{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R701",
  "content_hash": null,
  "slug": "sat-attempt-growth-order",
  "type": "attempt",
  "title": "Determine the growth order of the triangular-game P-positions",
  "summary": "The density question is settled, while the size of \\(P\\cap[1,N]\\) remains a sharper quantitative target.",
  "relevance": "This records a line of attack on the target problem, including what it establishes and where it stops.",
  "relevance_source": "derived",
  "body": "Let \\(T(N)\\) be the number of positive triangular numbers at most \\(N\\). Maximality of the P-position set gives \\((L(N)+1)(T(N)+1)\\geq N+1\\), hence \\(L(N)=\\Omega(\\sqrt N)\\). Polynomial recurrence gives \\(L(N)=o(N)\\). The computation through 200,000 is compatible with a much smaller order than the linear scale. Establishing a sharp exponent or asymptotic for this greedy triangular-difference-free set is a possible replacement question. Its novelty needs a separate literature review.",
  "status": "open_strategy",
  "evidence_grade": "proposed",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://oeis.org/A030194",
      "locator": "Quantitative refinement of the sequence recorded as OEIS A030194"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://oeis.org/A030194",
    "locator": "Quantitative refinement of the sequence recorded as OEIS A030194"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R702",
      "title": "Subtract-a-triangular losing positions have density zero",
      "object_type": "claim",
      "relation": "uses",
      "direction": "outgoing"
    },
    {
      "slug": "R700",
      "title": "Exact subtract-a-triangular sweep through 200,000",
      "object_type": "artifact",
      "relation": "uses",
      "direction": "outgoing"
    },
    {
      "slug": "subtract-a-triangular-density",
      "title": "subtract a triangular density",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}