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Subject and independence

Role. The reviewer is an independent reviewer in a fresh context, commissioned for refutation, who took no part in writing the page under review or its two input pages and received nothing but the assignment. The page's author is identified only by role, as the page author; no contact took place.

Subject. wiki/research/erdos_1221/ko26a_theorem_1_1_reconstruction.md as it stood on 2026-09-28T05:03:27Z (the reconstruction page), read in full from the committed text.

Artifact. The folder-name PDF held by the library card: S. Korsky, An improved lower bound for the de Bruijn--Erdős consecutive gap problem, arXiv:2605.30959v1, eight physical pages; physical and printed page numbers coincide, the title page being printed page 1. Read: p. 2 (Theorem 1.1 and the remarks after it) and p. 7 (Section 5, the proof of the theorem) clause by clause against the page, in the text layer and on the rendered image; pp. 3--6 (Lemma 2.1, the mean identity and (2.1), Lemma 3.1 with (3.1)--(3.4), Section 4 and Proposition 4.1 with its proof) clause by clause for the statements of the imported results and for the structure of their proofs, in the text layer and on the images; p. 1 (the definitions) and p. 8 (Section 6, Conjecture 6.1, the references) in the text layer. Page images: all eight pages rendered at 110 dpi; pages 2--7 read as images, and every displayed formula on those pages checked on the image. The canonical conversion beside the PDF was read in full; it agrees with the PDF on every display used here, and it drops the equation labels, so that it closes the proof of Lemma 3.1 with "This is (3.1)" where the PDF (p. 4) has "This is (3.2)"; the PDF decided.

Allowed material read. The Statement sections of the Lemma 3.1 page and the Proposition 4.1 page in the same state, together with their Definitions and Source notes, which carry the hypotheses the theorem page must meet; the Statement sections of the (5.7) page and the later preprint's Theorem 1.1 page, which the page cites for comparison; the provenance paragraph of the library card and the Statement section of the result page; the statement paragraphs of Problem 1221 (the site wording, the Formulation paragraph and the references); the "Whole-claim report" and "Audit checklist" sections of docs/verification.md (both the shared list and the Erdos-specific list), "Source fidelity" in docs/evidence.md, and docs/math_authoring.md.

Exposures. (1) The two input pages were printed in full, so their Proof sections reached the reviewer; they were read but not audited, and no finding rests on them. (2) The library card was printed in full: its Read-status paragraph ("nothing here is independently reviewed"), its Overview and its Bears-on paragraph reached the reviewer, and so did the result page's Proof pointer, Dependencies and Bears-on sections. (3) The statement region of the Problem 1221 page includes a Status paragraph (the site label, the later preprint's claim and a dated search), which reached the reviewer; it concerns the problem's status, not this page's result, and no finding rests on it. Nothing under any evidence/ folder, no folder index, no Current assessment or Known results text, nothing outside the repository and no web search was read.

Restatement

Fix an integer $r\ge2$. Let $x_1,x_2,\ldots$ be pairwise distinct points of $\mathbb T=\mathbb R/\mathbb Z$ (circumference $1$). For each $n\ge1$ the points $x_1,\ldots,x_n$ cut $\mathbb T$ into $n$ arcs of positive length, the gaps at time $n$, listed in cyclic order; an $r$-block at time $n$ is the union of $r$ cyclically consecutive gaps, one block for each of the $n$ starting positions (for $n<r$ the cyclic sums repeat gaps, which is irrelevant to a limit superior). $M_n^{(r)}$ and $m_n^{(r)}$ are the largest and smallest lengths of an $r$-block at time $n$, both positive, and $R_n=M_n^{(r)}/m_n^{(r)}\ge1$. The claim: for every such $r$ and every such sequence,

$$\limsup_{n\to\infty}R_n\ \ge\ 1+\frac r{r^2-1}.$$

The inequality is not strict; nothing is asserted about the limit inferior; no uniformity in $r$ is asserted; sequences with coincident points are outside the statement. The stated consequences are the arithmetic identity $r/(r^2-1)=1/r+1/(r(r^2-1))$, the value $5/3$ at $r=2$, and, for the infimum $\mu_r$ of $\limsup_nR_n$ over sequences of distinct points, $r(\mu_r-1)\ge1+1/(r^2-1)$.

