Fable 5, Opus 5
AI discovered theorems, formalized them and wrote a draft of the paper from the Lean.
Provider: Anthropic · Prompt public: unknown · Independence: unknown
Problem detail · source-aware
Mahler's $Z$-number problem in its generalized form
VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Precise 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.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.
What AI did
Verification boundary
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 Not peer reviewed and no independent expert endorsement.
Correctness: supported · statement fidelity: unaudited · peer review: none
Timeline
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ResearchGate preprint
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.
Known method families
computation (source-reported)
Source-reported tools: computation.
Independent: unknown · difference confidence: 0
What remains uncertain
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.
- 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.