{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R700",
  "content_hash": null,
  "slug": "sat-artifact-sweep-200000",
  "type": "artifact",
  "title": "Exact subtract-a-triangular sweep through 200,000",
  "summary": "Exact dynamic programming finds 2,325 positive P-positions through 200,000 and a density of 0.011625 at the endpoint.",
  "relevance": "This supplies a computation, dataset, or executable check used to assess the target problem.",
  "relevance_source": "derived",
  "body": "The program computes every Sprague-Grundy value in increasing heap order. The P-position counts are 376 through 10,000, 998 through 50,000, 1,523 through 100,000, and 2,325 through 200,000. The first positive P-positions are 2, 4, 9, 11, 13, 18, 20, 22, 27, 29, 31, and 51. The maximum Grundy value in the range is 144, first reached at 198,787.\n\nThe four reported densities decrease with the cutoff. The ratio itself can rise at an individual P-position, so the data supports a trend rather than monotonicity.",
  "status": "available",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "every starting heap size from 0 through 200000",
    "bounds": {
      "n": {
        "min": 0,
        "max": 200000
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "partial",
    "kind": "python_computation",
    "source": "research/candidate-screening/multi2.py",
    "citation": {
      "url": "https://github.com/philipfweiss/theoremdb/blob/main/research/candidate-screening/multi2.py",
      "locator": "research/candidate-screening/multi2.py, deeper subtraction-game sweep; independently rerun on 2026-07-24"
    },
    "missing": [
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://github.com/philipfweiss/theoremdb/blob/main/research/candidate-screening/multi2.py",
    "locator": "research/candidate-screening/multi2.py, deeper subtraction-game sweep; independently rerun on 2026-07-24"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R702",
      "title": "Subtract-a-triangular losing positions have density zero",
      "object_type": "claim",
      "relation": "tests",
      "direction": "outgoing"
    },
    {
      "slug": "R701",
      "title": "Determine the growth order of the triangular-game P-positions",
      "object_type": "attempt",
      "relation": "uses",
      "direction": "incoming"
    },
    {
      "slug": "subtract-a-triangular-density",
      "title": "subtract a triangular density",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}