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Subject and independence

The reviewer worked in a fresh context from the commissioned assignment alone, took no part in writing the page or any page of its folder, and had read none of them before this review. The charge was refutation.

Frozen subject: wiki/research/erdos_1221/ko26b_proposition_3_1_reconstruction.md as it stood on 2026-09-28T05:03:27Z, read in full from the committed text.

Artifact: the retained PDF of arXiv:2609.07196v2 (16 pages) under the card Korsky 2026, resolution. Physical pages 6 and 7 (Section 3: the statement and proof of Proposition 3.1, displays (3.1) to (3.4)) were read in full, from the text layer and from page images rendered at 130 dpi; every displayed formula on those pages was read from the images. Physical pages 4 and 5 (Section 2: display (2.1), the definitions of $U_t$ and $V_t$, and the statement and proof of Lemma 2.1) were read the same way, because the page imports Lemma 2.1 and its proof. Section 3 of the canonical conversion beside the PDF was read and compared with the PDF; the two agree, and the PDF decided.

Allowed material actually read: the page in full; the Lemma 2.1 reconstruction page in the same state, its Definitions and Statement sections and its proofs of (2.2) and (2.3), the last two because the page's final comparison cites those proofs "before the supremum"; the retained-artifact paragraph of the library card; the Statement of the problem page wiki/problems/analysis/E1221/_index.md; docs/verification.md "Whole-claim report" and "Audit checklist" (the shared list and the repository-specific list); docs/evidence.md "Source fidelity"; and docs/math_authoring.md in full.

Exposures: the library card's index page was printed whole, so its read-status paragraph (which ends in a standing sentence), its overview and its relation section (which holds an acceptance sentence) were seen; none of it was used, and every verdict below rests on the PDF, the page and the Lemma 2.1 page. The printed slice of the problem page ran past the Statement into the Formulation paragraph, which discusses the literal wording; not used. A directory listing showed the file names of the other review reports in this folder; none was opened. No web search was made, and nothing among the private working files or outside the repository was read.

Restatement

Setting, from the source's Section 2 and the Lemma 2.1 page. The points $x_1,x_2,\ldots$ are distinct points of $\mathbb T=\mathbb R/\mathbb Z$ and $r\in\mathbb N$ is fixed. For real $t\ge1$, $P_t=\{x_1,\ldots,x_{\lfloor t\rfloor}\}$, and $N_t(I)$ counts the points of $P_t$ in the oriented half-open arc $I$. The $r$-spans $S_i(t)$ are the clockwise distances from a point of $P_t$ to the point $r$ places after it in the cyclic order of $P_t$. For $D>0$, $U_t(D)=D^{-1}\sup_xN_t((x,x+D/t])$ and $V_t(D)=D^{-1}\inf_xN_t((x,x+D/t])$. Hypothesis (2.1): a number $A\ge1$ is fixed, and for every sufficiently large $t$ there are $a_t,b_t\ge0$ with $a_t+b_t\le A$ such that every $r$-span of $P_t$ lies in $[(r-a_t)/t,(r+b_t)/t]$. Convention: every time considered is large enough that every interval used has length below $1$ and that $|P_t|$ exceeds the number of places moved.

Claim. There are absolute constants $C_0,C_1>0$ such that: if (2.1) holds for all sufficiently large $t$ with $A\ge1$ and $r\ge C_0A$, and $\Lambda=\log(r/A)$, $S=\sqrt{Ar}/\Lambda^2$, then there is a time threshold, which may depend on $r$, $A$ and the sequence but not on $x$ or $D$, beyond which for every $x\in\mathbb T$ and every real $D$ with $0\le D\le S$

$$\bigl|N_t((x,x+D/t])-D\bigr|\ \le\ 3A+\frac{C_1A}\Lambda .$$

The bound holds for the double supremum over $x$ and $D$ at once, and the implied constants of the proof's $O(\cdot)$ terms are absolute.

