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Mahler's $Z$ -number problem in its generalized form

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.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 12, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: preprint. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Mahler's $Z$-number problem in its generalized form · Z(p,q)-numbers

Confidence: Not scored

Registry verification: lean checked · preprint · partial

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VibeMathed
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Fable 5
model · ai model contributor · Anthropic

Opus 5
model · ai model contributor · Anthropic

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Named evidence

VibeMathed record: Mahler's $Z$ -number problem in its generalized form

specialist registry · passed

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.

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Mahler's $Z$ -number problem in its generalized form

preprint · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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witness · pending

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paper · pending

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