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Subject and independence

The reviewer is an independent examiner in a fresh context, given only the commissioning assignment, who took no part in writing the page, the sibling reconstructions or the library card, and who read no other review of any of them. Charge: refutation.

Subject: wiki/research/erdos_1221/ko26b_lemma_7_2_reconstruction.md as it stood on 2026-09-28T05:03:27Z (the page), read in full.

Artifact: the PDF held by the library card (arXiv:2609.07196v2, 16 pages; physical page numbers equal printed page numbers). Physical pages 13 and 14 (Section 7: the definition of $D_{\mathcal P}$, Theorem 7.1, Lemma 7.2 and its proof) were read in full, every displayed formula from page images rendered at 130 dots per inch and the prose from the text layer. Page 15 (Section 8) was read in full the same way, for the page's "Role in the argument" paragraph; the rendered images are pages 13, 14 and 15. Page 16 (references) was read in the text layer for entry [10]. Pages 1--2 (setup and Theorem 1.1), 4--5 (Section 2: $P_t$, $N_t$, Lemma 2.1) and 10--12 (Section 6 through the statement of Proposition 6.4) were read in the text layer for the notation and for the hypothesis that the chain supplies. The canonical conversion beside the PDF was read for Section 7 only; the PDF decided.

Allowed material actually read: the Definitions and Statement sections of the sibling pages for Lemma 2.1 and Proposition 6.4 and the whole page for Theorem 1.1, all in the same state; the library card's provenance paragraph; the "Whole-claim report" and "Audit checklist" sections of the verification guide, the "Source fidelity" section of the evidence guide, and the math authoring guide. Not read: the folder index, anything under any evidence/ folder other than this report's own path, the problem page, other reviews, the web.

Exposures: (1) the library card's index was read in full, so its "Read status" and "Relation to Problem 1221" paragraphs (standing and acceptance text) reached the reviewer; (2) the sibling pages' "Standing" paragraphs precede their Statement sections and were read with them, and the Theorem 1.1 page was read in full, including its "Imported inputs and gaps" and "Readings addressed" sections; (3) the verification guide's "Durable reports and current standing" section was printed together with the two requested sections. None of this material reviews the page under examination or bears on the mathematics checked, and none of it was used.

Restatement

Setting: a sequence $(x_n)_{n\ge1}$ of distinct points of $\mathbb T=\mathbb R/\mathbb Z$; $P_n=\{x_1,\dots,x_n\}$ and $N_n(I)=\#(P_n\cap I)$ for an oriented half-open arc $I=(x,x+\ell]$ of length $\ell<1$, with $P_0=\varnothing$. For a finite set $\mathcal P\subset[0,1]^2$ of $M$ points, $D_{\mathcal P}(u,v)=\#(\mathcal P\cap((0,u]\times(0,v]))-Muv$.

Imported input (Theorem 7.1). There is an absolute $c_H>0$ such that every set $\mathcal P$ of $M\ge2$ points of $[0,1]^2$ satisfies $\iint_{[0,1]^2}|D_{\mathcal P}|\ge c_H\sqrt{\log M}$.

The lemma. There exist absolute constants $c_4>0$ and $S_0$ such that: for every real $S\ge S_0$, every real $B\ge1$, and every sequence as above for which there is an integer $n_0$ with

$$\int_{\mathbb T}\bigl|N_n((x,x+D/n])-D\bigr|\,dx\ \le\ B \quad\text{for every integer }n\ge n_0\text{ and every real }D\in[0,S],$$

one has $B\ge c_4\sqrt{\log S}$. The threshold $n_0$ is one integer serving all $D$ at once (a reading; see F1); $\log$ is the natural logarithm; the conclusion is an inequality between the two given numbers and involves no limit.

