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Subject and independence

The reviewer is an independent reviewer working in a fresh context, given only this assignment and charged with refutation; the reviewer took no part in writing the page under review and had not seen it, its input pages or the folder before the assignment. Nothing was read outside the allowed set below except the two exposures disclosed at the end of this section.

Frozen subject: wiki/research/erdos_1221/ko26b_proposition_6_4_reconstruction.md as it stood on 2026-09-28T05:03:27Z, the page Proposition 6.4 reconstruction, read whole.

Artifact: the PDF held under the library card Korsky 2026, resolution (arXiv:2609.07196v2, 16 pages; the physical page numbers equal the printed ones, page 1 being printed as 1). Physical pages 10--13 were read in full in the text layer, and page images of pages 10, 11, 12 and 13 were rendered at 130 dots per inch and read; every displayed formula of Proposition 6.4 and its proof (pp. 12--13), of Lemma 6.2 and its proof (p. 11), of Lemma 6.3 and its proof (p. 12) and of (6.1)--(6.4) (pp. 10--11) was read from the images. The canonical conversion beside the PDF was consulted at the Proposition 6.4 statement and proof as a secondary check; the PDF decided.

Allowed material actually read: the page; the input pages Lemma 6.2 and Lemma 6.3 in the same state, whose Statement sections were compared with the source and whose interfaces this review consumes, the closing sentence of the Lemma 6.2 proof (its description of the threshold) being used for the uniformity deduction; the library card's provenance paragraph ("Retained artifact"); the Statement paragraph of the problem page wiki/problems/analysis/E1221/_index.md; docs/verification.md "Whole-claim report" and "Audit checklist"; docs/evidence.md "Source fidelity"; and docs/math_authoring.md.

Exposures: (1) the Lemma 6.2 and Lemma 6.3 pages were printed whole, so their Standing paragraphs (each describing itself as an author-recorded reconstruction) and the rest of their proofs were seen; only the Statement sections and the Lemma 6.2 threshold sentence were relied upon, and neither proof was checked. (2) The library card's "Read status" paragraph, printed together with the provenance paragraph, contains one sentence on the source's standing in the corpus; it played no role. No evidence folder, no folder index, no assessment or status text, no other review and no web search was read.

Restatement

Setting, from Section 6 of the source (pp. 10--12): a sequence of distinct points on the circle $\mathbb T=\mathbb R/\mathbb Z$; $P_t$ the set of the first $\lfloor t\rfloor$ points at a real time $t$; $N_t(I)$ the number of points of $P_t$ in an arc $I$; $r$ a fixed positive integer; for $D\ge0$,

$$\Delta_t(x,D)=N_t\bigl((x,x+D/t]\bigr)-D,\qquad Z_t(D)=\int_{\mathbb T}\bigl(\Delta_t(x,D)\bigr)_+\,dx,$$

and $z_t(D)=Z_t(D)/D$ for $D>0$. Hypothesis (6.1): a constant $A\ge1$ is fixed, and one fixed alternative among

$$nM_n^{(r)}-r\le A\qquad\text{or}\qquad r-nm_n^{(r)}\le A$$

holds for every sufficiently large integer $n$, where $M_n^{(r)}$ and $m_n^{(r)}$ are the largest and the smallest $r$-span (sum of $r$ consecutive gaps) of the first $n$ points.

The proposition. There are absolute constants $C_2,C_3>0$, depending on nothing, such that the following holds for every sequence of distinct points, every positive integer $r$ and every $A\ge1$ with $r\ge C_2A$ for which (6.1) holds. With $\Lambda=\log(r/A)$ (natural logarithm) and $S=\sqrt{Ar}/\Lambda^2$, there is a threshold $N$, which may depend on $r$, $A$ and the sequence but not on $D$, such that for every integer $n\ge N$ and every real $D$ with $0\le D\le S$,

$$\int_{\mathbb T}\Bigl|N_n\bigl((x,x+D/n]\bigr)-D\Bigr|\,dx\ \le\ C_3A ;$$

that is, for each $n\ge N$ the supremum over $0\le D\le S$ of the left side is at most $C_3A$. Conventions: arcs $(x,x+D/n]$ are half-open arcs of the circle of length $D/n$; the conclusion is at integer times, the lemmas at real times; the implied constants are absolute.

