{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R626",
  "content_hash": null,
  "slug": "qafp-structural-restrictions",
  "type": "claim",
  "title": "Any solution has at most one nonidentity automorphism and tightly fixed intersections",
  "summary": "Peer-reviewed results reduce the automorphism group to the trivial group or one specified involution and fix the solid intersection distributions.",
  "relevance": "For A binary q-analog of the Fano plane, record qafp-structural-restrictions (“Any solution has at most one nonidentity automorphism and tightly fixed intersections”) records a bound, answer, status fact, or structural consequence. The record states: Peer-reviewed results reduce the automorphism group to the trivial group or one specified involution and fix the solid intersection distributions.",
  "relevance_source": "recorded",
  "body": "Kiermaier, Kurz, and Wassermann proved that the full automorphism group of a binary q-Fano plane is either trivial or cyclic of order two. In the order-two case, its conjugacy class in \\(\\operatorname{GL}(7,2)\\) is fixed. This restriction explains why symmetry-based searches cannot finish the unrestricted problem: a solution may be rigid.\n\nIntersection-number equations impose further conditions. A 4-subspace contains at most one block. If it contains none, the numbers of blocks meeting it in dimensions \\(0,1,2,3\\) are\n\\[\n(136,210,35,0).\n\\]\nIf it contains a block, the vector is\n\\[\n(128,224,28,1).\n\\]\nAmong the 11,811 solids, 6,096 have the first type and 5,715 have the second.\n\nAt every point, the 21 incident blocks induce a line spread of \\(\\operatorname{PG}(5,2)\\). A point is called an alpha-point when this spread is geometric. Kiermaier's 2022 theorem implies that every hyperplane contains a non-alpha-point, so the non-alpha-points form a hyperplane-blocking set.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "necessary structure of any binary 2-(7,3,1)_2 subspace design",
    "bounds": {
      "field_order": {
        "min": 2,
        "max": 2
      },
      "blocks": {
        "min": 381,
        "max": 381
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1007/s10623-017-0360-6",
      "locator": "Kiermaier, Kurz, and Wassermann, The order of the automorphism group of a binary q-analog of the Fano plane is at most two, Theorem 1; Kiermaier and Pavcevic, Intersection numbers for subspace designs, Journal of Combinatorial Designs 23 (2015), section 4; Kiermaier, On alpha-points of q-analogs of the Fano plane, Theorem 2"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1007/s10623-017-0360-6",
    "locator": "Kiermaier, Kurz, and Wassermann, The order of the automorphism group of a binary q-analog of the Fano plane is at most two, Theorem 1; Kiermaier and Pavcevic, Intersection numbers for subspace designs, Journal of Combinatorial Designs 23 (2015), section 4; Kiermaier, On alpha-points of q-analogs of the Fano plane, Theorem 2"
  },
  "models": [],
  "relations": [
    {
      "slug": "R621",
      "title": "Exact enumeration of the 11,811 by 2,667 cover matrix",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R625",
      "title": "Existence of the binary q-Fano plane remains open",
      "object_type": "claim",
      "relation": "constrains",
      "direction": "outgoing"
    },
    {
      "slug": "R622",
      "title": "The 2017 triviality claim was withdrawn after a computational error",
      "object_type": "attempt",
      "relation": "qualifies",
      "direction": "incoming"
    },
    {
      "slug": "q-analog-fano-plane",
      "title": "q analog fano plane",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}