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Subject and independence

The reviewer is an independent reviewer working in a fresh context from the commissioned assignment alone, charged with refutation. The reviewer took no part in writing the page, any page in its folder, or the library card it cites, and had no contact with the page's author. Nothing was read beyond the material listed here.

Frozen subject: wiki/research/erdos_1221/ko26a_lemma_3_1_reconstruction.md as it stood on 2026-09-28T05:03:27Z (the page), read whole from the committed text.

Artifact: the PDF beside the library card Korsky 2026, improved lower bound (S. Korsky, An improved lower bound for the de Bruijn--Erdős consecutive gap problem, arXiv:2605.30959v1, 8 pages; physical page equals printed page). The file in the worktree is byte-identical to the file in the subject state. The text layer of all eight pages was read. Pages 3 and 4, holding Lemma 2.1, the mean identity, Lemma 3.1 and its proof, were read clause by clause against page images rendered at 130 dots per inch, every displayed formula included. Pages 1, 2 and 5 were read as page images for the definitions, the Section 2 notation and the use of the lemma in Section 4; pages 6 to 8 were read in the text layer only, for the use of the lemma's constant in Sections 4 and 5. The canonical conversion beside the PDF was read from Lemma 2.1 to the end of Section 3 and compared with the PDF; the PDF decided.

Allowed material read: the library card's _index.md and its result page theorem_1_1.md (see the exposures); the problem page wiki/problems/analysis/E1221/_index.md, frontmatter and the first part of its untitled statement region (the page has no Statement heading; lines 15 to 60 in that state were read, which cover the site's statement and the start of the Formulation paragraph); docs/verification.md sections "Audit checklist -- the canonical failure modes", "Whole-claim report" and "Audit checklist"; docs/evidence.md section "Source fidelity"; docs/math_authoring.md whole. The two reconstruction pages linked from the page's "Role in the argument" section are consumers, not inputs, and were not read; their existence in that state was confirmed by a tree listing.

Exposures: (1) the card's _index.md and theorem_1_1.md were printed whole, so their read-status, overview, proof-pointer, dependencies and bears-on paragraphs were seen beyond the provenance paragraph and the statement section; they carry read-status sentences about the source's Theorem 1.1 (unrefereed, not independently reviewed) and no text about the page under review; (2) the problem page's frontmatter, including its status field and its description, was printed while the statement region was being located. No Current assessment, Known results, acceptance text, other review, evidence folder, workspace content or web search was read. None of the exposed text bears on the lemma's mathematics or affected a verdict.

Restatement

Setting and conventions. Distinct points $x_1,x_2,\ldots$ lie on the circle of circumference $1$. At time $n$ the first $n$ points cut the circle into $n$ gaps (the arcs between cyclically consecutive points), all of positive length. An integer $r\ge2$ is fixed. For $n\ge r$ an $r$-block at time $n$ is the union of $r$ distinct cyclically consecutive gaps; $M_n$ and $m_n$ are the largest and smallest $r$-block lengths at time $n$, and $R_n=M_n/m_n\ge1$. The step $n\to n+1$ is the insertion of $x_{n+1}$: it splits one gap $\ell$ into two gaps $x,y$ with $x+y=\ell$ and changes no other gap and no cyclic adjacency among the other gaps. Convention, fixed by the page's proof but not written in its Statement: a gap is split at time $T$ when $x_T$ lands in it, that is, in the step $T-1\to T$, so that its two pieces are gaps at time $T$.

Lemma 2.1 as the page states it: for every $n\ge r$, $M_{n+1}\le M_n$.

Mean identity as the page states it: for $n\ge r$, $m_n\le r/n\le M_n$; hence $R_n\le\rho$ gives $M_n\le\rho r/n$, and $M_n\to0$ when $R_n$ is bounded for all large $n$.

Lemma 3.1 as the page states it. Let $\rho>1$, $\eta\in(0,1)$, and let $n$ satisfy $n+1\ge2r$, $M_{n+1}\ge(1-\eta)M_n$ and $R_{n+1}\le\rho$. Let $\ell=x+y$ be the gap split in the step $n\to n+1$, and let $h_1,\ldots,h_{2r}$ be the $2r$ distinct consecutive gaps at time $n+1$ with $h_r=x$, $h_{r+1}=y$, so $r-1$ gaps lie on each side of $\ell$. Put

$$q=\frac{1-\eta}\rho,\qquad \alpha=1-q=\frac{\rho-1+\eta}\rho,\qquad \beta=(r-1)(\rho-1+\eta)=\rho(r-1)\alpha .$$

Then (1) $h_j\le\alpha M_n$ for every $1\le j\le2r$; and (2) for every $T>n+1$: if $x_T$ lands in one of $h_1,\ldots,h_{2r}$, none of $x_{n+2},\ldots,x_{T-1}$ landed in any of them, and $R_T\le\rho$, then $M_T\le\beta M_n$.

