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Subject and independence

Role: independent reviewer in a fresh context, commissioned with the charge of refutation and given only the assignment. The reviewer took no part in writing the page, the pages it cites, or the library card, and had no earlier contact with any of them.

Subject: wiki/research/erdos_1221/ko26a_proposition_4_1_reconstruction.md as it stood on 2026-09-28T05:03:27Z, read whole from the committed text.

Artifact: the folder-name PDF beside the library card (arXiv:2605.30959v1; physical page numbers equal the printed ones). Physical pages read: pp. 5--6 (Section 4, Proposition 4.1 and its proof) in full, clause by clause, in the text layer and as page images; pp. 3--4 (Lemma 2.1, the mean identity with (2.1), Lemma 3.1 with (3.1)--(3.2)) in full for the imported results, text layer and images; p. 7 (Section 5) in the text layer for the use of the proposition only; the remaining pages in the text layer for orientation. Page images: all eight pages rendered at 130 dpi, pages 3--6 read as images, and every displayed formula on pp. 3--6 checked against the image. The canonical conversion beside the PDF was compared with the PDF over Section 4; they agree, and the PDF decided.

Allowed material actually read: the page, whole; the Lemma 3.1 reconstruction page in the same state, its Definitions, Preliminaries and Statement; the library card's provenance paragraph; the Statement section of wiki/problems/analysis/E1221/_index.md; docs/verification.md "Whole-claim report" and both "Audit checklist" sections; docs/evidence.md "Source fidelity"; docs/math_authoring.md, whole. Not read: the folder _index.md, any evidence/ content, the card's result page (not linked from the page's Source paragraph), the other reconstruction pages, other reviews, anything under the private working files, the web.

Exposures:

1. E1221.md: the lines after the Statement section printed the "Status" paragraph and the start of the "Source" paragraph, which are excluded status text. They concern the problem's site label and a later preprint, not the source of the page, and played no part in this review.
2. The library card _index.md printed whole, so its "Read status", "Overview" and "Bears on" paragraphs were seen besides the provenance paragraph; the Overview summarizes the source's argument in one paragraph, including one sentence on Proposition 4.1. No verdict on the page appears there.
3. The Lemma 3.1 reconstruction page printed whole, so its Standing paragraph and its Proof section were seen; the review used its Definitions, Preliminaries and Statement and did not audit its proof.
4. A working-tree listing showed the name of an untracked evidence directory of another problem folder; it was not opened.

Restatement

Setting (the page's Definitions and the Lemma 3.1 page): $x_1,x_2,\ldots$ are distinct points of $\mathbb T=\mathbb R/\mathbb Z$; after $n$ insertions the circle is cut into $n$ gaps of positive length in cyclic order; inserting $x_{n+1}$ splits exactly one gap into two and leaves every other gap, and the cyclic adjacency of the unsplit gaps, unchanged. For a fixed integer $r\ge2$, $M_n$ and $m_n$ are the largest and smallest sums of $r$ cyclically consecutive gaps at time $n$, and $R_n=M_n/m_n$.

Parameters: $\rho>1$, $\eta\in(0,1)$, and $\beta=(r-1)(\rho-1+\eta)$ with $0<\beta<1$. Hypothesis: there is $N_1$ with $R_n\le\rho$ for every $n\ge N_1$. A step $n\to n+1$ is slow when $M_{n+1}\ge(1-\eta)M_n$ and fast otherwise. For $N\ge N_1$,

$$N^+=\min\{\,n>N:\ M_n\le\beta M_N\,\},$$

which exists because $M_n\le\rho r/n\to0$ for $n\ge N_1$ while $\beta M_N>0$.

Claim: there is a constant $C$ depending only on $r$, $\eta$ and $\rho$, not on the sequence, on $N_1$ or on $N$, such that for every $N\ge\max(N_1,2r)$,

$$N^+\ \le\ \Bigl(1+\frac1r\Bigr)N+C ;$$

the page's proof gives $C=(2r+3)B_0+1$ with $B_0=\lceil\log\beta/\log(1-\eta)\rceil$.