Checklist

• Quantifiers and scope. Pass. The page keeps "for all sufficiently large $t$" as one threshold, states that it may depend on $r$, $A$ and the sequence and not on $x$ or $D$, and justifies this (finitely many predetermined comparison times; the last comparison at $(1\pm\theta)t$; Lemma 2.1's uniformity clause on the bounded range $(0,S]$). The supremum runs over real $0\le D\le S$; $D=0$ is the empty arc; the lower branch of the final comparison is split at the sign of the positive part, and both branches are carried out. No "almost all" is upgraded and no exceptional set is dropped.
• Circularity. Pass. The conclusion (3.1) is never used. The inputs are (2.1), the two counting arguments behind (3.2), and Lemma 2.1, whose statement does not involve (3.1).
• Model and convention changes. Pass for the statement and the proof: oriented half-open arcs, lifts to $\mathbb R$, real times, the nested sets $P_t$, the choice $k=\lceil D/K\rceil$ and the doubling chains are the source's own, and the page's $U_t$, $V_t$ are the source's. One summary-level substitution is flagged: the frontmatter desc restates (2.1) as a symmetric bound, which is a weaker hypothesis (F1).
• Finite and statistical overreach. Inapplicable. No finite check or heuristic average stands in for a proof; the averaging over $u$ inside Lemma 2.1 is an exact identity of integrals and lives on the Lemma 2.1 page.
• Uniformity. Pass. After $C_0$ is fixed every constant is absolute: the numbers of comparisons obey $h_U,h_V\le(2+1/(2\log2))\Lambda$; the $O(\theta\Lambda)$ constants rest on $\theta\Lambda\le\log(144)/12<0.42$ and $\theta\Lambda^2\le\log^2(144)/12<2.1$ on $r/A\ge144$ (both functions of $u=r/A$ decrease beyond $e^2$ and $e^4$); the threshold's dependence is stated as in the source. See the derivations below.
• Extremal conclusions. Pass. $U_t(D)$ and $V_t(D)$ are a supremum and an infimum over $x$ of integer counts bounded by $|P_t|$, so they are finite and attained. No sharpness or attainment is claimed for (3.1).
• Consequences and composition. Pass for the proof. The chain (3.2) to (3.3) to (3.4) to (3.1) was rederived step by step below, and every consumed clause of Lemma 2.1 is supplied at its stated strength ($q<1$ at every use). The two sentences of "Role in the argument" (Section 5 supplies (2.1) with $A=(\log r)/100$; Lemma 4.2 consumes (3.1)) lie outside the pages this review read and are not checked here.
• Computation. Inapplicable. The page carries no computation and no evidence folder is in scope.
• Reproduction. Inapplicable. No rerun command or coverage claim appears on the page.
• Source and verdict fidelity. Pass. The Statement matches p. 6 clause by clause: the constants $C_0,C_1>0$, the hypotheses "(2.1) for all sufficiently large $t$", $A\ge1$, $r\ge C_0A$, the definitions of $\Lambda$ and $S$, the two suprema, the bound $3A+C_1A/\Lambda$, and the sentence on the threshold's dependence. "Absolute" is the section preamble's own qualification ("All constants in this section are absolute", p. 6). The locators (Section 3, Proposition 3.1, pp. 6--7; Lemma 2.1, p. 4; labels (2.1) to (2.3) and (3.1) to (3.4)) are correct. The Standing paragraph claims author-recorded only and names the source as an unrefereed preprint, which the card's provenance paragraph confirms.

Weakest steps

1. The terminal bounds (3.2). The source gives two one-line sketches (p. 6) and the page expands them. Rederivation. Take $t$ large enough that (2.1) holds at $t$, $|P_t|\ge r+1$, and $(r+A)/t<1$. Let $I=(x,x+(r-A)/t]$ and suppose $N_t(I)\ge r+1$. List the points of $P_t$ in $I$ by their lifts, $p_1<p_2<\cdots$. Every point of $P_t$ on the arc $[p_1,p_{r+1}]$ lies in $I$ and so is one of $p_1,\ldots,p_{r+1}$; hence $p_{r+1}$ is the point $r$ places after $p_1$ in the cyclic order of $P_t$, and the $r$-span at $p_1$ is $p_{r+1}-p_1<|I|=(r-A)/t\le(r-a_t)/t$, using $a_t\le a_t+b_t\le A$. This contradicts the lower bound in (2.1), so $N_t(I)\le r$ for every $x$ and $U_t(r-A)\le r/(r-A)$. Next let $I=(x,x+(r+A)/t]$, let $p_0$ be the largest lift of a point of $P_t$ with $p_0\le x$, and let $p_1<\cdots<p_r$ be the lifts of the next $r$ points clockwise. Then $p_1>x$ by the choice of $p_0$, and $p_r-p_0$ is the $r$-span at $p_0$, so $p_r\le p_0+(r+b_t)/t\le x+(r+A)/t$; the $r$ distinct points $p_1,\ldots,p_r$ lie in $I$, so $N_t(I)\ge r$ and $V_t(r+A)\ge r/(r+A)$. Composition: (3.2) anchors both chains at their terminal scales and terminal times, and nothing else is known about counts before the iteration starts.

