{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R31",
  "content_hash": null,
  "slug": "asq-claim-h6-vector-construction",
  "type": "claim",
  "title": "A six-letter morphic construction works over Z^2",
  "summary": "Rao and Rosenfeld prove that the fixed point of h6 has no adjacent equal-length blocks with equal vector sums under six explicit weights in Z^2.",
  "relevance": "For Infinite additive-square avoidance over a finite integer alphabet, record asq-claim-h6-vector-construction (“A six-letter morphic construction works over Z^2”) records a bound, answer, status fact, or structural consequence. The record states: Rao and Rosenfeld prove that the fixed point of h6 has no adjacent equal-length blocks with equal vector sums under six explicit weights in Z^2.",
  "relevance_source": "recorded",
  "body": "Let h6 map a->ace, b->adf, c->bdf, d->bdc, e->afe, and f->bce. Give a,b,c,d,e,f the respective Z^2 weights (0,0),(1,1),(2,1),(0,1),(2,0),(1,0). Rao and Rosenfeld use the augmented map with leading coordinate 1 to enforce equal block lengths, and prove that h6^omega(a) contains no additive square for these vector weights. This theorem is over Z^2 and supplies candidate one-dimensional projections.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "family",
    "statement": "the fixed point h6^omega(a) under the six displayed two-dimensional integer weights",
    "family": "Rao-Rosenfeld h6 fixed point with Z^2 weights"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1511.05875",
      "locator": "Michael Rao and Matthieu Rosenfeld, Avoiding two consecutive blocks of same size and same sum over Z^2, arXiv:1511.05875v2, Results section, displayed h6 morphism and subsection Additive-square-free words on Z^2"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1511.05875",
    "locator": "Michael Rao and Matthieu Rosenfeld, Avoiding two consecutive blocks of same size and same sum over Z^2, arXiv:1511.05875v2, Results section, displayed h6 morphism and subsection Additive-square-free words on Z^2"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R27",
      "title": "Audit sources, variants, duplicates, and target integrity",
      "object_type": "attempt",
      "relation": "reports",
      "direction": "incoming"
    },
    {
      "slug": "R26",
      "title": "Screen rational projections of the Z^2 construction",
      "object_type": "attempt",
      "relation": "informs",
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    },
    {
      "slug": "R23",
      "title": "The direct h6 template criterion fails by dimension",
      "object_type": "attempt",
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    {
      "slug": "additive-square-finite-alphabet",
      "title": "additive square finite alphabet",
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}