{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:mahler-s-z-number-problem-in-its-generalized-form",
  "title": "Mahler's $Z$-number problem in its generalized form",
  "canonical_statement": "Let $p>q>1$ be coprime integers and let $Z{p/q}(s,s+t)$ be the set of $\\xi>0$ whose fractional parts $\\{\\xi(p/q)^{n}\\}$ all lie in $[s,s+t)$. The question for which parameters $Z$ is empty remains open.",
  "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "partial",
  "solution_events": [
    {
      "id": "vibemathed:mahler-s-z-number-problem-in-its-generalized-form:event",
      "problem_id": "vibemathed:mahler-s-z-number-problem-in-its-generalized-form",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-09-12T00:00:00.000Z",
      "title": "ResearchGate preprint",
      "summary": "Theorem E gives $Z_{3/2}(\\tfrac27,\\tfrac57)=\\emptyset$: a window of length $\\tfrac37=0.428571\\ldots$ at base $3/2$, beating the $31/81$ of [Dub19, Thm. 1.2], and with no assumption on the arithmetic nature of $\\xi$, where that result needs algebraicity. Mahler's problem itself is untouched: for which parameters $Z$ is empty remains open, and this is a record window rather than a classification.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:mahler-s-z-number-problem-in-its-generalized-form:attempt"
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        "vibemathed:mahler-s-z-number-problem-in-its-generalized-form:method"
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      ]
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      "problem_id": "vibemathed:mahler-s-z-number-problem-in-its-generalized-form",
      "model": "Fable 5, Opus 5",
      "provider": "Anthropic",
      "attempted_at": "2026-09-12T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "AI discovered theorems, formalized them and wrote a draft of the paper from the Lean.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Registry link",
          "url": "https://palomar-registry.org/entry?id=PALOMAR-2026-09-12-000003",
          "kind": "palomar",
          "license": null
        },
        {
          "label": "Bulk d/l",
          "url": "https://github.com/rwst/Confinement-Certificates/blob/main/paper2.pdf",
          "kind": "paper",
          "license": null
        },
        {
          "label": "Challenge2.lean: the trusted statement of record, Mathlib only",
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          "kind": "lean-statement",
          "license": null
        },
        {
          "label": "PRIOR_ART.md: the literature search, with novelty recorded as unknown",
          "url": "https://github.com/rwst/Confinement-Certificates/blob/main/PRIOR_ART.md",
          "kind": "other",
          "license": null
        }
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      "sources": [
        {
          "label": "ResearchGate preprint",
          "url": "https://www.researchgate.net/publication/414097146_Confinement_schemas_and_the_reach_of_block_certificates_for_powers_of_rational_numbers_modulo_one",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Mahler's $Z$-number problem in its generalized form",
          "url": "https://vibemathed.com/problem/mahler-s-z-number-problem-in-its-generalized-form",
          "kind": "source-assertion",
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        }
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      "cost_usd": null,
      "independence": "unknown"
    }
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      "id": "vibemathed:mahler-s-z-number-problem-in-its-generalized-form:verification",
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      "solution_event_id": "vibemathed:mahler-s-z-number-problem-in-its-generalized-form:event",
      "level": "lean_checked_statement_unaudited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Filed at Lean-checked, as submitted. Challenge2.lean is a trusted statement of record that imports nothing but Mathlib, redeclares verbatim every definition occurring in the certified theorems, and states the paper's lettered results with sorry proofs; comparator2.json names fourteen of them and, as configured, checks constant identity across the challenge and solution environments, restricts axioms to propext, Quot.sound and Classical.choice, and requires the Lean kernel to re-accept the solution from a fresh export with no olean loaded. That configuration reads correctly. It is not lifted to Lean-verified because no CI run was found on the repository and the comparator run was not reproduced here, so the machine check rests on the author's report - which is what Lean-checked means. PRIOR_ART.md is unusually candid and worth reading: it records that the search was targeted rather than exhaustive, that the whole of Section 3 was found in print with Theorems 3.5 and 3.6 a machine-checked fragment of [Bug04, Thm. 1], and concludes \"Novelty is therefore recorded as unknown\". Not peer reviewed and no independent expert endorsement.",
      "sources": [
        {
          "label": "VibeMathed: Mahler's $Z$-number problem in its generalized form",
          "url": "https://vibemathed.com/problem/mahler-s-z-number-problem-in-its-generalized-form",
          "kind": "source-assertion",
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          "kind": "primary-mathematical-source",
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      "label": "ResearchGate preprint",
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      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Mahler's $Z$-number problem in its generalized form",
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      "kind": "source-assertion",
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  "opportunity_signals": [
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      "id": "vibemathed:mahler-s-z-number-problem-in-its-generalized-form:signal:recent_partial_result",
      "problem_id": "vibemathed:mahler-s-z-number-problem-in-its-generalized-form",
      "kind": "recent_partial_result",
      "reason": "A source-reported partial result may support bounded expansion.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}