{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R92",
  "content_hash": null,
  "slug": "c7p5-claim-certified-lower-bound-367",
  "type": "claim",
  "title": "The certified lower bound is 367 words",
  "summary": "A certified 367-word independent set in \\(C_7^{\\boxtimes5}\\) passes all 67,161 pair checks; whether an independent set of size 368 exists remains open.",
  "relevance": "For A 368-word code in the fifth strong power of the 7-cycle, record c7p5-claim-certified-lower-bound-367 (“The certified lower bound is 367 words”) records a bound, answer, status fact, or structural consequence. The record states: A certified 367-word independent set in \\(C_7^{\\boxtimes5}\\) passes all 67,161 pair checks; whether an independent set of size 368 exists remains open.",
  "relevance_source": "recorded",
  "body": "Let \\(R\\) be the public file `c7/R367.txt`. It contains 367 distinct members of \\(\\mathbb Z_7^5\\). For each unordered pair \\(x,y\\in R\\), the exact checker finds a coordinate \\(i\\) with\n\\[\n\\min\\bigl((x_i-y_i)\\bmod 7,(y_i-x_i)\\bmod 7\\bigr)\\geq2.\n\\]\nThus \\(R\\) is independent in \\(C_7^{\\boxtimes5}\\), and\n\\[\n\\alpha(C_7^{\\boxtimes5})\\geq367.\n\\]\nThe check covers all \\(\\binom{367}{2}=67{,}161\\) pairs and reproduces the construction in Polak and Schrijver.\n\nThe capacity comparison is exact before taking roots:\n\\[\n367^2=134689<134753<135424=368^2.\n\\]\nConsequently, the 2026 ten-dimensional construction improves the capacity bound obtained from 367 words, while a 368-word fifth-power construction would improve the ten-dimensional bound.",
  "status": "reproduced",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "the explicit 367-word set R in Z_7^5 distributed with the cited papers",
    "bounds": {
      "cycle_length": {
        "min": 7,
        "max": 7
      },
      "strong_power": {
        "min": 5,
        "max": 5
      },
      "code_cardinality": {
        "min": 367,
        "max": 367
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1808.07438v2",
      "locator": "Sven C. Polak and Alexander Schrijver, New lower bound on the Shannon capacity of C7 from circular graphs, Information Processing Letters 143 (2019), 37-40, Section 3 and Appendix: explicit code; DOI 10.1016/j.ipl.2018.11.006; independently checked by c7p5-artifact-r367-and-local-exchanges"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1808.07438v2",
    "locator": "Sven C. Polak and Alexander Schrijver, New lower bound on the Shannon capacity of C7 from circular graphs, Information Processing Letters 143 (2019), 37-40, Section 3 and Appendix: explicit code; DOI 10.1016/j.ipl.2018.11.006; independently checked by c7p5-artifact-r367-and-local-exchanges"
  },
  "relations": [
    {
      "slug": "R90",
      "title": "Exact R367 verifier and radius-three exchange certificate",
      "object_type": "artifact",
      "relation": "reproduces",
      "direction": "incoming"
    },
    {
      "slug": "R91",
      "title": "The checked sources leave 368 unresolved",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "c7-fifth-power-independent-368",
      "title": "c7 fifth power independent 368",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}