{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R505",
  "content_hash": null,
  "slug": "mdsc-claim-sdp-upper-bound-and-integer-fallback",
  "type": "claim",
  "title": "The SDP upper bound is 388, with an integer-only fallback of 394",
  "summary": "Heinlein and Ihringer prove A₂(7,4) ≤ 388 using semidefinite programming. Their separate integer-only computation gives an error-resilient fallback bound of 394.",
  "relevance": "For Exact mixed-dimension subspace-code number A_2(7,4), record mdsc-claim-sdp-upper-bound-and-integer-fallback (“The SDP upper bound is 388, with an integer-only fallback of 394”) records a bound, answer, status fact, or structural consequence. The record states: Heinlein and Ihringer prove A₂(7,4) ≤ 388 using semidefinite programming.",
  "relevance_source": "recorded",
  "body": "Theorem 1.1 states the binary upper bound 388. Lemma 4.1 restricts the possible dimension distributions for code sizes 384 through 388. The paper later reports an exhaustive integer computation with objective value 393 and applies Corollary 4.6 to obtain A₂(7,4) ≤ 394. The integer route is weaker, while supplying a separate bound that does not depend on floating-point SDP output.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "upper bounds for binary mixed-dimension subspace codes in ambient dimension 7 with minimum distance 4",
    "bounds": {
      "field_order": {
        "min": 2,
        "max": 2
      },
      "ambient_dimension": {
        "min": 7,
        "max": 7
      },
      "minimum_subspace_distance": {
        "min": 4,
        "max": 4
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1809.09352",
      "locator": "Heinlein and Ihringer, arXiv:1809.09352v2, Theorems 1.1 and 1.2 on PDF pp. 2-3, Lemma 4.1 on p. 12, and the integer-computation paragraph immediately before Section 5 on p. 17"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1809.09352",
    "locator": "Heinlein and Ihringer, arXiv:1809.09352v2, Theorems 1.1 and 1.2 on PDF pp. 2-3, Lemma 4.1 on p. 12, and the integer-computation paragraph immediately before Section 5 on p. 17"
  },
  "relations": [
    {
      "slug": "R501",
      "title": "Audit the primary sources, current bounds table, and production record",
      "object_type": "attempt",
      "relation": "reports",
      "direction": "incoming"
    },
    {
      "slug": "R502",
      "title": "The dated interval is 334 ≤ A₂(7,4) ≤ 388",
      "object_type": "claim",
      "relation": "supports",
      "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"
    }
  ]
}