Conventions used by the proof: the steps of the epoch are $k\to k+1$ for $N\le k\le N^+-1$; the terminal step is $N^+-1\to N^+$; "before the terminal step" means $N\le k\le N^+-2$. The $N$ gaps at time $N$ are the initial gaps; every gap at a time in $[N,N^+]$ is a piece of exactly one initial gap, an unsplit gap being its own piece, and the descendants of an initial gap are its pieces. Cyclic distance between initial gaps is measured along the $N$-cycle of initial gaps. Lemma 3.1's "split at a later time $T$" places $T$ after the split, so it names the step $T-1\to T$.

Checklist

• Quantifiers and scope: pass. "Sufficiently large $N$" is $N\ge\max(N_1,2r)$, the hypothesis is eventual ($n\ge N_1$) exactly as in the source, and $C$ is uniform over sequences. The boundary case $N^+=N+1$, with no step before the terminal step, is covered, but only through the supplied hypothesis $\beta<1$; see F1.
• Circularity: pass. The frozen-block step uses the minimality of $N^+$ and Lemma 3.1, never the proposition; the count is a direct bound.
• Model and convention changes: pass with notes. The descendant and cyclic-distance conventions are the source's; "split at time $T$" is used in two ways on the page, each correct in place (F4); the definition of an active gap needs its time range made explicit (F2), and the descendant relation needs the unsplit gap included from time $N$ on (F3). No relaxed or transformed system is substituted for the gap process.
• Finite and statistical overreach: inapplicable; no finite check or heuristic average is used anywhere on the page.
• Uniformity: pass. $B_0=\lceil\log\beta/\log(1-\eta)\rceil$ and $C=(2r+3)B_0+1$ depend on $r,\eta,\rho$ only, the dependence the source claims.
• Extremal conclusions: pass. The only extremal object is the first-passage time $N^+$, whose existence is proved from $M_n\to0$; the bound $N/r$ for an $r$-separated subset of an $N$-cycle is proved in the claim's own units.
• Consequences and composition: pass. Every "so", "hence" and "in particular" was re-derived (Weakest steps); Lemma 2.1, the mean identity and Lemma 3.1 are consumed at the strength stated on the Lemma 3.1 page, which matches the source (Premises).
• Computation: inapplicable; the page and this review are noncomputational.
• Reproduction: inapplicable; the page states no rerun command or coverage claim.
• Source and verdict fidelity: pass with one note. The statement matches Proposition 4.1 on p. 5 clause by clause; the two quoted phrases match p. 6; the count "two sentences" is inaccurate (F5); the Standing sentence claims author-recorded status and nothing more.

Weakest steps

1. Frozen blocks (first proof paragraph). Let $k\to k+1$ be slow with $N\le k\le N^+-2$ and let $B$ be its block of $2r$ gaps at time $k+1$; Lemma 3.1 applies, since $R_{k+1}\le\rho$ ($k+1>N_1$) and $k+1\ge2r+1$. Suppose a gap of $B$ is split at some step $T-1\to T$ with $T\le N^+-1$, and take the first such $T$. Then $T\ge k+2$, no gap of $B$ was split before, so all $2r$ are present and adjacent just before the split, and $R_T\le\rho$; Lemma 3.1(2) gives $M_T\le\beta M_k$. Lemma 2.1 gives $M_k\le M_N$, and $\beta>0$, so $M_T\le\beta M_N$ with $N<T<N^+$, against the choice of $N^+$ as the first time after $N$ with $M\le\beta M_N$. Hence no gap of $B$ is split by a step before the terminal step, and all $2r$ gaps of $B$ are present at time $N^+-1$. Composition: this is the only mechanism that makes a block inert; it feeds the active-gap count (a protected gap is never split before the terminal step) and the arc argument (the descendants of $J'$ cannot be split).

2. The active-gap count (the corpus's expansion). For $N\le k\le N^+-1$ let $a_k$ be the number of gaps at time $k$ that descend from a bad initial gap and lie in no block marked at an epoch slow step $k'\to k'+1$ with $N\le k'$ and $k'+1\le k$. Then $a_N\le(2r+1)B_0$, since $|F|\le b\le B_0$ and each gap of $F$ has at most $2r+1$ initial gaps within cyclic distance $r$. A gap never turns from inactive to active: its ancestor is fixed, and a marked block keeps its gaps through time $N^+-1$ (step 1). At a step $k\to k+1$ before the terminal step that splits a gap $g$: if $g$ is inactive, then either $g$ descends from a good gap, and so do its pieces, or $g$ lies in a marked block, which step 1 forbids; so $a_{k+1}\le a_k$. If $g$ is active and the step is fast, $g$ is replaced by two pieces, so $a_{k+1}\le a_k+1$. If $g$ is active and the step is slow, both pieces lie in the block marked at this step and $g$ disappears, so $a_{k+1}\le a_k-1$. Summing over the $N^+-1-N$ steps before the terminal step, with $s$ the number of slow steps among them that split an active gap,