Checklist

• Quantifiers and scope. Pass. The statement's quantifiers match the source (p. 13): constants first, then $S\ge S_0$ and $B\ge1$, then "for all sufficiently large integers $n$" with $D$ ranging over the real interval $[0,S]$. The proof uses one threshold $n_0$ for all $D$; this is the source's reading (p. 14, "Let $n_0$ be a threshold for (7.1)") and the version Proposition 6.4 supplies (p. 12), and it is unlabeled on the page (F1). Boundary cases: $D=0$ is trivial in (7.1); $M_a\ge2$ is secured on the good set by $L\ge4$; $u=0$ or $v<1/N$ give empty counts (F3).
• Circularity. Pass. The conclusion $B\ge c_4\sqrt{\log S}$ is never assumed; (7.1) is consumed only at time $N$ with $D=L$ and at times $n=\lfloor Nv\rfloor\ge n_0$ with $D=nuw$.
• Model and convention changes. Pass. The passage from arc counts to the planar set $\mathcal P_a$ is an exact identity, rederived below (weakest step 1) with the half-open conventions matched on both sides; nothing is transferred by analogy.
• Finite and statistical overreach. Inapplicable. No finite case or heuristic average stands in for a proof; the Markov step is a measure inequality with its constant tracked. The reviewer's own finite random check of two identities is a sanity check, not evidence.
• Uniformity. Pass. $c_H$ is absolute, so $c_4$ is; $S_0$ is chosen after $c_4$ and depends only on $c_H$; the limit $N\to\infty$ is taken with $L$, $n_0$ and $B$ fixed; the threshold $n_0$ is uniform in $D$ (F1).
• Extremal conclusions. Pass, narrowly applicable. No attained extremum is claimed; the supremum over $D$ in (6.7) is consumed as a uniform bound, which is its actual strength.
• Consequences and composition. Pass. Each "hence" was rederived: (7.2) from (7.1); (7.3) from Theorem 7.1 and the identity; the measure bound from Markov's inequality and $B<L/4$; (7.4) from (7.3), $|G|\ge\tfrac12$ and (7.2); (7.5) from (7.1) at every $n\in[n_0,N]$ and the strip bound; the conclusion from $N\to\infty$; the final constant from $L=\lfloor S\rfloor$. The "Role in the argument" paragraph agrees with pp. 12 and 15.
• Computation. Inapplicable. The page retains no computation.
• Reproduction. Inapplicable. The page states no rerun commands or coverage claims.
• Source and verdict fidelity. Pass. The statement, Theorem 7.1, the displays (7.1)--(7.5), the constants $\tfrac14$, $\tfrac12$, $\tfrac32$, $\tfrac54$, the locators (Theorem 7.1 p. 13, Lemma 7.2 pp. 13--14) and the citation of [10] (p. 16) match the PDF. The standing sentence claims author-recorded status only.

Weakest steps

1. The localization identity. Fix $a\in\mathbb T$, an integer $N>L$, and $w=L/N<1$. For $x_i\in P_N\cap(a,a+w]$ let $d_i\in(0,w]$ be the clockwise distance from $a$ to $x_i$, and form $(d_i/w,\,i/N)\in(0,1]^2$; the map is injective because the indices $i$ differ, so $\mathcal P_a$ has exactly $M_a=N_N((a,a+w])$ points. For $(u,v)\in[0,1]^2$ the point lies in $(0,u]\times(0,v]$ exactly when $0<d_i\le uw$ and $i\le Nv$, that is, when $x_i\in(a,a+uw]$ and $i\le\lfloor Nv\rfloor$ (as $i$ is an integer). Hence $\#(\mathcal P_a\cap((0,u]\times(0,v]))=N_{\lfloor Nv\rfloor}((a,a+uw])$ exactly, for every $(u,v)$, with $N_0\equiv0$; under these conventions the "null set" caveat of the source and the page is not even needed. Subtracting $M_auv$ and adding and subtracting $Luv$ gives $D_{\mathcal P_a}=G_a+(L-M_a)uv$. Composition: with $\iint_{[0,1]^2}uv\,du\,dv=\tfrac14$ and Theorem 7.1 (which needs $M_a\ge2$ and a set in $[0,1]^2$, both met), $\|G_a\|_1\ge\|D_{\mathcal P_a}\|_1-\tfrac14|L-M_a|\ge c_H\sqrt{\log M_a}-\tfrac14|M_a-L|$, which is (7.3).

