{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R836",
  "content_hash": null,
  "slug": "wfci127-claim-weight-85-witness",
  "type": "claim",
  "title": "A weight-five unit has inverse weight 85",
  "summary": "Exact cyclic multiplication verifies the supplied 127-bit inverse.",
  "relevance": "For Densest inverse of a weight-five binary cyclic polynomial, record wfci127-claim-weight-85-witness (“A weight-five unit has inverse weight 85”) records a bound, answer, status fact, or structural consequence. The record states: Exact cyclic multiplication verifies the supplied 127-bit inverse.",
  "relevance_source": "recorded",
  "body": "Set\n\\[\nf=1+x^{45}+x^{49}+x^{53}+x^{94}.\n\\]\nIn hexadecimal coefficient encoding, with the coefficient of \\(x^i\\) in bit \\(i\\),\n```\nf       = 0x400000000022200000000001\nf^{-1}  = 0x3bf17593bf175d3fb175d3fb377f3fb3\n```\nThe inverse has 85 set bits. Exact carryless cyclic multiplication gives\n\\[\nf\\,f^{-1}\\equiv1\\pmod{x^{127}+1}.\n\\]\nThe executable record repeats this check using integer bit operations. It also verifies the equivalent complement identity.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "the residue with support {0,45,49,53,94} in F_2[x]/(x^127+1)",
    "bounds": {
      "length": {
        "min": 127,
        "max": 127
      },
      "input_weight": {
        "min": 5,
        "max": 5
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://eprint.iacr.org/2012/409",
      "locator": "wfci127-artifact-exact-witness-verifier, executed 2026-07-25"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://eprint.iacr.org/2012/409",
    "locator": "wfci127-artifact-exact-witness-verifier, executed 2026-07-25"
  },
  "models": [],
  "relations": [
    {
      "slug": "R831",
      "title": "Exact 127-bit cyclic-product verifier",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "R833",
      "title": "The maximum inverse weight lies between 85 and 101",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "weight-five-cyclic-inverse-127",
      "title": "weight five cyclic inverse 127",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}