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Claim. For a sequence of distinct points on the circle of circumference $1$, write $M_n^{(r)}$ and $m_n^{(r)}$ for the largest and smallest sums of $r$ consecutive gaps cut by the first $n$ points, so that the mean such sum is $r/n$. Theorem 1.1 of S. Korsky, A resolution of the de Bruijn–Erdős consecutive-gap problem, arXiv:2609.07196 (v2, 9 September 2026), recorded on its library card with the theorem's page, asserts absolute constants $c>0$ and $r_0$ such that for every $r\ge r_0$ and every such sequence

$$\limsup_{n\to\infty}\bigl(nM_n^{(r)}-r\bigr)\ge c\sqrt{\log r},\qquad \limsup_{n\to\infty}\bigl(r-nm_n^{(r)}\bigr)\ge c\sqrt{\log r},\qquad \limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}}\ge1+\frac{\log r}{100\,r}.$$

Taking the infimum or supremum over sequences of distinct points, and writing $\Lambda_r^{\mathrm{dist}}$, $\lambda_r^{\mathrm{dist}}$ and $\mu_r^{\mathrm{dist}}$ for the constants over that family, these say that $\Lambda_r^{\mathrm{dist}}-r$, $r-\lambda_r^{\mathrm{dist}}$ and $r(\mu_r^{\mathrm{dist}}-1)$ all tend to infinity. Together with the upper bound $\mu_r\le1+c'\log r/r$ of Clément and Steinerberger, proved by sequences of distinct points and so valid for $\mu_r^{\mathrm{dist}}$ as well as $\mu_r$, the third bound fixes the order of $\mu_r^{\mathrm{dist}}-1$ at $\log r/r$. These are the three parts of the corrected Statement of Problem 1221, restricted to sequences of distinct points. The problem's constants $\Lambda_r$, $\lambda_r$ and $\mu_r$ are taken over all sequences, and since $\Lambda_r\le\Lambda_r^{\mathrm{dist}}$, $\lambda_r\ge\lambda_r^{\mathrm{dist}}$ and $\mu_r\le\mu_r^{\mathrm{dist}}$, both conclusions transfer to them only if the constants over the two families agree, as the Scope paragraph below records. The author also notes that each one-sided bound alone implies that the ratio excess is unbounded, the weaker form of the third part.

Submission note. Posted to erdosproblems.com as a proof claim by Samuel Korsky (account SamKorsky) on 8 September 2026, giving "GPT Astra" as the AI used:

We prove all three de Bruijn--Erdős conjectures. Writing $M_n^{(r)}$ and $m_n^{(r)}$ for the largest and smallest $r$-spans after $n$ points, we obtain

$$\limsup_n(nM_n^{(r)}-r),\ \limsup_n(r-nm_n^{(r)})\gg\sqrt{\log r}, > \qquad \limsup_n\frac{M_n^{(r)}}{m_n^{(r)}}\ge 1+\frac{\log r}{100r}.$$

The

proof compares forward and backward walks by $kr$ places at nearby insertion times, converting span bounds into estimates for point counts in short intervals. Larcher’s finite-prefix discrepancy bound yields the ratio result. For the one-sided bounds, the fact that the average $r$-span is exactly $r/n$ converts one-sided span control into $L^1$ control; averaging the same comparison and applying Halász’s planar $L^1$ discrepancy theorem then forces the stated $\sqrt{\log r}$ growth. Note that the one-sided bounds already implies the third conjecture, but the stronger estimate for the ratio here is actually sharp asymptotically (matching upper bounds of Steinerberger and Clement). Notes: I plan to revise the document to make it more human-readable; the only reason I posted now in its half-baked form is because I have an arxiv pre-print proving only the third limit coming out today or tomorrow night. The full resolution follows from similar ideas (Astra was able to derive the one-sided inequalities quickly after I fed in my existing argument for the third limit). Thus I expect the problem to otherwise fall to Erdős hunters shortly, hence the rush.

Method, as the paper describes it. Counts of points in short arcs at nearby insertion times are compared by walking $kr$ places forward and backward along the cyclic order, which turns control of the $r$-spans into control of those counts. For the ratio bound the comparison is closed by a finite-prefix discrepancy inequality extracted from Larcher's proof of Schmidt's theorem. For the one-sided bounds the identity that the average $r$-span equals $r/n$ converts one-sided span control into an $L^1$ bound, and Halász's planar $L^1$ discrepancy theorem then forces the $\sqrt{\log r}$ growth. Version 1 of the preprint (7 September 2026) carried the ratio bound only; version 2 added the two one-sided bounds. The problem page's research folder holds author-recorded reconstructions of the argument; they are reading aids with no standing of their own and are not acceptance evidence.

Scope. The theorem is stated for sequences of distinct points, while the site's wording and the 1949 source allow coincident points; whether the constants over the two families agree is not settled in the sources, as the problem page's Formulation records. The claim is registered on the site's proof-claim tab as a full proof of the problem (submitted 2026-09-08, with the author's note that the document was posted ahead of the arXiv version). The theorem covers sequences of distinct points only, a narrower family than the Statement's, and this page records it at that scope.

Covers. The three parts of the corrected Statement for sequences of distinct points: $\Lambda_r^{\mathrm{dist}}-r$, $r-\lambda_r^{\mathrm{dist}}$ and $r(\mu_r^{\mathrm{dist}}-1)$ tend to infinity. Not covered: the constants over all sequences, coincident points allowed, which the Statement takes.

Claimant and assistance. The claimant is the author, who submitted the claim on the site, where the proof-claim tab registers it as made using GPT Astra. The paper's acknowledgments (p. 15) state that GPT Astra was used for the literature search that identified Larcher's quantitative discrepancy bound and to complete the argument from the author's ideas, to develop and audit the $L^1$ transport and localization argument, and to review the proof and revise the exposition, and that the author independently checked the arguments and calculations. The claim's notes on the tab credit the two one-sided inequalities to the system. Of the two comments under the entry (10 September 2026), one concerns the division of credit, and the claimant's reply says that with GPT Astra they found the right application of the known discrepancy results, and that the system was solely responsible for the two one-sided bounds, building on their own work on the ratio bound.

Acceptance. None found. The preprint is unrefereed, with no journal reference on arXiv and no published version in Crossref; the site keeps the problem OPEN with the claim pending; the community database marks the entry ambiguous and open; no independent review by a named mathematician exists in the sources searched. The two comments under the site's proof-claim entry (10 September 2026) concern the speed of posting and the attribution of the AI system's role, not the mathematics, and an automated review posted on a public review site carries no human sign-off and counts for nothing here. The page is dated by the site submission of 2026-09-08, the first posting that claimed all three parts.

Depends on. Korsky, Theorem 1.1. The preprint's own argument rests on nothing else in this wiki: the two results it imports, Larcher's discrepancy bound and Halász's planar $L^1$ theorem, are literature it cites.