{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R11",
  "content_hash": null,
  "slug": "ac0123-claim-only-four-letter-class-open",
  "type": "claim",
  "title": "The four-term progression is the remaining four-letter equivalence class",
  "summary": "Lietard and Rosenfeld settle every size-four complex alphabet outside the additive-equivalence class of {0,1,2,3}; Andrade and Mol still list this class as open in February 2025.",
  "relevance": "For Additive-cube avoidance on the alphabet zero through three, record ac0123-claim-only-four-letter-class-open (“The four-term progression is the remaining four-letter equivalence class”) records a bound, answer, status fact, or structural consequence. The record states: Lietard and Rosenfeld settle every size-four complex alphabet outside the additive-equivalence class of {0,1,2,3}; Andrade and Mol still list this class as open in February 2025.",
  "relevance_source": "recorded",
  "body": "Lietard and Rosenfeld call two same-size numerical alphabets equivalent when a letter map preserves and reflects equal-length, equal-sum equivalence for every pair of finite words. Their Lemma 1 shows that every affine map \\(x\\mapsto ax+b\\), with \\(a\\ne0\\), preserves this equivalence. Their Main Theorem and Corollary 1 give an infinite additive-cube-free word over every complex alphabet of size four outside the class of \\(\\{0,1,2,3\\}\\). In the integer setting, the exceptional class consists of four-term arithmetic progressions under an affine image.\n\nTheir closing Question 1 leaves \\(\\{0,1,2,3\\}\\) open and reports an exhaustive failure for morphisms whose four images each have length at most seven. Andrade and Mol state in their February 2025 introduction that it is still unknown whether an infinite additive-cube-free word exists over \\(\\{0,1,2,3\\}\\). This later explicit status is the latest direct statement located in the audit.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "family",
    "statement": "the sourced additive-cube avoidability classification of numerical alphabets of cardinality four",
    "family": "finite subsets of the complex numbers having cardinality four"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1007/978-3-030-48516-0_15",
      "locator": "Florian Lietard and Matthieu Rosenfeld, Avoidability of Additive Cubes over Alphabets of Four Numbers, DLT 2020, Lemma 1, Main Theorem, Corollary 1, and Question 1; Jonathan Andrade and Lucas Mol, arXiv:2408.15390v2, Introduction, PDF page 2"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1007/978-3-030-48516-0_15",
    "locator": "Florian Lietard and Matthieu Rosenfeld, Avoidability of Additive Cubes over Alphabets of Four Numbers, DLT 2020, Lemma 1, Main Theorem, Corollary 1, and Question 1; Jonathan Andrade and Lucas Mol, arXiv:2408.15390v2, Introduction, PDF page 2"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R7",
      "title": "A 2026 source and live-state audit leaves the infinite case open",
      "object_type": "attempt",
      "relation": "reports",
      "direction": "incoming"
    },
    {
      "slug": "R6",
      "title": "Search beyond image length seven with a decision certificate",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "additive-cube-four-term-progression-alphabet",
      "title": "additive cube four term progression alphabet",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}