Both parts are universal over the sequence, $r$, $\rho$, $\eta$, $n$ and $T$ subject to the hypotheses; $\alpha$ and $\beta$ depend on $r$, $\rho$, $\eta$ only. Departures from the source, all on the safe side: the source fixes $\rho$ in its Section 3 preamble without a range and the page adds $\rho>1$; the source's Lemma 2.1 has no range and the page restricts it to $n\ge r$; the hypothesis $n+1\ge2r$ is supplied by the page and labeled as such.

Checklist

• Quantifiers and scope. Pass, with two labeling corrections (F1, F2) and one convention gap (F3). The lemma is a single-step statement with no limit, so no eventual-versus-all or limit-inferior-versus-superior issue arises. Boundary cases: $r=2$ is handled on the page (the block at time $T$ is the two pieces alone); $n+1\ge2r$ is supplied, labeled, and satisfied where the source uses the lemma (Proposition 4.1 holds "for all sufficiently large $N$", p. 5, and the slow steps considered have $k\ge N$); $T>n+1$ is stated; $\rho=1$ is excluded by the page though not by the source (F1); $n<r$ is excluded by the page's Lemma 2.1 though not by the source's (F2).
• Circularity. Pass. Nothing equivalent to either conclusion is assumed. Part 2 consumes part 1 at time $n+1$ and the ratio hypothesis at time $T$, both hypotheses of the lemma; neither part assumes anything about $M_T$.
• Model and convention changes. Pass, with F3. The page argues on the actual gap configuration. The page's definition of an $r$-block as $r$ distinct gaps differs from the source's cyclic-index formula (p. 1) only for $n<r$, outside the range used (F2). The one convention, which step "split at time $T$" names, is fixed by the page's proof, agrees with the source's proof (p. 4) and with the source's use of the lemma (p. 5), and is load-bearing (see Strongest attack), so it belongs in the Statement (F3).
• Finite and statistical overreach. Inapplicable. No finite case or average stands in for a proof; the mean identity is an exact counting identity used only through the inequality $m_n\le r/n\le M_n$.
• Uniformity. Pass. The constants $\alpha$ and $\beta$ depend only on $r$, $\rho$, $\eta$, as the page states; both bounds hold for every sequence and every $n$, $T$ under the hypotheses; no limit or sum is exchanged.
• Extremal conclusions. Inapplicable in the sharpness sense: the page claims no infimum, supremum or sharpness. The extremal quantities it uses, $M_n$ and $m_n$, are a maximum and a minimum over finitely many $r$-blocks, and $m_T$ is bounded above by the length of one exhibited $r$-block at time $T$, the correct direction.
• Consequences and composition. Pass. Each "so" was rederived: $m_{n+1}\ge qM_n$ from $R_{n+1}\le\rho$ and the slow-step hypothesis; $W_i\ge qM_n$ as $r$-blocks at time $n+1$; $U_i\le M_n$ by merging; the two telescoping identities; the $r$-block at time $T$; $M_T\le\rho m_T$. The "Role in the argument" sentences match the source: slow steps in one epoch number at most $N/r+O(1)$ (p. 6) and the main proof chooses $\beta<r/(r+1)$ (p. 7). The remark $M_n\to0$ under a bounded ratio follows from $M_n\le\rho r/n$. The description's "none can be split again until" is the source's own gloss (p. 4), and its clause "the ratio stays below rho" carries the hypothesis $R_T\le\rho$ that the gloss needs.
• Computation. Inapplicable. The page has no computation and no evidence folder is in the read set.
• Reproduction. Inapplicable. The page states no rerun command and no coverage claim.
• Source and verdict fidelity. Pass, with corrections. The locators were verified against the page images: Lemma 2.1 and the mean identity on p. 3 (Section 2), Lemma 3.1's statement on pp. 3 and 4 and its proof on p. 4 (Section 3), arXiv identifier and version as on the PDF's first page. The Standing paragraph claims author-recorded, unreviewed status only. The Source paragraph's method statement is accurate; the canonical conversion agrees with the PDF on every formula of the lemma and its proof, with one label slip at the end of the proof that the page does not inherit (F4).

