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Claim. S. Ringer, Local gap statistics, telescoping, and normality: a local-pattern approach to Erdős problem 251, manuscript dated 11 September 2026, 31 pages with four appendices, in the author's GitHub repository StefanRinger/erdos-251 (the preprint link, pinned to the commit of 2026-09-13 that holds the final text). Its Corollary 1.2 states that, under the positive-comparison hypothesis (19) of Theorem 1.1 with parameter $\kappa\ge1/\log B$, which the averaged one-sided Hardy–Littlewood condition $(\mathrm{AHL}_\kappa)$ implies and Kuperberg's Conjecture 1.3 implies in turn (Section 5.4), the series $\sum_{n\ge1}c_np_nB^{-n}$ is normal to base $B$ for every integer $B\ge2$ and every nonzero rational periodic sequence $c_n$; at $B=2$ and $c_n\equiv1$ this says that

$$S=\sum_{n\ge1}\frac{p_n}{2^n}$$

is normal to base $2$, hence irrational, which answers Problem 251 with more than it asks. The corollary follows by a finite Abel transformation from Theorem 1.1, which classifies geometrically weighted periodic polynomial series in consecutive prime gaps, of normal-form degree at most $\kappa\log B$, as rational exactly when a cyclic normal form vanishes and normal otherwise. The source card ringer_2026_local_gap_statistics_telescoping_normality holds the digest, and its result pages Corollary 1.2 and Theorem 1.1 give the statements as the paper prints them; this outline is a reading aid, not proof coverage.

Submission note. Posted to erdosproblems.com as a proof claim by Stefan Ringer (account stefanringer) on 13 September 2026, giving "GPT 6 Astra, Fable 5.1" as the AI used:

Conditional claim: Under Kuperberg's uniform Hardy–Littlewood conjecture, for each fixed integer $B\ge2$, $\sum_{n\ge1}p_nB^{-n}$ is normal to base $B$, with orbit star discrepancy $D_N^*\ll_B(\log\log N)^{-1/2}$. Periodic rational local gap-polynomial series are rational exactly when they telescope, and normal otherwise. A computable normal form determines all rational linear relations; independent classes have jointly equidistributed orbits. Unconditional claim: The classification holds for suitably rough integers. Also, $\sum_{n\ge1}p_nB^{-S_n}$ is normal for $S_n=\sum_{j=1}^n\lceil\log_B\log(j+3)\rceil$. Proof: The algebra identifies telescopes as precisely those polynomials invisible to every one-point variation. Otherwise, resampling one point in the sieve model yields absolute continuity. A one-sided comparison transfers this to actual orbit limits; the carry recurrence makes them invariant, hence Lebesgue. No local equidistribution is needed. Notes: I believe there is much more to explore here. Unfortunately, I won't be able to invest much more time into this problem at the moment. While I tried to highlight connections to existing literature, possible transfers, and the high level idea of the proof, I think it would still benefit from a proper digestion. If someone wants to build on this or turn this into a formal paper, I'd be open to a collaboration.

Hypothesis. The claim is conditional on Kuperberg's Conjecture 1.3 (conjecture_1_3), a Hardy–Littlewood prime-tuples conjecture with one power-saving error term uniform over admissible sets of at most $(\log\log x)^3$ shifts in $[0,(\log x)^2]$, or on the weaker positive-comparison condition (19) the paper states. Neither is proved. Acceptance of the claim would establish an implication, not the irrationality, and would leave the problem open. The manuscript's unconditional results, a classification for the gaps of suitably rough integers and the normality of a series with a stretched clock $B^{-S_n}$ in place of $B^{-n}$, do not cover the problem's series, as the paper itself states.

Postings. The author announced the result on the problem's discussion thread on 2026-09-07; the repository's first commit is dated 2026-09-10 and the card digests the text at the commit of 2026-09-13; the author registered the result on 2026-09-13 as the problem's single proof claim on the site's proof-claims page, labeled partial, with the AI systems GPT 6 Astra and Fable 5.1 named in the claim's title and an attached summary stating the conditional normality, the discrepancy bound and the unconditional companions; the claim's notes say the work would benefit from a proper digestion. The partial label matches the paper's own scope, an implication from an unproved conjecture, and the repository README says the original problem remains open unconditionally. The paper's provenance paragraph says that GPT 6 Astra led the mathematical development and Fable 5.1 acted as a sparring partner. The repository's lean folder (the formalization link, at the same pinned commit) holds the author's Lean 4 package, whose audited theorem types carry the conjecture as an explicit premise; the author reports a local build and axiom audit and disclaims a clean independent reproduction.

Standing. Claimed. The manuscript is not refereed and not on arXiv, no outside reader is recorded as having reviewed it, the site's curator has not acknowledged it beyond listing the claim under the site's standard disclaimer, and the card digests its statements, not its proof; the author's Lean package is not part of this repository's audited Lean, so it gives no formalized evidence, and no Lean statement of the repository has been compared with the manuscript. Land claims the same implication by a different argument; the manuscript names that draft as an independent earlier conditional proof and states that no implication between the two specialized hypotheses is claimed.

Depends on. Nothing in this wiki.