2. From the explicit iteration bounds to (3.3). Chains: with $T$ the terminal scale ($r-A$ or $r+A$), put $D_i=2^iK$ for $i<h$ and $D_h=T$, where $h$ is the least integer with $2^hK\ge T$; then $D_{h-1}<T\le 2D_{h-1}$, so $K\le D_i<D_{i+1}\le2D_i$ at every step, and $2^{h-1}K<T$ gives $h<\log_2(T/K)+1\le\log_2(2r/K)+1=\Lambda/(2\log2)+2\le (2+1/(2\log2))\Lambda$, using $A\le r$ and $\Lambda\ge\log144>1$. One step with $k=\lceil D/K\rceil$: since $D/K\ge1$, $D/K\le k\le D/K+1\le 2D/K$, so $q=E/(kr)\le2DK/(Dr)=2\theta$, $3kA/D\le6A/K=6\theta$ and $kA/D\le2\theta$. The upper multiplier of (2.2) is at most $(1+6\theta)(1+2\theta)=1+8\theta+12\theta^2\le1+9\theta$ when $12\theta\le1$; the lower multiplier of (2.3) is at least $1-2\theta-2\theta\cdot3=1-8\theta\ge1/3>0$, so its positive part is inactive. Iterating, $U_t(D_0)\le(1+9\theta)U_{t_1}(D_1)\le\cdots\le (1+9\theta)^hU_{t_h}(T)$ with $t_{i+1}=(1+q_i)t_i\ge t$, and (3.2) at $t_h$ closes the upper chain; the lower chain runs the same way with $t_{i+1}=(1-q_i)t_i\ge(5/6)^ht$, a fixed positive multiple of $t$, so one threshold on $t$ makes every comparison and both terminal bounds apply. For the asymptotics put $u=r/A\ge144$. The function $u^{-1/2}\log u$ decreases for $u>e^2$, so $\theta\Lambda\le\log(144)/12<0.42$; hence $(1+9\theta)^h\le\exp(9C\theta\Lambda)\le1+9C\theta\Lambda\,e^{9C\cdot0.42}=1+O(\theta\Lambda)$ with $C=2+1/(2\log2)$. Also $r/(r-A)=1/(1-\theta^2)\le1+(144/143)\theta^2$ and $\theta^2\le\theta\le\theta\Lambda$; Bernoulli's inequality, valid because $8\theta\le2/3<1$, gives $(1-8\theta)^h\ge1-8\theta h\ge1-8C\theta\Lambda$; and $r/(r+A)=1/(1+\theta^2)\ge1-\theta^2$. Multiplying out gives (3.3) with absolute constants. Composition: (3.3) is consumed at the two times $(1\pm\theta)t$ in the final comparison, which is legitimate because (3.3) holds at every sufficiently large time and $(1-\theta)t\ge(11/12)t$.

3. The final comparison and the choice of $S$. Apply Lemma 2.1 with $E=K$, $k=1$, $q=K/r=\theta<1$. For $I=(x,x+D/t]$, the definitions give $N_t(I)\le DU_t(D)$ and $N_t(I)\ge DV_t(D)$, so multiplying (2.2) and (2.3) by $D$,

$$N_t(I)\le(D+3A)(1+\theta)\,U_{(1+\theta)t}(K),\qquad N_t(I)\ge\bigl(D(1-\theta)-(3-\theta)A\bigr)_+V_{(1-\theta)t}(K),$$