Weakest steps

W1, the telescoping bound of part 1. Fix $1\le i\le r+1$. The gaps $h_i,\ldots,h_{i+r-1}$ are $r$ distinct consecutive gaps at time $n+1$ (distinct because the whole run $h_1,\ldots,h_{2r}$ is, as $n+1\ge2r$), so $W_i=h_i+\cdots+h_{i+r-1}$ is an $r$-block at time $n+1$ and

$$W_i\ \ge\ m_{n+1}\ \ge\ \frac{M_{n+1}}\rho\ \ge\ \frac{1-\eta}\rho M_n=qM_n .$$

Fix $1\le i\le r$. The run $h_i,\ldots,h_{i+r}$ has $r+1$ gaps and contains $h_r=x$ and $h_{r+1}=y$ because $i\le r$ and $i+r\ge r+1$. Replacing the adjacent pair $x,y$ by $\ell$ gives $r$ consecutive gaps of time $n$: the other $r-1$ gaps are untouched by the step and $\ell$ sits between them in the same cyclic order. So $U_i=h_i+\cdots+h_{i+r}\le M_n$. For $1\le j\le r$, $W_{j+1}=h_{j+1}+\cdots+h_{j+r}$ is defined ($j+1\le r+1$) and $U_j-W_{j+1}=h_j$, so $h_j\le M_n-qM_n=\alpha M_n$. For $r+1\le j\le2r$, $1\le j-r\le r$, $U_{j-r}=h_{j-r}+\cdots+h_j$ and $W_{j-r}=h_{j-r}+\cdots+h_{j-1}$, so $U_{j-r}-W_{j-r}=h_j\le\alpha M_n$. Composition: this is the only place where the slow-step hypothesis and $R_{n+1}\le\rho$ enter; part 2 consumes only the output $h_j\le\alpha M_n$ and the fact that unsplit gaps keep their lengths.

W2, the block at time $T$ in part 2. By hypothesis $x_{n+2},\ldots,x_{T-1}$ land outside the marked run, so at time $T-1$ the $2r$ marked gaps are present with their time-$(n+1)$ lengths and are still consecutive, since a split outside the run inserts no gap inside it. Then $x_T$ lands in $h_j$, producing pieces $p,p'$ with $p+p'=h_j$, adjacent to each other and, on the two sides, to $h_{j-1}$ and $h_{j+1}$ where these exist. For $j\le r$ take $p,p',h_{j+1},\ldots,h_{j+r-2}$: the indices are at most $2r-2$, so all these gaps are marked, and there are $r-2\ge0$ of them. For $j\ge r+1$ take $h_{j-r+2},\ldots,h_{j-1},p,p'$: the indices are at least $3$. Either choice is $r$ distinct consecutive gaps at time $T$ (there are $T>2r$ gaps), hence an $r$-block at time $T$, and its length is at most

$$h_j+(r-2)\alpha M_n\ \le\ (r-1)\alpha M_n ,$$

so $m_T\le(r-1)\alpha M_n$, and with $R_T\le\rho$, $M_T=R_Tm_T\le\rho(r-1)\alpha M_n=(r-1)(\rho-1+\eta)M_n=\beta M_n$. For $r=2$ the block is $p,p'$ alone and the bound is $h_j\le\alpha M_n$. The case split on $j$ matters: for $j\le r$ the left side holds only $j-1$ marked gaps, which can be fewer than $r-2$, so the neighbors must be taken on the right. Composition: the conclusion needs $R_T\le\rho$ at time $T$ only; no ratio hypothesis between times $n+1$ and $T$ is used, matching the source. The step depends on the time convention, examined under Strongest attack.

W3, the merged case of Lemma 2.1 and its range. Let $n\ge r$ and consider an $r$-block at time $n+1$ ($r$ distinct consecutive gaps). If it contains both $x$ and $y$, they are adjacent inside it; merging them gives $r-1$ consecutive gaps of time $n$, and since $n-(r-1)\ge1$ there is a further gap of time $n$ adjacent to that run; adjoining it gives an $r$-block at time $n$ longer than the new block by that gap's positive length. If it contains $x$ but not $y$, its other $r-1$ gaps are gaps of time $n$ and remain consecutive with $\ell$ in place of $x$ ($y$ lies beyond $x$, outside the block), so replacing $x$ by $\ell=x+y$ gives an $r$-block at time $n$ at least as long; the same with $x$ and $y$ exchanged. Hence every $r$-block at time $n+1$ has length at most $M_n$, so $M_{n+1}\le M_n$. Composition: the page uses the lemma only in the remark $M_n\to0$ and hands it to the downstream pages; the range $n\ge r$ is the page's, not the source's (F2).

