{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R833",
  "content_hash": null,
  "slug": "wfci127-claim-certified-interval",
  "type": "claim",
  "title": "The maximum inverse weight lies between 85 and 101",
  "summary": "An explicit unit gives 85, while a complement argument gives the universal upper bound 101.",
  "relevance": "For Densest inverse of a weight-five binary cyclic polynomial, record wfci127-claim-certified-interval (“The maximum inverse weight lies between 85 and 101”) records a bound, answer, status fact, or structural consequence. The record states: An explicit unit gives 85, while a complement argument gives the universal upper bound 101.",
  "relevance_source": "recorded",
  "body": "Let \\(M\\) be the requested maximum. The support\n\\[\n\\{0,45,49,53,94\\}\n\\]\nhas the inverse displayed in `wfci127-claim-weight-85-witness`, so \\(M\\geq85\\). The argument in `wfci127-claim-complement-upper-bound` proves \\(M\\leq101\\). Therefore\n\\[\n85\\leq M\\leq101.\n\\]\nBoth endpoints are odd, as every inverse of an odd-weight polynomial evaluates to 1 at \\(x=1\\). The exact maximum remains open in this fixture. Its complete normalized search space has\n\\[\n\\binom{126}{4}=10{,}009{,}125\n\\]\nsupports.",
  "status": "supported",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "all normalized weight-five residues 1+x^a+x^b+x^c+x^d in F_2[x]/(x^127+1)",
    "bounds": {
      "length": {
        "min": 127,
        "max": 127
      },
      "input_weight": {
        "min": 5,
        "max": 5
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://eprint.iacr.org/2012/409",
      "locator": "Exact witness replay and complement proof in this fixture, 2026-07-25"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://eprint.iacr.org/2012/409",
    "locator": "Exact witness replay and complement proof in this fixture, 2026-07-25"
  },
  "models": [],
  "relations": [
    {
      "slug": "R836",
      "title": "A weight-five unit has inverse weight 85",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R831",
      "title": "Exact 127-bit cyclic-product verifier",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R834",
      "title": "Every inverse has weight at most 101",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R832",
      "title": "Exact exhaustive-search template",
      "object_type": "artifact",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R835",
      "title": "The exact normalized extremum was not located in the audited literature",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "weight-five-cyclic-inverse-127",
      "title": "weight five cyclic inverse 127",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}