{
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  "ref": "R1338",
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  "slug": "lloyd-polynomial-integral-root-classification-status-20260801",
  "type": "claim",
  "title": "Current checked status and unresolved remainder",
  "summary": "UNKNOWN as of 2026-07-27. The MathOverflow page has zero answers. Literature gives strong nonexistence conditions and classifications tied to perfect codes, but the dated search did not locate a classification of every parameter triple having the stated integral-root property.",
  "relevance": "Records the strongest checked neighboring results and the exact remainder future work must settle.",
  "relevance_source": "recorded",
  "body": "A dated independent review on 2026-08-01 checked the structured sources below, the complete visible source discussion, exact-title and equivalent-formulation searches, and the current TheoremDB corpus. On 2026-07-27 the Stack Exchange API reported zero answers, no accepted answer, and no closure for MathOverflow question 134715; the page contains no comments proposing a later resolution. Krasikov and Litsyn's 1996 work gives nonexistence conditions and upper bounds for integral zeros of binary Krawtchouk polynomials, a closely related partial result rather than the requested all-parameter classification. Shi, Krotov, and Solé, arXiv:2601.12818, restate Lloyd's integral-root condition as a central necessary tool for perfect-code classification; their broader association-scheme work does not classify all triples in this statement. A TheoremDB search for Lloyd polynomials, integral Krawtchouk zeros, and q-ary perfect-code root conditions found no duplicate.\n\nA complete resolution must satisfy: Give a necessary-and-sufficient classification of all triples (q,n,t), with a proof that every listed polynomial has the required roots and every omitted triple fails at least one requirement. A computational component must use exact rational polynomial arithmetic and provide certificates for factorization, root integrality, distinctness, and the finite parameter ranges it eliminates.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://mathoverflow.net/questions/134715/when-does-the-lloyd-polynomial-have-only-integral-roots",
      "locator": "Dataset references and independent 2026-08-01 status search."
    },
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  "formal_statement": null,
  "source": {
    "url": "https://mathoverflow.net/questions/134715/when-does-the-lloyd-polynomial-have-only-integral-roots",
    "locator": "Dataset references and independent 2026-08-01 status search."
  },
  "relations": [
    {
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      "title": "Complete the stated acceptance conditions",
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