Strongest attack

The strongest attack targets the time convention in part 2, because the page's Statement says "at a later time $T$ one of $h_1,\ldots,h_{2r}$ is split" without saying which step that names. Under the reading "the step $T\to T+1$", that is, $x_{T+1}$ lands in $h_j$ while $M_T$, $m_T$, $R_T$ refer to the configuration before the split, the statement is false. Witness: $r=2$, $\rho=6/5$, $\eta=2/5$, so $q=\alpha=1/2$ and $\beta=3/5$. At time $3$ let the gaps be $1/2,1/4,1/4$, so $M_3=3/4$ and $m_3=1/2$. Let the step $3\to4$ split the gap $1/2$ into $1/4+1/4$; then all four gaps at time $4$ equal $1/4$, $M_4=1/2\ge(3/5)(3/4)=9/20$ (a slow step), $R_4=1\le\rho$, and $n+1=4=2r$, so $h_1,\ldots,h_4$ are the four gaps, each $1/4\le\alpha M_3=3/8$ (part 1 holds). Let $x_5$ land anywhere; every gap is marked. Under the alternative reading $T=4$: $R_4=1\le\rho$ but $M_4=1/2>\beta M_3=9/20$, so the conclusion fails. Under the page's reading $T=5$: splitting a gap $1/4$ into $s$ and $1/4-s$ leaves a $2$-block of length $1/4$ beside $2$-blocks of length $1/2$, so $R_5\ge2>\rho$, the hypothesis $R_T\le\rho$ fails and the lemma asserts nothing, which is consistent. The page's proof ("after the split of $h_j$ there is an $r$-block at time $T$ consisting of the two pieces") fixes the reading under which the statement is proved; the source's proof (p. 4) uses the same reading, and the source's use of the lemma (p. 5: no marked gap can be split at a later time $T<N^+$, since otherwise $M_T\le\beta M_k$) is consistent with it. The attack therefore does not refute the page; it shows that the convention is load-bearing and belongs in the Statement (F3).

Secondary attacks, all failed. Making the exhibited block at time $T$ fail to be an $r$-block: impossible, since the two pieces are adjacent, the $r-2$ neighbors are consecutive marked gaps on one side, and the gap count at time $T$ exceeds $r$. Breaking part 1 at the ends of the run, $j=1$ or $j=2r$: the telescoping uses $U_1,W_2$ and $U_r,W_r$, whose indices stay in range. Dropping $n+1\ge2r$: for $n+1<2r$ the "$2r$ consecutive gaps" repeat gaps and the lemma is not even well posed; the page excludes this by a labeled supplied hypothesis that the source's Section 4 satisfies. Weakening the ratio hypothesis to a time before $T$: the argument uses $R_T$ only at time $T$, and the witness above shows that $R_{T-1}\le\rho$ would not do. A randomized simulation of the splitting process (random sequences, $r\in\{2,3,4\}$, several values of $\rho$ and $\eta$) found no violation of either part under the page's convention; it is a heuristic probe, carries no evidential weight, and is not retained.

Premises

• Lemma 2.1 (source p. 3, Section 2). Interface: $M_{n+1}\le M_n$. Source held; read clause by clause against the page image. The page names it as the source's and proves it for $n\ge r$; its standing on the page is the page's own, author-recorded.
• Mean identity and (2.1) (source p. 3, between Lemma 2.1 and Section 3). Interface: $m_n\le r/n\le M_n$, and $M_n\le\rho r/n$ when $R_n\le\rho$. Source held; read clause by clause. The page proves it by the counting argument, each gap lying in exactly $r$ of the $n$ blocks.
• Lemma 3.1 (source pp. 3 and 4, Section 3). The page's subject; source held; statement and proof read clause by clause against the page images.
• No other theorem is imported, no native claim is consumed, and there is no batch acceptance order.
• Explicit assumptions: the points are distinct, so every gap is positive and each insertion splits exactly one gap; $r\ge2$ is fixed; $\eta\in(0,1)$; $\rho>1$ on the page, unrestricted in the source, with $\rho\ge1$ forced by $R_{n+1}\le\rho$; $n+1\ge2r$, supplied by the page and labeled; the time convention of F3.
• Standing: the page's Standing paragraph records an author-recorded, unreviewed reconstruction; the source is an unrefereed arXiv preprint (v1, 29 May 2026), as the card's provenance paragraph records.