Checklist

• Quantifiers and scope. Pass. "Every integer $r\ge2$", "every sequence of distinct points" and the limit superior match p. 2; the contradiction hypothesis is the exact negation $\limsup_nR_n<1+r/(r^2-1)$; the eventual bound $R_n\le\rho$ ($n\ge N_1$) is used only at times $\ge N_0\ge N_1$; there is no exceptional set and no boundary case beyond the strictness of $\rho<1+r/(r^2-1)$, which is what the non-strict conclusion needs.
• Circularity. Pass. The target is assumed false and the contradiction comes from Proposition 4.1 and the mean identity, neither of which involves the theorem.
• Model and convention changes. Pass. The objects are the source's gaps, $r$-blocks and epochs; the epoch time $N^+$ is defined as on p. 7; the mean identity is exact, not an averaging heuristic.
• Finite and statistical overreach. Inapplicable: no finite case, sample or heuristic average enters.
• Uniformity. Pass. The theorem is a fixed-$r$ statement and the page says so; the one uniformity the argument needs, that the constant $C_0$ of Proposition 4.1 is the same for every epoch, holds because $C_0=C(r,\eta,\rho)$ does not depend on $N$ (see Premises). The Source note on constant dependence is loose about $C_0$ (F2).
• Extremal conclusions. Pass. The limit superior bound is proved in the ratio's own units by contradiction; no sharpness or attainment is claimed.
• Consequences and composition. Pass. Each "hence" was checked separately (the identity for $r/(r^2-1)$, the value $5/3$, the geometric bound on $N_j$, the final contradiction, and $r(\mu_r-1)\ge1+1/(r^2-1)$); the interface of Proposition 4.1 is supplied at the strength used, and the input pages' standing hypotheses ($\rho>1$, $\eta\in(0,1)$, $\beta\in(0,1)$, $N\ge\max(N_1,2r)$) are all met, $\rho>1$ silently (F3).
• Computation. Inapplicable: the page carries no computation or evidence code.
• Reproduction. Inapplicable: the page states no rerun command or coverage claim.
• Source and verdict fidelity. Pass. The statement, labels and page numbers match the PDF; the characterization of Section 6 and Conjecture 6.1 matches p. 8; the Standing paragraph claims only author-recorded status. One comparison with the 1949 note is not checkable from the page's cited source (F4), and the pointer to the later preprint carries no standing word (F5).

Weakest steps

1. The parameter choice. Given $1<\rho<1+r/(r^2-1)$, the page needs $\eta\in(0,1)$ with $0<\beta=(r-1)(\rho-1+\eta)<r/(r+1)$. Rederivation: $r^2-1=(r-1)(r+1)$, so $\rho-1<r/((r-1)(r+1))$ and $(r-1)(\rho-1)<r/(r+1)$; put $\delta=r/(r+1)-(r-1)(\rho-1)>0$ and take $\eta=\min(1/2,\ \delta/(2(r-1)))$. Then $\beta=(r-1)(\rho-1)+(r-1)\eta\le(r-1)(\rho-1)+\delta/2<r/(r+1)<1$, and $\beta>0$ because $\rho>1$ and $\eta>0$; $\rho>1$ holds because $R_n\ge1$ for every $n$, so $\limsup_nR_n\ge1$ and $\rho$ exceeds it. Composition: $\beta<r/(r+1)$ is the only place where the value $r/(r^2-1)$ enters, and it is exactly what step 3 consumes; at the excluded boundary $\rho=1+r/(r^2-1)$ one would have $(r-1)(\rho-1)=r/(r+1)$ and no $\eta>0$ would do, which is why the theorem gives $\ge$ and not $>$.

2. Geometric growth of the epoch times. Proposition 4.1 gives, for every $N\ge N_0$, $N^+\le(1+1/r)N+C_0$ with one constant $C_0=C(r,\eta,\rho)$. The epoch times exist and increase strictly: for $n\ge N_1$, $M_n\le\rho m_n\le\rho r/n\to0$ while $M_{N_j}\ge r/N_j>0$, so a first time $n>N_j$ with $M_n\le\beta M_{N_j}$ exists, and it is not $N_j$ itself because $\beta<1$. Hence every $N_j\ge N_0$ and the proposition applies at each with the same $C_0$. Put $a_j=N_j+rC_0$; then $a_{j+1}\le(1+1/r)N_j+C_0+rC_0=(1+1/r)(N_j+rC_0)=(1+1/r)a_j$, because $(1+1/r)\,rC_0=rC_0+C_0$. By induction $a_j\le(1+1/r)^ja_0$ with $a_0=N_0+rC_0=C_1$, and $N_j\le a_j$ because $C_0\ge0$ (the Proposition 4.1 page gives $C=(2r+3)B_0+1>0$; any constant may be replaced by its positive part). Composition: this is the source's "for some constant $C_1$" made explicit and feeds $r/N_j\ge(r/C_1)(r/(r+1))^j$.