2. The lower bound (7.4). Assume $B<L/4$ and $L\ge4$. Markov's inequality for $f(a)=|M_a-L|\ge0$ with (7.2) gives $|\{a:f(a)>L/2\}|\le\frac2L\int_{\mathbb T}f\le\frac{2B}L<\frac12$, so the complement $G=\{a:L/2\le M_a\le3L/2\}$ has $|G|\ge\tfrac12$ (in fact $>\tfrac12$). On $G$, $M_a\ge L/2\ge2$ and $\sqrt{\log M_a}\ge\sqrt{\log(L/2)}\ge0$. Integrating (7.3) over $G$,

$$\int_{\mathbb T}\|G_a\|_1\,da\ \ge\ \int_G\|G_a\|_1\,da \ \ge\ c_H\sqrt{\log(L/2)}\,|G|-\tfrac14\int_G|M_a-L|\,da \ \ge\ \tfrac{c_H}2\sqrt{\log(L/2)}-\tfrac B4 ,$$

the first inequality because $\|G_a\|_1\ge0$, the last by $|G|\ge\tfrac12$ and (7.2) over all of $\mathbb T$. Composition: the case $B\ge L/4$ is disposed of separately (below).

3. The upper bound (7.5). Fix $(u,v)\in[0,1]^2$ and put $n=\lfloor Nv\rfloor$, so $0\le v-n/N<1/N$ and $n\le N$. If $n\ge n_0$, put $D=nuw$; then $0\le D\le Nuw=Lu\le L\le S$ and $uw=D/n$, so (7.1) at time $n$ gives $\int_{\mathbb T}|N_n((a,a+uw])-nuw|\,da\le B$. Since $nuw=Lu\cdot n/N$, $G_a(u,v)=\bigl(N_n((a,a+uw])-nuw\bigr)+Lu\,(n/N-v)$, and the second term is at most $Lu/N\le w$ in absolute value; so $\int_{\mathbb T}|G_a(u,v)|\,da\le B+w$. If $n<n_0$ (the strip $0\le v<n_0/N$, of area $n_0/N$), then $0\le N_n(\cdot)\le n<n_0$ and $0\le Luv\le L$, so $|G_a(u,v)|\le n_0+L$. By Tonelli's theorem (the integrand $|G_a(u,v)|$ is nonnegative and measurable in $(a,u,v)$), integrating first in $a$ and then over $(u,v)$,

$$\int_{\mathbb T}\|G_a\|_1\,da\ \le\ (B+w)\cdot1+\frac{n_0}N(n_0+L) =B+\frac LN+\frac{n_0(n_0+L)}N ,$$

which is (7.5) and tends to $B$ as $N\to\infty$ with $L$ and $n_0$ fixed. This step is where the uniformity of $n_0$ in $D$ is consumed: $D=nuw$ sweeps $[0,Lu]$ as $v$ varies, so a threshold depending on $D$ would not leave a single strip.

Conclusion and constants. Combining, $\tfrac{c_H}2\sqrt{\log(L/2)}-\tfrac B4\le B+o(1)$, so $\tfrac54B\ge\tfrac{c_H}2\sqrt{\log(L/2)}$. With $L\ge S-1$ and $S\ge6$, $(S-1)/2\ge\sqrt S$ (equivalent to $\sqrt S\ge1+\sqrt2$), so $\log(L/2)\ge\tfrac12\log S$ and $B\ge\tfrac{2c_H}5\cdot\tfrac1{\sqrt2}\sqrt{\log S}=\tfrac{\sqrt2\,c_H}5\sqrt{\log S}$. Take $c_4=\sqrt2\,c_H/5$. In the case $B\ge L/4\ge(S-1)/4$ the same $c_4$ works once $S_0$ satisfies $(S_0-1)/4\ge c_4\sqrt{\log S_0}$, a condition on $c_H$ alone and hence absolute, since $(S-1)/4-c_4\sqrt{\log S}$ is increasing for large $S$. Both cases need $L\ge4$, which $S_0\ge6$ covers. This reproduces the page's "after adjusting the constants" with explicit values.

Strongest attack

The attack aimed at the quantifier structure of hypothesis (7.1). The upper bound (7.5) applies (7.1) at every integer time $n\in[n_0,N]$ and, for each such $n$, at every real $D=nuw\in[0,Lu]$ at once; if "for all sufficiently large $n$" were read with a threshold allowed to depend on $D$, the set of $(u,v)$ where (7.1) is available at $n=\lfloor Nv\rfloor$ would no longer be a strip, the bound $n_0(n_0+L)/N$ on the exceptional region would be unavailable, and the limit $N\to\infty$ would fail. The attack fails: the source reads (7.1) with one threshold ("Let $n_0$ be a threshold for (7.1)", p. 14), the page does the same, and the only supplier of (7.1) in the chain, Proposition 6.4, states that its threshold "may depend on $r$, $A$, and the sequence, but not on $D$" (p. 12). The reconstruction uses the hypothesis at exactly the strength at which it is supplied; what survives of the attack is a labeling request (F1).