$$0\ \le\ a_{N^+-1}\ \le\ a_N+b-s\ \le\ (2r+1)B_0+B_0-s ,$$

so $s\le(2r+2)B_0$. A slow step before the terminal step that splits a descendant of a bad gap splits an active gap, because the alternative, a descendant lying in a marked block, is never split before the terminal step. Composition: contributes the $(2r+2)B_0$ term of $C$ and needs only step 1 and the fast-step bound.

3. The arc argument. Let two slow steps before the terminal step be attached to good initial gaps $I,J$ with cyclic distance $d\le r-1$, $I=J$ allowed. The shorter arc $\mathcal A$ from $I$ to $J$ has $d+1\le r$ gaps, each within distance $d\le r-1$ of $I$; since $I$ is good, no gap of $\mathcal A$ is in $F$, so no descendant of a gap of $\mathcal A$ is split at a fast step before the terminal step. Let $k\to k+1$ be the first slow step before the terminal step that splits a descendant of a gap of $\mathcal A$; both chosen steps qualify, so it exists, and at least one chosen step, at a step $T\to T+1$ with $k<T\le N^+-2$, comes later; let $J'\in\{I,J\}$ be its attached gap. Before step $k\to k+1$ no descendant of a gap of $\mathcal A$ was split, so at time $k$ the gaps of $\mathcal A$ are their own only descendants, present and still adjacent to each other, and the split gap $G$ is one of them. Every other gap of $\mathcal A$ is within $r-1$ places of $G$, hence among the $r-1$ gaps to the left or to the right of $G$; so at time $k+1$ the gap $J'$, or both pieces of $G$ when $J'=G$, lies in the block of that step. By step 1 that block is unsplit through time $N^+-1$, so the descendants of $J'$ at any time in $[k+1,N^+-1]$ are $J'$ itself or the pieces of $G$, and the split of one of them at the step $T\to T+1$, with $T+1\le N^+-1$, contradicts step 1. So the attached gaps are pairwise distinct with mutual cyclic distance at least $r$; sorting the $m$ of them cyclically, the $m$ runs between consecutive ones sum to $N$ and are each at least $r$, so $m\le N/r$ (for $m=1$ this reads $r\le N$). Composition: gives the $N/r$ term; with steps 1 and 2 and the fast-step bound $b\le B_0$, from $(1-\eta)^bM_N\ge M_{N^+-1}>\beta M_N$, the epoch has $N^+-N\le1+B_0+(2r+2)B_0+N/r$ steps.

Strongest attack

The attack aimed at the corpus's own expansion, the active-gap count, through the phrase "a protected block marked at a slow step before time $k$". If that phrase admits slow steps before time $N$, the count breaks: a block marked by a slow step before the epoch is not covered by the frozen-block paragraph, which treats only $N\le k\le N^+-2$, so a descendant of a bad gap sitting in such a block would be inactive and could still be split before the terminal step, by a fast step (two active pieces appear, a rise of two, against "rises by at most one") or by a slow step (a slow step attached to a bad gap that the count never sees, so $(2r+2)B_0$ would not follow). The attack fails against the page as written: "protected block" is introduced on the page only in the first proof paragraph, for slow steps with $N\le k\le N^+-2$, and the count's sentence "a protected gap is not split before the terminal step at all" is the conclusion of that paragraph, so the page's usage is the epoch-restricted one, under which every inequality of step 2 above holds. The residue is the wording finding F2.

