{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R677",
  "content_hash": null,
  "slug": "rsbm10-claim-certified-count-interval",
  "type": "claim",
  "title": "The singular count is between 126,174,821,830,345,268,667,240,568,576 and 901,210,462,928,281,273,073,900,978,176",
  "summary": "A two-term inclusion-exclusion count supplies the lower bound. Invertibility over F_2 supplies the upper bound.",
  "relevance": "For Number of singular ten by ten binary matrices over the reals, record rsbm10-claim-certified-count-interval (“The singular count is between 126,174,821,830,345,268,667,240,568,576 and 901,210,462,928,281,273,073,900,978,176”) records a bound, answer, status fact, or structural consequence. The record states: A two-term inclusion-exclusion count supplies the lower bound.",
  "relevance_source": "recorded",
  "body": "Let S_10 be the number in the question and let T=2^100=1,267,650,600,228,229,401,496,703,205,376. The certified interval is\n\\[\n126{,}174{,}821{,}830{,}345{,}268{,}667{,}240{,}568{,}576\n\\leq S_{10}\\leq\n901{,}210{,}462{,}928{,}281{,}273{,}073{,}900{,}978{,}176.\n\\]\nEquivalently, for a uniform random binary matrix,\n\\[\n\\frac{492870397774786205731408471}{4951760157141521099596496896}\n\\leq \\Pr(\\det A=0)\\leq\n\\frac{25613941912987493}{36028797018963968}.\n\\]\nThe decimal endpoints are approximately 0.09953438416518605 and 0.7109297015802510.\n\nFor the lower bound, first count every matrix having a zero row or two equal rows. Its complement consists of ten ordered, distinct, nonzero vectors chosen from 1,023 possibilities, so this first family has size T-(1023)_10, where (a)_k=a(a-1)\\cdots(a-k+1). Among matrices outside that family, consider the 55 column events consisting of ten zero-column events and 45 equal-column events. Each single event leaves 511 possible nonzero row patterns, giving (511)_10 row-distinct matrices. Any two distinct column events impose two independent binary equations on a row, giving (255)_10 matrices. The first Bonferroni lower bound for their union is 55(511)_10-1485(255)_10. Adding the disjoint row-degenerate family gives the stated lower endpoint. Every counted matrix has determinant zero over the reals.\n\nFor the upper bound, every binary matrix invertible over F_2 has odd integer determinant and is invertible over the reals. There are\n\\[\n|\\operatorname{GL}(10,2)|=\\prod_{i=0}^{9}(2^{10}-2^i)\n=366{,}440{,}137{,}299{,}948{,}128{,}422{,}802{,}227{,}200\n\\]\nsuch matrices. Subtracting this from T gives the upper endpoint.\n\nThe interval leaves the exact value unresolved.",
  "status": "open",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "all labeled 10 by 10 matrices with entries in {0,1}, with singularity taken over the real numbers",
    "bounds": {
      "order": {
        "min": 10,
        "max": 10
      },
      "matrices": {
        "min": 1.2676506002282294e+30,
        "max": 1.2676506002282294e+30
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1016/S0021-9800(67)80006-1",
      "locator": "The zero-or-equal row and column family follows the Metropolis-Stein construction; arithmetic is reproduced by rsbm10-artifact-bounds-and-regression"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/S0021-9800(67)80006-1",
    "locator": "The zero-or-equal row and column family follows the Metropolis-Stein construction; arithmetic is reproduced by rsbm10-artifact-bounds-and-regression"
  },
  "models": [],
  "relations": [
    {
      "slug": "R674",
      "title": "Bareiss regression and exact order-ten bound verifier",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R678",
      "title": "The current exact table ends at order nine",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R676",
      "title": "The binary order-n singular probability equals the sign-matrix order-(n+1) singular probability",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R675",
      "title": "The exact order-ten count was not located",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "real-singular-binary-matrices-ten",
      "title": "real singular binary matrices ten",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}