{
  "schema_version": 1,
  "problem_number": "OPG-37656",
  "title": "Three-Colourability of Random Lifts of K_5",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let L_n be the random n-lift of K_5 obtained by independent uniform perfect matchings over its ten edges. We prove that P(chi(L_n)=3) tends to one. The main ingredient is an exact-marginal edge entropy inequality for pairs of fibrewise balanced three-colourings. A sharp three-coordinate fractional-power inequality yields the edge bound through a spectral absolute-value estimate. This gives the unique global maximum in the degree-four second-moment calculation. We derive the Gaussian constants and integer-lattice factors, treat every lift size by bounded affine-lattice shifts, and apply classical small subgraph conditioning. An odd-cycle argument gives the matching lower bound. The exact K_5 question in OPG-37656 is answered; the corresponding question for arbitrary complete bases is not addressed. Prior random-lift moment machinery is credited to Nir and Perez-Gimenez. This is an unrefereed, AI-assisted preprint with reproducible exact checks, not independent or formal verification, and makes no absolute priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "keywords": [
    "random lifts",
    "graph colouring",
    "chromatic number",
    "entropy method",
    "small subgraph conditioning",
    "OPG-37656"
  ],
  "manuscript_version_date": "2026-10-17",
  "publication_date": "2026-10-17",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-17",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/opg-37656/",
  "pdf_url": "https://eulersolve.org/papers/opg-37656/paper.pdf?v=0358e036aef0",
  "doi": "10.5281/zenodo.23430098",
  "zenodo_record_url": "23430098",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Resolves the exact frozen K5 concentration question through all positive integer lift sizes tending to infinity, not the broader background question for arbitrary complete base graphs. AI-assisted, self-audited and unrefereed; no independent expert review, formal verification or absolute-priority certification is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "0358e036aef0a419f62fea807c84348773e2ced3a2f53f5bbb60c0e56107584d"
    },
    "source.zip": {
      "sha256": "691e8e76d43cf8d35e5c64a38bdd6da7e5382e5f6b1c63075efb66dc466e6e02"
    },
    "verification_report.md": {
      "sha256": "44766cb9c614feadf405d5fd3781188a74812a44aabfea1a3e553931cfb0552e"
    }
  },
  "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."
}