Two further attacks failed outright. The boundary $N^+=N+1$: the page's "by the minimality of $N^+$, $M_{N^+-1}>\beta M_N$" then reads $M_N>\beta M_N$, which minimality does not give; it holds because the page assumes $\beta<1$ (F1), and then $b=0$ and the total is $1\le N/r+C$. The arc argument with a bad gap strictly inside $\mathcal A$, or with the first $\mathcal A$-step attached to a third gap of $\mathcal A$: neither matters, since the argument uses only $\mathcal A\cap F=\emptyset$, which the goodness of $I$ alone gives, and the freezing of the block marked at the first $\mathcal A$-step, which contains every descendant of $\mathcal A$ present at time $k+1$.

Premises

• Lemma 3.1 (source pp. 3--4, statement and (3.1)--(3.2) read in the text layer and the image; corpus statement on the Lemma 3.1 page). Interface as consumed: for a step $n\to n+1$ with $\eta\in(0,1)$, $n+1\ge2r$, $M_{n+1}\ge(1-\eta)M_n$ and $R_{n+1}\le\rho$, the $2r$ consecutive gaps at time $n+1$ centered on the two pieces satisfy $h_j\le\alpha M_n$, and if the first split of one of them happens at a time $T$ (after the split) with $R_T\le\rho$, then $M_T\le\beta M_n$ with $\beta=(r-1)(\rho-1+\eta)$. The corpus statement matches the source's, with $n+1\ge2r$ supplied and labeled there. Source held; the page names it as imported; its proof was not audited here.
• Lemma 2.1 (source p. 3, read whole): $M_{n+1}\le M_n$. The corpus states it for $n\ge r$, which covers every use, since $n\ge N\ge2r$. Source held.
• Mean identity and (2.1) (source p. 3): $m_n\le r/n\le M_n$, hence $M_n\le\rho r/n$ when $R_n\le\rho$; used only for the existence of $N^+$. Source held.
• Elementary facts used without citation: an insertion elsewhere leaves unsplit gaps and their mutual adjacency unchanged (the Lemma 3.1 page's Definitions); an $r$-separated subset of an $N$-cycle has at most $N/r$ elements; a gap of $F$ has at most $2r+1$ initial gaps within cyclic distance $r$.
• Explicit assumptions of the page: distinct points; $r\ge2$; $\rho>1$; $\eta\in(0,1)$; $0<\beta<1$; $R_n\le\rho$ for all $n\ge N_1$; $N\ge\max(N_1,2r)$. Of these, $\eta<1$, $\beta<1$ and $N\ge2r$ are supplied relative to the source's Section 4 setup; only $N\ge2r$ is labeled (F1).
• Standing of consumed material: the page presents Lemma 2.1 and Lemma 3.1 as imported from the held source through the corpus reconstruction, author-recorded; no tier is claimed for either. No batch acceptance order applies.

Findings

F1. Severity: suggested. Location: "Fix $\rho>1$ and $\eta\in(0,1)$ with" and "$N^+>N$ because $\beta<1$" (Definitions); "By the minimality of $N^+$, $M_{N^+-1}>\beta M_N$" (Fast steps). Defect: $\eta<1$ and $\beta<1$ are supplied hypotheses that are not labeled as such. The source's Section 4 (p. 5) says "Choose $\eta>0$ and define $\beta=(r-1)(\rho-1+\eta)$" and imposes (4.1) only "eventually"; $\eta<1$ is Lemma 3.1's hypothesis (p. 3, "Fix $\eta\in(0,1)$"). The page cites $\beta<1$ where it is not needed, since $N^+>N$ holds by the definition "first time after $N$", and not where it is needed: when $N^+=N+1$, the minimality of $N^+$ says nothing about $M_{N^+-1}=M_N$, and $M_N>\beta M_N$ is exactly $\beta<1$. The source's own line "Before time $N^+$, we have $M_n>\beta M_N$" (p. 5) has the same boundary gap at $n=N$; for $\beta\ge1$ the proposition is trivial, since then $N^+=N+1$. Witness: p. 5, the Section 4 preamble and the first line of the proof; p. 3, the first words of Lemma 3.1. Replacement: in Definitions write "and $N^+>N$ by definition"; in Fast steps write "By the minimality of $N^+$ when $N^+-1>N$, and by $\beta<1$ when $N^+-1=N$, $M_{N^+-1}>\beta M_N$"; add a source note: "The source's Section 4 takes any $\eta>0$ and states $\beta<r/(r+1)$ only as the eventual choice; $\eta<1$ is Lemma 3.1's hypothesis, and $\beta<1$ is what $M_{N^+-1}>\beta M_N$ needs when $N^+=N+1$. Both are supplied here; for $\beta\ge1$ the proposition is trivial, since $N^+=N+1$."