Two further attacks were tried and failed. (a) Theorem 7.1 on the bad set of $a$: for $a$ with $M_a\le1$ the theorem does not apply and (7.3) is unavailable; the page never integrates (7.3) over such $a$, and the passage from $\int_G$ to $\int_{\mathbb T}$ uses only $\|G_a\|_1\ge0$. (b) The constants: the page's "$\log(L/2)\ge\tfrac12\log S$, say" and "after adjusting the constants" were recomputed above with explicit $c_4$ and $S_0$; no hidden dependence on $N$, on the sequence or on $B$ appears. A finite random check of the identities in weakest steps 1 and 3 (random distinct points, random $N$, $L$, $a$, $u$, $v$; 300 trials) found no failure; it is a sanity check only.

Premises

• Theorem 7.1 (Halász). Interface as used: an absolute constant $c_H>0$; for every set $\mathcal P$ of $M\ge2$ points in $[0,1]^2$, $\iint_{[0,1]^2}|D_{\mathcal P}(u,v)|\,du\,dv\ge c_H\sqrt{\log M}$ with the unnormalized $D_{\mathcal P}$. Held source: none; the 1981 paper (the source's reference [10], G. Halász, On Roth's method in the theory of irregularities of point distributions, in Recent Progress in Analytic Number Theory, vol. 2, Academic Press, 1981, pp. 79--94) is not held by the corpus. Reading depth: the statement was read on p. 13 of the held preprint (image and text layer) and its bibliography entry on p. 16; the page names it as imported and unchecked, and this review did not check it against the original either. Applied with $M=M_a\ge2$ on the good set, to a set of $M_a$ distinct points of $(0,1]^2$; hypotheses met.
• Hypothesis (7.1). A hypothesis of the lemma, not a premise of the page; in the chain it is supplied by Proposition 6.4 with $B=C_3A$ and $S=\sqrt{Ar}/\log^2(r/A)$ and a threshold independent of $D$. Read at Statement depth on the Proposition 6.4 page in the frozen state and on p. 12 of the PDF; its proof was not examined here.
• Definitions. $P_t$, $N_t$, the oriented half-open arcs and the distinct-point setting, from the Definitions of the Lemma 2.1 page and p. 4 of the PDF. Explicit assumption added by this review: $P_0=\varnothing$, so $N_0\equiv0$ (F3). Distinctness of the $x_i$ is not needed for the planar set to have $M_a$ points, since the second coordinates $i/N$ already differ; it is part of the setting throughout.
• Standard tools. Markov's inequality; Tonelli's theorem for the nonnegative integrand $|G_a(u,v)|$; $\iint_{[0,1]^2}uv\,du\,dv=\tfrac14$. No source needed.
• Threshold. $n_0\ge1$ is implicit, since $N_n$ is defined for $n\ge1$; the page's choice $N>\max(L,n_0)$ makes (7.2) and the strip bound available.

Findings

F1. Severity: suggested. Location: "let $n_0$ be a threshold for (7.1)". Defect: the hypothesis "for all sufficiently large integers $n$ ... $(0\le D\le S)$" is used with one threshold serving every $D\in[0,S]$, in (7.2) and throughout the upper bound, where $D=nuw$ varies with $(u,v)$; this reading is essential (see Strongest attack) and is not labeled. Witness: source p. 14, "Let $n_0$ be a threshold for (7.1), and put $n=\lfloor Nv\rfloor$. For $n\ge n_0$, set $D=nuw$"; Proposition 6.4, p. 12, "The time threshold may depend on $r$, $A$, and the sequence, but not on $D$." Proposed text, after the Statement: "Reading. The threshold in 'for all sufficiently large integers $n$' is a single integer $n_0$ serving every $D\in[0,S]$; the proof applies (7.1) at times $n=\lfloor Nv\rfloor\ge n_0$ with $D=nuw$ depending on $(u,v)$, and Proposition 6.4 supplies (6.7) with exactly this uniformity."