Checklist

• Quantifiers and scope. Correction needed (F1). The page states the eventual quantifier ("for all sufficiently large integers $n$") and the clause that the threshold does not depend on $D$ exactly as the source does, and handles the boundary case $D=0$. The conclusion is proved at integer times only, as the source proves it, through identity (6.4). The clause "not on $D$" is true; the page's justification of it is inaccurate as written.
• Circularity. Pass. The bound descends from Lemma 6.3 at scale $r$ through Lemma 6.2 to scale $K$ and then to $D\le S$; nothing equivalent to (6.7) is assumed.
• Model and convention changes. Pass. The normalization $z_t=Z_t/D$, the doubling chain and the real-time counting set $P_t$ are the source's own objects; no averaged or relaxed system is substituted, and the transfer from real times to integer times is the identity (6.4).
• Finite and statistical overreach. Inapplicable. No finite case or heuristic average appears.
• Uniformity. Pass in the mathematics, correction needed in the exposition (F1). The constant $C=18$ in (6.8) is verified below; the factor $(1+C\theta)^h$ is bounded by an absolute constant because $C_2$ imposes $\theta<1/2$ and $\theta\Lambda\le1$; the threshold in $t$ behind (6.9) does not involve $D$; the threshold of the descent step is uniform over $0<D\le S$ for the reason given under Weakest steps, which the page does not state.
• Extremal conclusions. Pass. The only extremal object is the supremum over $D$ in (6.7); the bound is proved for each $D$ with a constant free of $D$, so the supremum is bounded. No infimum, attainment or sharpness is claimed.
• Consequences and composition. Pass, with F2 recorded. Each deduction was re-derived: (6.8) from (6.5) with $k=\lceil D/K\rceil$; (6.9) by $h$ applications of (6.8) and Lemma 6.3; the descent inequality from (6.5) with $E=K$, $k=1$ and (6.9) at time $(1+\theta)t$; the bound $D\theta\Lambda\le A/\Lambda\le A$; the passage to integer times. The inputs are consumed at their stated strength except for the uniformity in $D$ of the Lemma 6.2 threshold, which the Lemma 6.2 Statement does not grant and the page does not derive (F1).
• Computation. Inapplicable. The page contains no computation.
• Reproduction. Inapplicable. The page states no rerun commands and makes no computational claim.
• Source and verdict fidelity. Pass, with F2 recorded. The Statement section agrees clause by clause with Proposition 6.4 on p. 12 of the artifact; the labels (6.1), (6.4), (6.5), (6.7), (6.8), (6.9), Lemma 6.2, Lemma 6.3 and the pages 12--13 are correct; the Standing paragraph claims only an author-recorded reconstruction. The source's reuse of the constant $C'$ in the descent display is repaired on the page without being recorded (F2).

Weakest steps

1. The iteration to (6.9). Let $D_0=K<D_1<\cdots<D_h=r$ with $D_i=2^iK$ for $i<h$ and $D_h=r$, where $h$ is the least integer with $2^hK\ge r$; then $D_{h-1}<r\le2D_{h-1}$, so consecutive scales satisfy $K\le D\le E\le2D$, and

$$h\le\log_2(r/K)+1=\frac{\Lambda}{2\log2}+1 ,$$

since $r/K=\sqrt{r/A}$; with $\Lambda>1$ this is at most $1.73\,\Lambda$. (When $K=r$ the chain is empty and (6.9) follows from Lemma 6.3 directly.) Put $t_0=t$ and $t_{i+1}=(1+q_i)t_i$, where $q_i=D_{i+1}/(k_ir)\le2\theta$ and $k_i=\lceil D_i/K\rceil$. Applying (6.8) at scale $D_i$ and time $t_i$ for $i=0,\ldots,h-1$ and substituting each bound into the previous one gives

$$z_t(K)\ \le\ (1+C\theta)^h z_{t_h}(r) +C\theta\sum_{i<h}(1+C\theta)^i +\sum_{i<h}(1+C\theta)^i\,\frac{4k_ir}{t_iD_i} .$$