which are the source's two displays (p. 7); the page reaches the same displays through the Lemma 2.1 proof (F2). Upper branch: with $U\le1+c\theta\Lambda$, $(1+\theta)(1+c\theta\Lambda)\le1+(1+2c)\theta \Lambda$ because $\theta\le\theta\Lambda$ and $\theta^2\Lambda\le\theta \Lambda$, so $N_t(I)-D\le3A+(1+2c)(D+3A)\theta\Lambda$. Lower branch: put $X=D(1-\theta)-(3-\theta)A=D-3A-\theta(D-A)$. If $X\le0$ then $D\le(3-\theta)A/(1-\theta)=3A+2\theta A/(1-\theta)\le3A+(24/11)\theta A$, and $N_t(I)\ge0$ gives $N_t(I)-D\ge-3A-(24/11)\theta A$. If $X>0$ then, with $V\ge1-c'\theta\Lambda$ (the sign of the right side is irrelevant), $N_t(I)\ge X(1-c'\theta\Lambda)\ge X-c'\theta\Lambda D\ge D-3A-\theta D-c'\theta\Lambda D$, using $0<X\le D$ and $-\theta(D-A)\ge-\theta D$. Both branches give (3.4), $|N_t(I)-D|\le3A+c''(D+A)\theta\Lambda$. For $0<D\le S=K/\Lambda^2$: $D\theta\Lambda\le K\theta/\Lambda=A/\Lambda$ because $K\theta=A$; and $A\theta\Lambda=(A/\Lambda)\,\theta\Lambda^2$ with $\theta\Lambda^2=u^{-1/2}\log^2u\le\log^2(144)/12<2.1$ on $u\ge144$ (the function decreases for $u>e^4$). So $|N_t(I)-D|\le3A+C_1A/\Lambda$ with $C_1=3.1\,c''$, absolute. $D=0$ is the empty arc. Composition: the threshold is the maximum of Lemma 2.1's uniform threshold for $D\in(0,S]$ at $E=K$, $k=1$, and the threshold of (3.3) divided by $1-\theta$; neither depends on $x$ or $D$.

Strongest attack

Two attacks were pressed hardest. First, on the additive constant: try to show that the lower branch of the final comparison loses more than $3A$ when $D$ is comparable to $A$, where the positive part is near zero. The exact lower coefficient is $X=D-3A-\theta(D-A)$; for $D\le A$ the term $-\theta(D-A)$ is nonnegative and helps, and for $A<D\le S$ it is at most $\theta D\le D\theta\Lambda\le A/\Lambda$ in size, so it is absorbed by $C_1A/\Lambda$ and never by $3A$. On the upper branch the corresponding term $(D+3A)\theta$ is at most $A/\Lambda^2+3A\theta\le (1+3\cdot0.42)A/\Lambda$. The constant $3A$ survives. Second, on the threshold's uniformity: try to make the time threshold depend on $D$ through the final comparison. The Lemma 2.1 proof at $E=K$, $k=1$ needs (2.1) at all times in $[(1-\theta)t,(1+\theta)t]$, $r<|P_{(1-\theta)t}|$, and the enlarged and shrunk intervals, of length at most $(D+3A)/t\le (S+3A)/t$, shorter than $1$; each condition is monotone in $t$ and independent of $x$ and of $D\le S$, and the lemma's uniformity clause says the same. The attack failed. A lesser attack on the hypotheses succeeded only at the level of the frontmatter summary (F1): the Statement section itself carries the source's shared budget $a_t+b_t\le A$.

Premises

• Lemma 2.1 (source p. 4, displays (2.2) and (2.3) and the uniformity clause "for fixed $E$ and $k$, the time threshold can be chosen uniformly for $D$ in any bounded range"). Interface: for $D,E>0$, integer $k\ge1$, $q=E/(kr)<1$, and all sufficiently large $t$, $U_t(D)\le(1+3kA/D)(1+q)U_{(1+q)t}(E)$ and $V_t(D)\ge(1-q-(kA/D)(3-q))_+V_{(1-q)t}(E)$. Held; read at pp. 4--5 from the page images and the text layer, statement and proof; the version on the Lemma 2.1 reconstruction page agrees with the source. Standing: imported, author-recorded reconstruction of an unrefereed preprint; the page names it as imported at every use. Hypotheses met where applied: $q\le2\theta\le1/6$ in the chains and $q=\theta\le1/12$ in the final comparison.
• Hypothesis (2.1) (source p. 4): $A\ge1$; for all sufficiently large $t$, $a_t,b_t\ge0$ with $a_t+b_t\le A$ and every $r$-span in $[(r-a_t)/t,(r+b_t)/t]$. Used directly in (3.2) and through Lemma 2.1.
• Explicit assumptions. $C_0\ge144$, which gives $\theta\le1/12$, $K<r-A$ (equivalent to $\theta<1-\theta^2$) and $\Lambda>1$; the source's $O(\cdot)$ constants absolute; every time large enough that the intervals used are shorter than $1$ and $|P_t|$ exceeds the places moved. The sequence has distinct points and $r\in\mathbb N$ is fixed.
• No local L-claim is consumed and no computation is used.

