e6021dc813e65cfd938adc26ec2e3/src/verify_certificates.py",
      "verifier_sha256": "93b63608d78e7b0a129c0cf262a560fb18e52e9278281bd793c71ab2761ec99f",
      "results": [
        "HJ(3,3): 1771 corner tuples checked, 0 monochromatic",
        "HJ(4,2): 1820 corner tuples checked, 0 monochromatic"
      ]
    },
    {
      "verifier_url": "https://raw.githubusercontent.com/ysmouhib/hj-certificates/6000a76f0e2e6021dc813e65cfd938adc26ec2e3/sat_campaign/verify_hj.py",
      "verifier_sha256": "affc29d133cbf40bcfea38ec54ebcb6e44948006c39a8df83b7ec2aca109c63c",
      "random_full_grid_lines_sampled_per_case": 200000,
      "monochromatic_sampled_lines": 0,
      "result": "ALL CHECKS PASSED"
    }
  ],
  "duplicate_audit": {
    "vibemathed_records_checked": 639,
    "matching_vibemathed_records": 0,
    "matching_existing_curated_bundles": 0,
    "terms": [
      "hales-jewett",
      "hales jewett",
      "2606.22155",
      "ysmouhib",
      "hj-certificates",
      "HJ(3,3)",
      "HJ(4,2)"
    ]
  },
  "limitations": [
    "The source is a recent arXiv preprint; peer-reviewed or journal acceptance was not established.",
    "This audit establishes the two stated lower bounds, not exact values for either Hales-Jewett number.",
    "Historical priority and the source's claims about previous records were not independently audited.",
    "The remaining theorems, SAT forcing claims, and broader census results in the source were outside this audit.",
    "The random full-grid samples are supplemental; the exhaustive simplex checks and reduction carry the result."
  ]
}