{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R255",
  "content_hash": null,
  "slug": "ffe-claim-census-below-500",
  "type": "claim",
  "title": "Exact orbit-cover counts for every prime below 500",
  "summary": "The census reproduces all candidate examples and finds powers of two at every prime with a solution.",
  "relevance": "For Power-of-two solution counts for a finite-field functional equation, record ffe-claim-census-below-500 (“Exact orbit-cover counts for every prime below 500”) records a bound, answer, status fact, or structural consequence. The record states: The census reproduces all candidate examples and finds powers of two at every prime with a solution.",
  "relevance_source": "recorded",
  "body": "The exact computation enumerates all cycles of \\(T_p\\) on the full set of \\(p^2\\) ordered pairs. It retains a cycle precisely when its first coordinates are distinct and exclude zero, then recursively counts disjoint projection covers of \\(\\mathbb F_p^*\\). A minimum-option branching rule changes search order only; memoization keys the uncovered field elements exactly.\n\nSelected results are\n\\[\nN(5)=1,\\ N(11)=2,\\ N(29)=4,\\ N(139)=8,\n\\]\n\\[\nN(199)=512,\\ N(211)=32,\\ N(281)=32,\\ N(461)=1024.\n\\]\nThe canonical record stores, for each prime, its count, eligible-cycle count, total \\(T_p\\)-cycle count, and number of memoized exact-cover states. Its SHA-256 digest is `c8ef45793aebcc55c45cb1ee99d55c67d9e418b867c5b5d6939f92e9493d2e37`.",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "all 95 primes p with 2<=p<500",
    "bounds": {
      "minimum_prime": {
        "min": 2,
        "max": 2
      },
      "maximum_prime": {
        "min": 499,
        "max": 499
      },
      "prime_count": {
        "min": 95,
        "max": 95
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://sites.math.rutgers.edu/~nussbaum/Pubs/dynamicsJDE.pdf",
      "locator": "Independent exact enumeration in ffe-artifact-primes-below-500, executed 2026-07-24"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://sites.math.rutgers.edu/~nussbaum/Pubs/dynamicsJDE.pdf",
    "locator": "Independent exact enumeration in ffe-artifact-primes-below-500, executed 2026-07-24"
  },
  "relations": [
    {
      "slug": "R256",
      "title": "Solutions are exactly projection-disjoint covers by Fibonacci-map cycles",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R253",
      "title": "Executable Fibonacci-cycle exact-cover census",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "R258",
      "title": "The power-of-two claim is verified below 500 and unresolved in general",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "fibonacci-functional-equation-prime-count",
      "title": "fibonacci functional equation prime count",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}