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Bailey-Crandall's mutual-nondivisibility question for Stoneham numbers: $\alpha_{6,4}$ is not $6$ -normal, and $\alpha_{b,c}$ is $b$ -normal iff some prime of $c$ misses $b$

Mutual nondivisibility is not enough: $b = 6$ , $c = 4$ is a counterexample, since every prime dividing 4 divides 6, and then the zero digit of $\alpha_{6,4}$ in base 6 has limiting frequency one. The exact criterion is that $\alpha_{b,c}$ is $b$ -normal if and only if some prime dividing $c$ does not divide $b$ . Sufficiency applies Bailey and Crandall's own exponential-sum estimate to the part of $c$ coprime to $b$ ; the repository's prior-art note records that Kano's 1993 theorem on Korobov-type constructions already reaches some non-coprime cases, $\alpha_{4,6}$ among them, so the classification rather than the method is what is new. Reversing the parameters, $\alpha_{4,6}$ is 4-normal, is also formalised.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 12, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-verified. Publication: announcement. 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: Bailey-Crandall's mutual-nondivisibility question for Stoneham numbers: $\alpha_{6,4}$ is not $6$-normal, and $\alpha_{b,c}$ is $b$-normal iff some prime of $c$ misses $b$ · Stoneham prime-support criterion

Confidence: Not scored

Registry verification: lean verified · announcement · resolved

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GPT-6 Astra
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Named evidence

VibeMathed record: Bailey-Crandall's mutual-nondivisibility question for Stoneham numbers: $\alpha_{6,4}$ is not $6$ -normal, and $\alpha_{b,c}$ is $b$ -normal iff some prime of $c$ misses $b$

specialist registry · passed

The posed question was read at source. Bailey and Crandall, Section 7, p. 544: "Instead of coprimality of b, c, just specify that neither integer divides the other. Can a result on b-normality then be effected?" The submission quotes it faithfully. The Lean development was read at commit 5919ad4 on 15 September 2026. `normal_iff` states the criterion for `stonehamConstant b c`, defined as $\sum_{k \ge 0} 1/(c^{k+1} b^{c^{k+1}})$ , the standard series; `counterexample` states $6 \nmid 4$ , $4 \nmid 6$ and non-normality of $\alpha_{6,4}$ in base 6; `NormalInBase` counts every finite word at every prefix length through a radix orbit and is the same definition audited for this author's two-sum entry. The audit file guards the transitive axioms of every headline theorem to exactly propext, Classical.choice and Quot.sound; no sorry, no native_decide. CI rebuilt the project on GitHub twice on 12 September, both green. Not read here: the analytic sufficiency proof line by line. The failing direction is elementary: when every prime of $c$ divides $b$ , $1/c^k$ terminates in base $b$ within $O(k)$ digits and the terms sit at positions $c^k$ , so the zero digit has limiting frequency one.

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Lean 4 development (BaileyCrandall.lean, audit file, verification logs)

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Bailey-Crandall's mutual-nondivisibility question for Stoneham numbers: $\alpha_{6,4}$ is not $6$ -normal, and $\alpha_{b,c}$ is $b$ -normal iff some prime of $c$ misses $b$

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Paper PDF (no byline; cite by title and repository)

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paper link supplied by the VibeMathed registry.

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Bailey and Crandall 2002, Section 7 poses the question (p. 544)

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Artifacts and verifiers

Lean 4 development (BaileyCrandall.lean, audit file, verification logs)

lean artifact · passed

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Lineage and corrections

VibeMathed record: Bailey-Crandall's mutual-nondivisibility question for Stoneham numbers: $\alpha_{6,4}$ is not $6$ -normal, and $\alpha_{b,c}$ is $b$ -normal iff some prime of $c$ misses $b$ evidence for this event

Bailey and Crandall 2002, Section 7 poses the question (p. 544) evidence for this event

Bailey-Crandall's mutual-nondivisibility question for Stoneham numbers: $\alpha_{6,4}$ is not $6$ -normal, and $\alpha_{b,c}$ is $b$ -normal iff some prime of $c$ misses $b$ parent of this event

Paper PDF (no byline; cite by title and repository) evidence for this event

Bailey-Crandall's mutual-nondivisibility question for Stoneham numbers: $\alpha_{6,4}$ is not $6$ -normal, and $\alpha_{b,c}$ is $b$ -normal iff some prime of $c$ misses $b$ evidence for this event

Lean 4 development (BaileyCrandall.lean, audit file, verification logs) evidence for this event

This event attributed to GPT-6 Astra

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