{
  "schema_version": 1,
  "problem_number": "AIM-PHYSICS-0025",
  "title": "More Colors Can Slow Mixing on a Four-Vertex Path",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Consider the discrete-time coloring chain that chooses a uniform vertex and a uniform color from all q colors, accepting the proposed recoloring when it is proper and otherwise staying in place. On the four-vertex path, the worst-start total-variation mixing time at threshold 1/e equals 8 for q=17 and 9 for q=18. Thus increasing the number of colors can increase the mixing time even on a connected tree with no isolated vertices. We give an exact finite certificate, verified both by full-state integer enumeration and by a color-symmetry quotient. An elementary comparison with independent coordinate refresh also proves that the mixing time is 9 for every q>=512. The result concerns the literal proposal chain and a fixed finite graph, not an asymptotic slowdown on growing lattices. The previously reported edgeless example is credited. This AI-assisted, self-audited preprint is unrefereed and makes no absolute priority claim.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "keywords": [
    "Glauber dynamics",
    "graph coloring",
    "total variation",
    "mixing time",
    "connected counterexample",
    "exact computation",
    "AIM-PHYSICS-0025"
  ],
  "manuscript_version_date": "2026-10-19",
  "publication_date": "2026-10-19",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-19",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-physics-0025/",
  "pdf_url": "https://eulersolve.org/papers/aim-physics-0025/paper.pdf?v=9bc71aa80e21",
  "doi": "10.5281/zenodo.23461791",
  "zenodo_record_url": "https://zenodo.org/records/23461791",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "This complete finite computer-assisted counterexample strengthens the previously reported edgeless example to a connected four-vertex path: t(P4,17)=8 and t(P4,18)=9 at threshold 1/e, with t(P4,q)=9 for all q>=512. Current-color proposals and rejections count as steps; this is not legal-color heat bath, continuous time, or sweep time. The upstream edgeless result is credited; the Hugging Face record is already marked solved. No new closure of its merged source record, no solution of the separate square-lattice rates or Questions 6-16, no minimality, and no sharp onset threshold are claimed. AI-assisted, self-audited and unrefereed; no external review, formal verification or absolute priority claim.",
  "files": {
    "paper.pdf": {
      "sha256": "9bc71aa80e216eccaca3de711099248326aaeb4d26f1dc3bba4aae3ec11448f4"
    },
    "source.zip": {
      "sha256": "88a79f8182628780ff0aa79e508016a20c70cb9231299781a1358b70a8ffd0f3"
    },
    "verification_report.md": {
      "sha256": "e4fdee23c759e1269b34aadcaeb07ea158fd1ad63a4faf5c9808b25ba45cadc5"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}