3. The two decay rates. Upper: $N_j=N_{j-1}^+$ means $M_{N_j}\le\beta M_{N_{j-1}}$ for $j\ge1$, so $M_{N_j}\le\beta^jM_{N_0}$ by induction. Lower: each gap at time $n$ lies in exactly $r$ of the $n$ blocks, so the blocks have total length $r$ and mean $r/n$, whence $M_n\ge r/n$; with step 2, $M_{N_j}\ge C_2(r/(r+1))^j$, $C_2=r/C_1>0$. Dividing $C_2(r/(r+1))^j\le\beta^jM_{N_0}$ by $C_2\beta^j>0$ gives $\bigl(r/((r+1)\beta)\bigr)^j\le M_{N_0}/C_2$ for every $j\ge0$; the base exceeds $1$ by step 1, so the left side is unbounded while the right side is one fixed number, a contradiction as soon as $j>\log(M_{N_0}/C_2)/\log\bigl(r/((r+1)\beta)\bigr)$. Composition: this closes the proof by contradiction and matches the two displays and the sentence "incompatible for sufficiently large $j$" on p. 7.

Strongest attack

The attack aimed at the iteration in step 2: if the constant or the threshold "sufficiently large $N$" of Proposition 4.1 were allowed to depend on the epoch index, the recursion $N_{j+1}\le(1+1/r)N_j+C_0$ would not give a geometric bound, and the whole contradiction would collapse. It failed: the source (p. 5) states one constant $C=C(r,\eta,\rho)$ for all sufficiently large $N$; the Proposition 4.1 page's interface is the same, with $C=(2r+3)B_0+1$, $B_0=\lceil\log\beta/\log(1-\eta)\rceil$, depending on nothing but $r$, $\eta$ and $\rho$, and its threshold is $N\ge\max(N_1,2r)$, fixed once $\rho$, $\eta$ and the sequence are fixed; since $N_0$ is taken above the threshold and $N_j$ increases strictly, every epoch uses the same constant. Secondary attacks also failed: the epoch times cannot stall ($\beta<1$ and $M_{N_j}>0$) or fail to exist ($M_n\to0$ under $R_n\le\rho$); no epoch starts before $N_1$ ($N_0\ge N_1$); a very fast drop within an epoch only strengthens $M_{N_j}\le\beta^jM_{N_0}$; and the boundary $\rho=1+r/(r^2-1)$ is excluded by the strict inequality in the choice of $\rho$, so the argument proves exactly the non-strict bound the source states.

Premises

• Lemma 2.1 (source p. 3): $M_{n+1}\le M_n$. Interface: monotonicity of the largest $r$-span. Not used on the theorem page itself; consumed inside Proposition 4.1. Source held; statement and proof read clause by clause.
• Mean identity (source p. 3, the unnumbered display $m_n\le r/n\le M_n$ and (2.1)): each gap lies in exactly $r$ of the $n$ blocks. Interface used on the page: $M_n\ge r/n$ for every $n$; used through the Proposition 4.1 page: $M_n\le\rho r/n\to0$ when $R_n\le\rho$. Source held; read clause by clause; rederived above.
• Lemma 3.1 (source pp. 3--4, (3.1) and (3.2)): consumed only inside Proposition 4.1. The corpus page adds the hypothesis $n+1\ge2r$ and records it as its own. Source held; statement read clause by clause, proof read for structure.
• Proposition 4.1 (source p. 5, proof pp. 5--6): under the standing assumptions of Section 4 (some $\rho$ with $R_n\le\rho$ for all sufficiently large $n$, some $\eta>0$, $\beta=(r-1)(\rho-1+\eta)$, and $N^+$ the first time after $N$ with $M_{N^+}\le\beta M_N$), there is $C=C(r,\eta,\rho)$ with $N^+\le(1+1/r)N+C$ for all sufficiently large $N$. The corpus page states it with $\rho>1$, $\eta\in(0,1)$, $\beta\in(0,1)$, $R_n\le\rho$ for $n\ge N_1$, and proves it for $N\ge\max(N_1,2r)$ with $C=(2r+3)B_0+1$. The page under review names it as the input, applies it at $N=N_j\ge N_0\ge\max(N_1,2r)$, and meets every hypothesis (with $\rho>1$ unstated, F3). Source held; statement read clause by clause on text layer and image, proof read for structure; the corpus page's own proof was not audited here. Standing as the input page records it: author-recorded reconstruction.
• The result page for Theorem 1.1: its Statement section matches the PDF and the page under review.
• Explicit assumptions. Distinct points, so every gap and every $m_n$ is positive and $R_n$ is defined; $r\ge2$ fixed throughout; the circle has circumference $1$ (the ratio does not depend on it).

