{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R506",
  "content_hash": null,
  "slug": "mdsc-claim-singer-invariant-maximum-255",
  "type": "claim",
  "title": "Full Singer invariance caps a mixed code at 255",
  "summary": "For multiplication by x modulo x⁷+x+1, exhaustive orbit enumeration gives maximum size 255 among all invariant mixed-dimension codes of minimum distance 4. Two compatible three-space orbits together with F₂⁷ attain the maximum.",
  "relevance": "For Exact mixed-dimension subspace-code number A_2(7,4), record mdsc-claim-singer-invariant-maximum-255 (“Full Singer invariance caps a mixed code at 255”) records a bound, answer, status fact, or structural consequence. The record states: For multiplication by x modulo x⁷+x+1, exhaustive orbit enumeration gives maximum size 255 among all invariant mixed-dimension codes of minimum distance 4.",
  "relevance_source": "recorded",
  "body": "The fixed Singer action has order 127 and partitions all 29,212 subspaces into the recorded orbit census. Exactly 72 three-space orbits and 72 four-space orbits satisfy the distance condition internally. Their compatibility graph has 144 vertices, 72 edges, and no triangles. Its edges stay within a single dimension. Adding the compatible singleton zero or whole-space orbit gives maximum total size 255. The listed bases A and B generate compatible three-space orbits, and adjoining F₂⁷ gives a checked 255-word code.",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "all unions of complete subspace orbits under multiplication by x in F_2[x]/(x^7+x+1), with minimum subspace distance 4",
    "bounds": {
      "ambient_subspaces": {
        "min": 29212,
        "max": 29212
      },
      "group_order": {
        "min": 127,
        "max": 127
      },
      "valid_nontrivial_orbits": {
        "min": 144,
        "max": 144
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "locator": "Exact Singer-action enumeration in mdsc-artifact-exact-f2-seven-census, executed 2026-07-28 UTC"
    },
    "runtime_seconds": 6.27,
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Exact Singer-action enumeration in mdsc-artifact-exact-f2-seven-census, executed 2026-07-28 UTC"
  },
  "relations": [
    {
      "slug": "R497",
      "title": "Exact F₂⁷ census, Appendix C replay, and Singer-orbit search",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R500",
      "title": "Extend the exact replacement search through six deletions",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "mixed-dimension-subspace-code-f2-7-d4",
      "title": "mixed dimension subspace code f2 7 d4",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}