{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R6",
  "content_hash": null,
  "slug": "ac0123-attempt-affine-morphism-certificate-search",
  "type": "attempt",
  "title": "Search beyond image length seven with a decision certificate",
  "summary": "Start with 23,298,600 incidence-valid image tuples in the smallest prefiltered affine-matrix orbit, use finite prefixes only as screening, and submit every survivor to the additive-power template decision algorithm.",
  "relevance": "For Additive-cube avoidance on the alphabet zero through three, record ac0123-attempt-affine-morphism-certificate-search (“Search beyond image length seven with a decision certificate”) documents a concrete method, search boundary, or failed route. The record states: Start with 23,298,600 incidence-valid image tuples in the smallest prefiltered affine-matrix orbit, use finite prefixes only as screening, and submit every survivor to the additive-power template decision algorithm.",
  "relevance_source": "recorded",
  "body": "Lietard and Rosenfeld report that every morphism over \\(\\{0,1,2,3\\}\\) whose four images have length at most seven fails to provide an additive-cube-free infinite fixed point. The next experiment should begin outside that published negative range.\n\nThe included exact prefilters cover affine image lengths 8 through 12. They leave 1,130 complement-conjugacy orbits of expanding length-sum matrices. Requiring each of the four image pools to contain an additive-cube-free word reduces the all-four-letters tranche to 588 orbits. Requiring a self-starting image whose incidence-reachable component contains all four letters gives an exact second ranking. The smallest orbit representative has matrix \\((8,1,2,2)\\), image lengths \\((8,9,10,11)\\), image sums \\((2,4,6,8)\\), and 23,298,600 retained tuples.\n\nEnumerate that representative first, then continue in the recorded all-four-reachability order. A one-million-symbol scan is an early rejection screen. For each surviving morphism, compute the short-factor bound required by Theorem 2.4 and record an exhaustive initial check through that bound. The million-symbol scan can serve as this initial check only when it reaches the computed bound. Next enumerate the finite template-ancestor set, then run the theorem's final template check. Preserve the first cube for every rejection. Fixed points omitting a letter form a separate tranche whose ternary subalphabet status should be audited first.\n\nThe template decision algorithm is due to Currie, Mol, Rampersad, and Shallit and implemented by Andrade and Mol. Theorem 2.4 makes it decisive for morphisms satisfying the stated affine and expanding hypotheses. Use a 16 GiB memory limit and a 12-hour wall limit per candidate. Preserve parameter regions, source digests, short-factor bounds, initial-check results, ancestor sets, final checks, first-cube witnesses, and resource exits so later searches can skip completed work.",
  "status": "next_experiment",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "a proposed certificate-oriented search over affine morphisms on {0,1,2,3} outside the published image-length-at-most-seven range",
    "bounds": {
      "image_length": {
        "min": 8,
        "max": 12
      },
      "alphabet_size": {
        "min": 4,
        "max": 4
      },
      "finite_screen_length": {
        "min": 1,
        "max": 1000000
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://arxiv.org/abs/2408.15390",
      "locator": "Andrade and Mol, arXiv:2408.15390v2, Section 2.4 and Theorem 2.4; Lietard and Rosenfeld, DLT 2020, Question 1; implementation commit 3b40fb14bd64f2d18455c024c4393b9e7142beca"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2408.15390",
    "locator": "Andrade and Mol, arXiv:2408.15390v2, Section 2.4 and Theorem 2.4; Lietard and Rosenfeld, DLT 2020, Question 1; implementation commit 3b40fb14bd64f2d18455c024c4393b9e7142beca"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R8",
      "title": "Direct ascending search stalls near the thesis hard boundary",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R11",
      "title": "The four-term progression is the remaining four-letter equivalence class",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R10",
      "title": "The reconstructed method produces a checked million-letter word",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R1",
      "title": "Exact affine-matrix prefilter for image lengths 8 through 12",
      "object_type": "artifact",
      "relation": "uses",
      "direction": "outgoing"
    },
    {
      "slug": "R2",
      "title": "Exact additive-cube-free image pools through length 12",
      "object_type": "artifact",
      "relation": "uses",
      "direction": "outgoing"
    },
    {
      "slug": "additive-cube-four-term-progression-alphabet",
      "title": "additive cube four term progression alphabet",
      "object_type": "problem",
      "relation": "recorded_for",
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    }
  ]
}