{
  "schema_version": 1,
  "problem_number": "AIM-PROBABILITY-0053",
  "title": "Total Variation Mixing of Weighted Quadratic Gibbs Samplers on a Cube",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We consider uniform random-scan coordinate Gibbs sampling on [0,1]^d for a density proportional to exp(-a sum_{i<j} c_ij (x_i-x_j)^2), with symmetric nonnegative weights. For every fixed nonzero weight matrix, we prove a worst-start total-variation upper bound of order a as a tends to infinity, with an explicit finite-parameter bound and logarithmic dependence on the requested accuracy. Together with the Wasserstein lower bound of Gerencser and Ottolini, this yields the sharp order Theta(a) at total-variation tolerance 1/4, including disconnected graphs and isolated vertices. The proof adapts the non-strongly-convex entropy interpolation of Ascolani, Lavenant and Zanella to the compact cube. A coordinate-coverage decomposition handles arbitrary deterministic starts without assigning finite entropy to the remaining singular branch. The result addresses the total-variation upper bound discussed for this weighted Gibbs model. It does not cover arbitrary Metropolis proposals or optimize dependence on dimension or graph parameters. This attributed application is AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority is claimed.",
  "result_type": "COMPLETE_EXPLICIT_WEIGHTED_GIBBS_TV_THEOREM",
  "categories": [
    "math.PR",
    "stat.CO"
  ],
  "keywords": [
    "Gibbs sampler",
    "total variation",
    "mixing time",
    "log concavity",
    "Knothe Rosenblatt transport",
    "entropy contraction",
    "weighted quadratic potential"
  ],
  "manuscript_version_date": "2026-10-11",
  "publication_date": "2026-10-11",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-11",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-probability-0053/",
  "pdf_url": "https://eulersolve.org/papers/aim-probability-0053/paper.pdf?v=ef00bb8c7606",
  "doi": "10.5281/zenodo.23297259",
  "zenodo_record_url": "https://zenodo.org/records/23297259",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete explicit theorem for exact uniform random-scan conditional Gibbs sampling on the weighted quadratic cube model. The entropy method is due to Ascolani, Lavenant and Zanella; the matching lower bound is due to Gerencser and Ottolini. The proposal-ambiguous original AIM wording is not marked unconditionally resolved. Arbitrary Metropolis proposals and optimal graph or dimension constants are not covered. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "ef00bb8c760662cc9ce318f7f330df87717d6c51a7bfb5d915de9860c7c17d50"
    },
    "source.zip": {
      "sha256": "bfa6874a2b8b908ea4e999e80e2e170a969877748f88d56c3d3148d492648349"
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    "verification_report.md": {
      "sha256": "acdc125489cb15eec96fe414ddfcafa6208abe8d4a41c28357be926c14a8bed4"
    }
  },
  "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."
}