{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R258",
  "content_hash": null,
  "slug": "ffe-claim-power-two-status",
  "type": "claim",
  "title": "The power-of-two claim is verified below 500 and unresolved in general",
  "summary": "All 46 positive counts among the 95 primes below 500 are powers of two; the orbit-cover model gives no general parity-factor proof yet.",
  "relevance": "For Power-of-two solution counts for a finite-field functional equation, record ffe-claim-power-two-status (“The power-of-two claim is verified below 500 and unresolved in general”) records a bound, answer, status fact, or structural consequence. The record states: All 46 positive counts among the 95 primes below 500 are powers of two; the orbit-cover model gives no general parity-factor proof yet.",
  "relevance_source": "recorded",
  "body": "For each prime \\(p<500\\), the exact orbit-cover computation determines \\(N(p)\\). Forty-six counts are positive and forty-nine vanish. Every positive count in this range is a power of two. The observed positive values have distribution\n\\[\n1^1,\\quad2^{30},\\quad4^7,\\quad8^3,\\quad32^3,\\quad512^1,\\quad1024^1,\n\\]\nwhere exponents record the number of primes attaining each value. The largest is \\(N(461)=1024\\).\n\nThe structural reduction in `ffe-claim-exact-orbit-cover` turns \\(N(p)\\) into a finite exact-cover count. Exact-cover counts can have odd factors in general. This audit found no additional involution or component theorem forcing the present counts to be powers of two for every prime. The universal question therefore remains unresolved in this entry.",
  "status": "reported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "the power-of-two question for every prime, with an exact census for all primes below 500",
    "bounds": {
      "verified_prime_upper_bound_exclusive": {
        "min": 500,
        "max": 500
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://sites.math.rutgers.edu/~nussbaum/Pubs/dynamicsJDE.pdf",
      "locator": "Exact census in ffe-artifact-primes-below-500 and structural analysis performed 2026-07-24"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://sites.math.rutgers.edu/~nussbaum/Pubs/dynamicsJDE.pdf",
    "locator": "Exact census in ffe-artifact-primes-below-500 and structural analysis performed 2026-07-24"
  },
  "relations": [
    {
      "slug": "R255",
      "title": "Exact orbit-cover counts for every prime below 500",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R257",
      "title": "Linear solutions come from the golden-ratio polynomial",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R254",
      "title": "Fibonacci iterative equations are studied over the reals",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R253",
      "title": "Executable Fibonacci-cycle exact-cover census",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "fibonacci-functional-equation-prime-count",
      "title": "fibonacci functional equation prime count",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}