{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R396",
  "content_hash": null,
  "slug": "hrt20-attempt-global-search-and-literature-audit",
  "type": "attempt",
  "title": "Global optimality still needs a Hamming-specific certificate",
  "summary": "Fixed-seed swap search found the beta-one 21 witness. Exact enumeration of its full one-point neighborhood proves strict local optimality, while the checked flag-complex theorem supplies only the global bound 81.",
  "relevance": "For Largest first Betti number of a connected Hamming Rips complex, record hrt20-attempt-global-search-and-literature-audit (“Global optimality still needs a Hamming-specific certificate”) documents a concrete method, search boundary, or failed route. The record states: Fixed-seed swap search found the beta-one 21 witness.",
  "relevance_source": "recorded",
  "body": "Two fixed-seed simulated-annealing runs proposed 2,600,000 one-point moves in total. They began with the candidate's beta-one 6 set and later restarted from improved incumbents. The best verified value rose successively through 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, and 21. The second run proposed 2,100,000 moves without improving 21. These heuristic totals do not count distinct 20-subsets and carry no global exclusion force.\n\nThe executable artifact replaces the heuristic's final neighborhood check with an exact one. Every changed one-point replacement is inspected, and each connected neighbor has beta one at most 20.\n\nBeers and Botnan prove the sharp extremal result for all flag complexes on 20 vertices, giving 81. Their theorem ignores Hamming realizability. No primary source located in this audit treats the exact induced 20-point problem in \\(\\{0,1\\}^8\\). Closing the remaining gap calls for a global branch-and-bound, an integer or semidefinite certificate that incorporates boundary rank, or a Hamming-specific homological inequality.",
  "status": "completed",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "searches for the exact maximum first Betti number in the candidate's 20-point Hamming-cube problem"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://doi.org/10.4230/LIPIcs.SoCG.2025.14",
      "locator": "Beers and Botnan, SoCG 2025, Theorem 10 and Corollary 11; local fixed-seed optimization and exact verifier run on 2026-07-25"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.4230/LIPIcs.SoCG.2025.14",
    "locator": "Beers and Botnan, SoCG 2025, Theorem 10 and Corollary 11; local fixed-seed optimization and exact verifier run on 2026-07-25"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R397",
      "title": "The certified interval is 21 through 81",
      "object_type": "claim",
      "relation": "contextualizes",
      "direction": "outgoing"
    },
    {
      "slug": "hamming-rips-twenty-beta-one",
      "title": "hamming rips twenty beta one",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}