Findings

F1. Severity: suggested. Location: "Adding $rC_0$ to both sides", "$C_1=N_0+rC_0$", "$C_2=r/C_1$", "Take $N_0\ge\max(N_1,2r)$". Defect: these are supplied steps and are not labeled as such. Witness: the source (p. 7) writes "It follows that, for some constant $C_1$, $N_j\le C_1(1+1/r)^j$", "for some $C_2>0$" and "Starting from a sufficiently large $N_0$", with no derivation and no explicit constants; the value $2r$ in the threshold is the corpus's Lemma 3.1 requirement, which the Proposition 4.1 page records as absent from the source. Proposed replacement: add a Source-notes bullet, "The source states the geometric bound on $N_j$ 'for some constant $C_1$', the lower bound on $M_{N_j}$ 'for some $C_2>0$' and starts 'from a sufficiently large $N_0$'; the induction with $C_1=N_0+rC_0$, the value $C_2=r/C_1$ and the threshold $N_0\ge\max(N_1,2r)$, whose $2r$ is the requirement of the Lemma 3.1 page, are the corpus's expansions."

F2. Severity: note. Location: Source notes, "The constants $N_0,C_0,C_1,C_2$ depend on $r$, $\rho$, $\eta$ and the sequence". Defect: $C_0$ does not depend on the sequence. Witness: the source (p. 7) and the page's own proof write $C_0=C(r,\eta,\rho)$, and the Proposition 4.1 page gives $C=(2r+3)B_0+1$. Proposed replacement: "$C_0=C(r,\eta,\rho)$ depends only on $r$, $\rho$ and $\eta$; $N_0$ (through $N_1$), $C_1$ and $C_2$ depend also on the sequence."

F3. Severity: note. Location: "Choose $\rho$ with" and "then also $0<\beta<1$, as Proposition 4.1 requires". Defect: the input pages fix $\rho>1$, and $\beta>0$ needs it, but the page does not say why $\rho>1$. Witness: $R_n=M_n/m_n\ge1$ for every $n$, so $\limsup_nR_n\ge1$ and $\rho>\limsup_nR_n\ge1$; the source (p. 7) leaves it implicit as well. Proposed replacement: after the choice of $\rho$, "(so $\rho>1$, since $R_n\ge1$ for every $n$)".

F4. Severity: note. Location: Source notes, "the splitting dynamics is otherwise the same as in the 1949 note". Defect: a comparison with a source the page does not cite in its Source paragraph and that this review's read set does not include; it is unverified here, not contradicted. Witness: the page's Source paragraph names only the 2026 note. Proposed replacement: anchor the clause with a link to the 1949 library card, or drop it.

F5. Severity: note. Location: Reading addressed, "that growth is the ratio part of [the later preprint]". Defect: the sentence points to the later preprint's Theorem 1.1 without a standing word, and can be read as asserting that the growth is established. Witness: the linked page's Statement section states the bound $1+\log r/(100r)$ as that preprint's theorem; the problem page's own description calls it a claim. Proposed replacement: "that growth is the ratio part of the claim in [the later preprint]".

Verdict

Source fidelity: faithful. The statement, its hypotheses, quantifiers, conclusion and consequences match Theorem 1.1 on p. 2, the proof follows Section 5 on p. 7 step for step, and every locator is right.

The argument as reconstructed: sound. Every deduction on the page was rederived above; the imported Proposition 4.1 is applied inside its hypotheses with a constant uniform over the epochs, and the contradiction between the decay rates $\beta^j$ and $(r/(r+1))^j$ is valid.

Limitations: the proofs on the Lemma 3.1 and Proposition 4.1 pages were not audited, only their statements were checked against the held PDF and their hypotheses checked at the point of use; the comparison with the 1949 note (F4) lies outside the read set; the review is noncomputational. Zero required corrections; one suggested labeling (F1) and four notes (F2--F5).

This focused review assigns no tier and changes no status.