Findings

F1. Severity: suggested. Location: Statement, "Fix $\rho>1$ and $\eta\in(0,1)$". Defect: the hypothesis $\rho>1$ is not in the source and is not labeled as supplied. Witness: Lemma 3.1 (p. 3) fixes only $\eta\in(0,1)$; $\rho$ enters from the Section 3 preamble on p. 3, "assume that, from some point onward, $R_n\le\rho$", with no range. The addition is harmless: $R_{n+1}\le\rho$ and $R_{n+1}\ge1$ force $\rho\ge1$, and the proof never uses $\rho>1$ (at $\rho=1$ one has $q=1-\eta$, $\alpha=\eta$, $\beta=(r-1)\eta$ and the same argument goes through). Proposed replacement: "Fix a real $\rho$ and $\eta\in(0,1)$ (the source fixes $\rho$ in the preamble of its Section 3 without a range; the hypothesis $R_{n+1}\le\rho$ below forces $\rho\ge1$, and the argument uses nothing more)".

F2. Severity: suggested. Location: Preliminaries, "Lemma 2.1 (p. 3). $M_{n+1}\le M_n$ for every $n\ge r$." Defect: the range $n\ge r$ is the page's, unlabeled. Witness: the source's Lemma 2.1 (p. 3) reads "The sequence $M_n$ is nonincreasing" with no range, and the source's formula for $M_n^{(r)}$ (p. 1, cyclic indices) defines it for every $n\ge1$, counting gaps with multiplicity when $n<r$, whereas the page's definition of an $r$-block as a union of $r$ consecutive gaps leaves $M_n$ undefined for $n<r$. Harmless: only $n\ge2r-1$ is used, and the mean identity carries the same implicit range (its $n$ blocks are the $n$ starting positions). Proposed replacement: append to the lemma's statement "(The source states the lemma for the whole sequence, with $M_n$ defined for every $n$ by the cyclic-index formula; under the definitions above an $r$-block needs $n\ge r$ gaps, and only $n\ge2r-1$ is used below. The same range is understood in the mean identity.)"

F3. Severity: suggested. Location: Statement, part 2, "If at a later time $T>n+1$ one of $h_1,\ldots,h_{2r}$ is split". Defect: the Statement does not say which step "split at time $T$" names, and the two readings give different statements; the proof uses the step $T-1\to T$ ($x_T$ lands in $h_j$, so the two pieces are gaps at time $T$ and $m_T$, $R_T$, $M_T$ refer to that configuration), and under the other reading the statement is false. Witness: the configuration under Strongest attack ($r=2$, $\rho=6/5$, $\eta=2/5$, gaps $1/2,1/4,1/4$ at time $3$, the long gap halved at the step $3\to4$, then any insertion): $R_4=1\le\rho$ but $M_4=1/2>\beta M_3=9/20$. The source (pp. 3 and 4) is equally implicit and its proof uses the page's reading, so this is a precision gap of the reconstruction, not a fidelity error. Proposed replacement: after "is split" insert ", that is, the point $x_T$ lands in it, so that its two pieces are gaps at time $T$,".

F4. Severity: note. Location: Source paragraph, "read in the canonical conversion beside the PDF and checked against the text layer". Defect: none on the page. Witness, for the record: the canonical conversion ends the proof of Lemma 3.1 with "This is (3.1)" where the PDF (p. 4) reads "This is (3.2)", and the conversion carries no equation numbers, so (2.1) and (3.1) to (3.4) cannot be located from it. The page cites no equation labels, so nothing on it is affected. No replacement text; a later edit that adds equation labels should take them from the PDF.

Verdict

Source fidelity: faithful with corrections. The hypotheses, conclusions, constants and locators of Lemma 2.1, the mean identity and Lemma 3.1 match the artifact at pp. 3 and 4; the corrections are two unlabeled narrowings (F1, F2) and one unstated convention (F3), all suggested, and zero required.

The argument as reconstructed: sound. Every deduction was rederived above; the supplied hypothesis $n+1\ge2r$ is labeled and is exactly what the distinctness used in both parts needs; the case $r=2$ is handled; the time convention under which part 2 is proved is the source's.

Limitations: the review covers this page and the source's own use of the lemma. The downstream reconstruction pages were not read, so the page's sentence that $n+1\ge2r$ "holds at every time considered in the later sections" was checked against the source's Sections 4 and 5 only. The page carries no computation, and the randomized probe mentioned above carries no weight. This focused review assigns no tier and changes no status.