F2. Severity: suggested. Location: "do not lie in a protected block marked at a slow step before time $k$". Defect: the range of slow steps is not stated. Read as including slow steps before time $N$ (the Lemma 3.1 page calls the block of any slow split a marked block), the sentences "a protected gap is not split before the terminal step at all" and "the count rises by at most one" do not follow, since the first proof paragraph freezes only blocks marked at steps $k\to k+1$ with $N\le k\le N^+-2$. Under the epoch-restricted reading every step of the count holds (Weakest steps, 2). Witness: the page's first proof paragraph, "Let $k\to k+1$ be a slow step with $N\le k\le N^+-2$"; the source (p. 6) counts "During the epoch" and refers to "the protected block created by that slow split". Replacement: "do not lie in a block marked at a slow step $k'\to k'+1$ with $N\le k'$ and $k'+1\le k$".

F3. Severity: suggested. Location: "the descendants of an initial gap are its pieces at later times". Defect: at time $N$ an initial gap is not a piece "at a later time", so by the letter the gap split at the step $N\to N+1$ descends from no initial gap; the set $F$, the active set at time $N$, the attachment of that step, and the three-way split of the steps in "Total" all need the unsplit initial gap to count as its own descendant from time $N$ on, which is how every later sentence reads the word. Witness: the page's Definitions; the source (p. 6) uses the same loose phrase "initial gaps whose descendants are split". Replacement: "the descendants of an initial gap at a time $k$ with $N\le k\le N^+$ are its pieces present at time $k$, the gap itself while it is unsplit".

F4. Severity: note. Location: "split at a time $T\le N^+-1$" (first proof paragraph) against "the gap split at time $k$" and "split as a descendant at a time $T$ with $k<T\le N^+-2$" (arc paragraph). Defect: two conventions for "split at time $T$" on one page. The first follows Lemma 3.1, where $T$ is the time after the split (the step $T-1\to T$), and its bound $T\le N^+-1$ is right; the arc paragraph uses the time before the split (the step $T\to T+1$), and its bound $T\le N^+-2$ is right. A reader who carries one convention into the other paragraph either admits the terminal step into the frozen window or drops the last step before it from the arc argument. Witness: source p. 4, "at a later time $T$ ... $M_T\le\beta M_n$", and p. 6, "after the split at time $k$", which mix the same two usages. Replacement: state in Definitions "a gap present at time $k$ is split at the step $k\to k+1$; Lemma 3.1's time $T$ is the time after the split, so a split at time $T$ in its sense is the step $T-1\to T$", and write the arc paragraph's later step as "at a step $T\to T+1$ with $k<T\le N^+-2$".

F5. Severity: note. Location: "is stated in two sentences" (Standing) and "in two sentences" (Source notes). Defect: the source's accounting of the slow splits attached to bad gaps is one paragraph of five sentences (p. 6, from "The slow steps associated to bad initial gaps contribute only $O_{r,\eta,\rho}(1)$." to "Thus the number of slow splits associated to bad initial gaps is $O_r(B_0)+B_0=O_{r,\eta,\rho}(1)$."). The quoted formula is accurate; the sentence count is not. Replacement: "in one short paragraph".

Verdict

Source fidelity: faithful. The statement on the page is Proposition 4.1 of p. 5 clause by clause, at the stated pages and label, with the supplied hypotheses $N\ge2r$ (labeled) and $\eta<1$, $\beta<1$ (unlabeled, F1) narrowing nothing the source uses; the proof follows the source's proof on pp. 5--6 step by step, and its one expansion, the active-gap count, is labeled as the corpus's and yields the explicit constant $C=(2r+3)B_0+1$.

The argument as reconstructed: sound. Each deduction was re-derived above; the three suggested findings are wording precision in the page's own definitions, and none changes a bound.

Limitations: a noncomputational review; Lemma 3.1 was checked as a statement against the source and not re-proved; Section 5's use of the proposition was not examined; the source was read only at the pages named. No required corrections. This focused review assigns no tier and changes no status.