{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R397",
  "content_hash": null,
  "slug": "hrt20-claim-certified-interval-21-81",
  "type": "claim",
  "title": "The certified interval is 21 through 81",
  "summary": "An exact 20-point witness gives first Betti number 21, improving the candidate incumbent 6. A sharp theorem for arbitrary 20-vertex flag complexes gives the upper bound 81.",
  "relevance": "For Largest first Betti number of a connected Hamming Rips complex, record hrt20-claim-certified-interval-21-81 (“The certified interval is 21 through 81”) records a bound, answer, status fact, or structural consequence. The record states: An exact 20-point witness gives first Betti number 21, improving the candidate incumbent 6.",
  "relevance_source": "recorded",
  "body": "Let \\(M\\) be the maximum in the candidate question. The point set certified in this record has 44 graph edges and four triangles. Its one-skeleton is connected. Exact elimination over \\(\\mathbb F_2\\) gives\n\\[\n\\operatorname{rank}\\partial_1=19,\\qquad\n\\operatorname{rank}\\partial_2=4,\n\\]\nand hence\n\\[\n\\beta_1=44-19-4=21.\n\\]\nThis raises the checked lower bound from 6 to 21.\n\nEvery clique complex is a flag complex. Theorem 10 and Corollary 11 of Beers and Botnan show that the largest first Betti number of any flag complex on 20 vertices is attained by the balanced Turán graph \\(K_{10,10}\\). Its clique complex is the graph itself and has\n\\[\n\\beta_1=(10-1)(10-1)=81.\n\\]\nThe Hamming Rips complexes in this problem form a subclass, so \\(M\\leq81\\). Combining both certificates gives\n\\[\n\\boxed{21\\leq M\\leq81}.\n\\]\nThe exact value remains open. The witness is a strict optimum under replacement of any one selected word, but that neighborhood certificate does not exclude a better set reached through two or more simultaneous replacements.",
  "status": "open",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "connected clique complexes of distance-at-most-two graphs induced by twenty distinct points of the eight-dimensional binary Hamming cube",
    "bounds": {
      "ambient_dimension": {
        "min": 8,
        "max": 8
      },
      "selected_points": {
        "min": 20,
        "max": 20
      },
      "distance_threshold": {
        "min": 2,
        "max": 2
      },
      "certified_beta_one_lower_bound": {
        "min": 21,
        "max": 21
      },
      "certified_beta_one_upper_bound": {
        "min": 81,
        "max": 81
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.4230/LIPIcs.SoCG.2025.14",
      "locator": "The lower endpoint is reproduced by hrt20-artifact-witness-and-neighborhood; the upper endpoint applies Beers and Botnan, Extremal Betti Numbers and Persistence in Flag Complexes, SoCG 2025, Theorem 10 and Corollary 11"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.4230/LIPIcs.SoCG.2025.14",
    "locator": "The lower endpoint is reproduced by hrt20-artifact-witness-and-neighborhood; the upper endpoint applies Beers and Botnan, Extremal Betti Numbers and Persistence in Flag Complexes, SoCG 2025, Theorem 10 and Corollary 11"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R395",
      "title": "Exact boundary ranks and exhaustive one-point neighborhood",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R396",
      "title": "Global optimality still needs a Hamming-specific certificate",
      "object_type": "attempt",
      "relation": "contextualizes",
      "direction": "incoming"
    },
    {
      "slug": "hamming-rips-twenty-beta-one",
      "title": "hamming rips twenty beta one",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}