{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R23",
  "content_hash": null,
  "slug": "asq-attempt-h6-template-criterion",
  "type": "attempt",
  "title": "The direct h6 template criterion fails by dimension",
  "summary": "The Rao-Rosenfeld sufficient decision criterion cannot apply to a scalar projection of h6: its expanding eigenspace has dimension 3, while the length-and-sum map has rank at most 2.",
  "relevance": "For Infinite additive-square avoidance over a finite integer alphabet, record asq-attempt-h6-template-criterion (“The direct h6 template criterion fails by dimension”) documents a concrete method, search boundary, or failed route. The record states: The Rao-Rosenfeld sufficient decision criterion cannot apply to a scalar projection of h6: its expanding eigenspace has dimension 3, while the length-and-sum map has rank at most 2.",
  "relevance_source": "recorded",
  "body": "Rao and Rosenfeld give the h6 incidence eigenvalues as 0 with algebraic multiplicity 3, together with 3, sqrt(3), and -sqrt(3). Thus the expanding eigenspace E_e(M_h6) has dimension 3. A scalar additive-square test uses Phi(letter)=(1,p*x(letter)+q*y(letter)), a linear map into a two-dimensional space. The restriction of Phi to the three-dimensional expanding eigenspace has a nonzero kernel by rank-nullity. Hence E_e(M_h6) intersects ker(Phi) nontrivially for every scalar direction (p,q), so the hypothesis of their finite-ancestor decision theorem fails. This blocks direct use of that sufficient theorem on h6. Other methods may still decide individual projections.",
  "status": "failed",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "family",
    "statement": "every scalar integer projection of the Rao-Rosenfeld h6 fixed point under the cited sufficient decision criterion",
    "family": "scalar integer projections of the h6 Z^2 weights"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://arxiv.org/abs/1511.05875",
      "locator": "Rao and Rosenfeld, arXiv:1511.05875v2, h6 Jordan decomposition in the Results section and the E_e(M_h) intersect ker(Phi) criterion in the Applications section; rank-nullity deduction checked 2026-07-28"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1511.05875",
    "locator": "Rao and Rosenfeld, arXiv:1511.05875v2, h6 Jordan decomposition in the Results section and the E_e(M_h) intersect ker(Phi) criterion in the Applications section; rank-nullity deduction checked 2026-07-28"
  },
  "models": [],
  "relations": [
    {
      "slug": "R31",
      "title": "A six-letter morphic construction works over Z^2",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R26",
      "title": "Screen rational projections of the Z^2 construction",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R24",
      "title": "Search for a low-expansion morphism and certify it",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "additive-square-finite-alphabet",
      "title": "additive square finite alphabet",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}