F2. Severity: suggested. Location: the Proof section, from "Put $L=\lfloor S\rfloor$" to "once $S_0$ is large enough". Defect: the reconstruction supplies several details beyond the source without marking them as supplied: the condition $N>n_0$ (source: "choose a large integer $N>L$", p. 13); the container $(0,1]^2$ for the planar points (source: $[0,1]^2$, p. 14); the reason for the case $B\ge L/4$ (source: "immediate for large $L$"); the Markov computation "$(2/L)B<1/2$"; the requirement "$L\ge4$"; "$D\le Nuw=Lu$"; "$Lu/N\le w$, since $0\le v-n/N<1/N$"; and "$\log(L/2)\ge\tfrac12\log S$, say" with the closing constant adjustment (source: "after adjusting the absolute constants"). All are correct and routine. Witness: pp. 13--14 as quoted. Proposed text, at the head of the Proof: "The source's proof is followed step by step; the reconstruction supplies the choice $N>n_0$, the justification of the case $B\ge L/4$, the Markov computation, the requirement $L\ge4$, the bound on the second term of the decomposition, and the final comparison $\log(L/2)\ge\tfrac12\log S$ with the resulting constant."

F3. Severity: note. Location: "Points, $P_n$ and $N_n(\cdot)$ are as on the Lemma 2.1 page; only integer times occur here." Defect: the cited definitions give $P_t$ for real $t\ge1$ only, while $G_a(u,v)$ and the box identity use $N_{\lfloor Nv\rfloor}$ with $\lfloor Nv\rfloor=0$ for $v<1/N$; the convention $P_0=\varnothing$ is needed there and is not stated (the source has the same gap). Witness: source p. 4, "For real $t\ge1$, write $P_t=\{x_1,\dots,x_{\lfloor t\rfloor}\}$"; p. 14, $G_a(u,v)=N_{\lfloor Nv\rfloor}((a,a+uw])-Luv$. Proposed text: append "with $P_0=\varnothing$, so that $N_0\equiv0$ on the strip $v<1/N$."

F4. Severity: note. Location: "Integrating over $(u,v)$ and then $a$". Defect: the bound on the first term of the decomposition is an integral in $a$ at fixed $(u,v)$, so the integration order that produces (7.5) is $a$ first and $(u,v)$ second, with Tonelli's theorem justifying the exchange; the phrase names the reverse order. The result is unaffected. Witness: source p. 14, "the integral in $a$ of the absolute value of the first is at most $B$ by (7.1)", followed by "Consequently" and (7.5). Proposed text: "Integrating over $a$ at fixed $(u,v)$ and then over $(u,v)$ (Tonelli),".

F5. Severity: note. Location: "Suppose $S\ge S_0$, $B\ge1$". Defect: none in fidelity; the hypothesis $B\ge1$ is stated by the source and reproduced, but neither the source's proof nor the page's uses it anywhere, and a reader is left to look for where it enters. Witness: pp. 13--14, no step invokes $B\ge1$; Section 8 (p. 15) arranges "$C_3A\ge1$" to meet it. Proposed text, after the Statement or in the Role paragraph: "The hypothesis $B\ge1$ is not used in the proof; Section 8 arranges $C_3A\ge1$ to meet it."

Verdict

Source fidelity: faithful. The statement of Lemma 7.2, the imported Theorem 7.1, the displays (7.1)--(7.5), the constants, the locators (Theorem 7.1 p. 13; Lemma 7.2 pp. 13--14) and the citation of the Halász paper agree with the held PDF; nothing the source proves is altered or strengthened, and the supplied details (F2) are correct.

The argument as reconstructed: sound. Every deduction was rederived above; the constants are absolute, with $c_4=\sqrt2\,c_H/5$ and an $S_0$ depending on $c_H$ alone as one explicit choice.

Limitations: Theorem 7.1 is not checked against the 1981 paper, which the corpus does not hold, so the lemma is verified here only relative to that imported statement; Proposition 6.4, which supplies (7.1) in the chain, was read at Statement depth only; the finite random check is not evidence of record. The findings are two labeling suggestions and three notes; there are no required corrections.

This focused review assigns no tier and changes no status.