Findings

F1

Severity: suggested.

Location: frontmatter desc, "when every r-span is within A/t of its mean".

Defect: the paraphrase states a weaker hypothesis than (2.1). "Within $A/t$ of its mean" reads as $a_t\le A$ and $b_t\le A$ separately, which gives only $a_t+b_t\le2A$; applying the proposition with $2A$ in place of $A$ yields $6A+2C_1A/\log(r/(2A))$, not the $3A$ the summary promises. The mean $r$-span at time $t$ is also $r/\lfloor t\rfloor$, while (2.1) is centered at $r/t$. The Statement section is correct; the summary propagates into the folder index, so it should carry the hypothesis's shape.

Witness: source p. 4, display (2.1) with "$a_t+b_t\le A$", and the sentence after it: keeping the sum under control "rather than bounding the two errors separately by $A$, is what gives the coefficient $3A$ below".

Replacement: "..., when the r-spans lie between (r - a_t)/t and (r + b_t)/t with a_t + b_t at most A."

F2

Severity: suggested.

Location: "Descent to short intervals", "the proof of (2.2) before the supremum gives".

Defect: the two displayed bounds follow from the statements (2.2) and (2.3) alone, since $N_t(I)\le DU_t(D)$ and $N_t(I)\ge DV_t(D)$ for every arc $I$ of length $D/t$; multiplying (2.2) and (2.3) by $D$ gives exactly the source's displays. The page instead routes through the internals of the Lemma 2.1 proof ($|J|$, $\ell$, $t_\pm$), which makes the deduction depend on that page's proof, while the Source paragraph declares only its "definitions and hypothesis (2.1)" as used. The detour is correct, so this is a dependency and clarity point, not an error.

Witness: source p. 7, "Apply Lemma 2.1 once more, with $E=K$ and $k=1$, so that $q=\theta$. For $D>0$ and $I=(x,x+D/t]$, it gives" the two displays.

Replacement: "For $D>0$ and $I=(x,x+D/t]$, the definitions give $N_t(I)\le DU_t(D)$ and $N_t(I)\ge DV_t(D)$, so multiplying (2.2) and (2.3) by $D$ gives" followed by the existing display.

F3

Severity: note.

Location: "Terminal bounds (3.2)", "if it contained r+1, the first and the last of them, in cyclic order, would be r places apart".

Defect: the count may exceed $r+1$, and then the first and the last of all the points in the interval are more than $r$ places apart. The argument needs "at least $r+1$" and the first $r+1$ of them; the rest of the sentence is then exact. The source's sketch has the same compression.

Witness: source p. 6, "$r+1$ points in such a half-open interval would give an $r$-span of strictly smaller length".

Replacement: "if it contained at least $r+1$, the first of them and the point $r$ places after it, which is the $(r+1)$-th point of the interval in cyclic order (all points of $P_t$ between them lie in the interval), would bound an $r$-span of length less than $(r-A)/t\le(r-a_t)/t$, contrary to (2.1)."

Verdict

Source fidelity: faithful. The Statement reproduces Proposition 3.1 of p. 6 with its hypotheses, quantifiers, constants and threshold sentence, and the locators are correct; the two suggested corrections concern the frontmatter summary and the route of one deduction, and the note a compressed sentence.

The argument as reconstructed: sound. Every deduction from (3.2) through (3.3) and (3.4) to (3.1) was rederived above, the imported Lemma 2.1 is applied inside its hypothesis $q<1$ at every use, and the threshold's independence of $x$ and $D$ holds as stated.

Limitations. The two sentences of "Role in the argument" (Section 5 and Lemma 4.2) lie outside the pages read and are not checked. The Lemma 2.1 reconstruction is not re-reviewed here beyond checking its statement against p. 4 and reading its proofs of (2.2) and (2.3) for the interface the page uses. The source is an unrefereed preprint, and this review says nothing about it beyond Section 3 as read. This focused review assigns no tier and changes no status.