Here $z_{t_h}(r)\le\theta^2$ by Lemma 6.3, since $t_h\ge t$ is late when $t$ is; $(1+C\theta)^i\le\exp(C\theta h)$, and with $C=18$, $\theta<1/2$ and $\theta\Lambda\le1$,

$$C\theta h\ \le\ 18\Bigl(\frac{\theta\Lambda}{2\log2}+\theta\Bigr)\ <\ 22 ,$$

so the factor is an absolute constant $M$. Also $k_i/D_i\le2/K$ and $t_i\ge t$, so the last sum is at most $8hr/(Kt)$. Hence

$$z_t(K)\ \le\ M\theta^2+MC\,h\theta+\frac{8Mhr}{Kt} \ \le\ C_a\theta\Lambda+\frac{C_bhr}{Kt},$$

using $\theta^2\le\theta\Lambda$ (as $\theta<1<\Lambda$) and $h\le1.73\,\Lambda$. For fixed $r$, $A$ and sequence the last term tends to $0$, so $z_t(K)\le(C_a+1)\theta\Lambda$ for all $t$ beyond a threshold that depends on $r$, $A$ and the sequence (through the $h$ Lemma 6.2 thresholds and the Lemma 6.3 threshold) and on nothing else. This is (6.9) with $C'=C_a+1$ absolute. It composes with the descent step by being applied at the time $(1+\theta)t\ge t$.

2. The threshold's independence from $D$. Fix $0<D\le S$. The descent bound $Z_t(D)\le(8+C'')A+4r/t$ at a time $t$ rests on: (a) (6.9) at the time $(1+\theta)t$, whose threshold is free of $D$; (b) Lemma 6.2 with the triple $(D,K,1)$, whose largeness requirement, by the closing sentence of the Lemma 6.2 page's proof, is that (6.1) hold at the integer parts of the times in $[t,(1+\theta)t]$, that $r<\lfloor t\rfloor$, and that the arcs of lengths $D/t$ and $K/((1+\theta)t)$ be shorter than $1$; only the first arc involves $D$, and $D/t\le S/t<1$ once $t>S$. At an integer time $n$, identity (6.4) needs $D/n\le1$ as well: for $D>n$ the arc is the whole circle, $\Delta_n\equiv n-D$ has nonzero mean and (6.4) fails. So with $N$ the largest of the (6.9) threshold divided by $1+\theta$, the (6.1) threshold, $r+1$, $K+1$ (which exceeds $S+1$, since $S<K$) and $4r/A$, every integer $n\ge N$ and every $0<D\le S$ give $\int|\Delta_n(\cdot,D)|=2Z_n(D)\le2(9+C'')A$, and $D=0$ gives $0$. The clause "not on $D$" of the statement follows. The page reaches the same conclusion with a reason that does not cover (b) (F1).

3. The single-step bound (6.8). For consecutive scales $D\le E\le2D$ with $D\ge K$ and $k=\lceil D/K\rceil$: $D/K\ge1$ gives $D/K\le k\le D/K+1\le2D/K$, so

$$q=\frac E{kr}\le\frac{2D}{kr}\le\frac{2K}r=2\theta,\qquad \frac{8kA}D\le\frac{16A}K=16\theta ,$$

and $q<1$ holds because $\theta<1/2$. Dividing (6.5) by $D>0$ and writing $Z_{(1+q)t}(E)=E\,z_{(1+q)t}(E)$,

$$z_t(D)\le(1+q)\,z_{(1+q)t}(E)+q+\frac{8kA}D+\frac{4kr}{tD} \le(1+18\theta)\,z_{(1+q)t}(E)+18\theta+\frac{4kr}{tD},$$

using $z\ge0$, $1+q\le1+2\theta$ and $q+8kA/D\le18\theta$. This is (6.8) with $C=18$, as the page states; the source leaves $C$ unnamed. The letter $D$ denotes the chain scale here and the short-interval length in the descent, as in the source.

Strongest attack

The attack aimed at the clause that the time threshold does not depend on $D$, the one part of the statement beyond the bound itself. The Lemma 6.2 Statement, on its page and in the source, fixes $D$ before saying "for all sufficiently large $t$", so its threshold may depend on $D$; the descent step invokes it once for each of the uncountably many $D\in(0,S]$, and if the threshold grew without bound as $D\to0$ or as $D\to S$, no single $n$ would serve all $D$ and (6.7) would fail as a supremum. The attack fails: the threshold's only dependence on $D$ is the requirement $D/t<1$, monotone in $D$ and met for all $D\le S$ by $t>S$; the transport error $8A+4r/t$ is free of $D$; (6.9) enters at the $D$-free time $(1+\theta)t$; and the integer-time identity (6.4) needs only $D\le n$, again met by $n>S$. A second attack tried to make the constant $C'$ of (6.9) depend on $r/A$ through the product $(1+C\theta)^h$ with $h$ growing like $\Lambda$; it fails because $C_2$ imposes $\theta\Lambda\le1$, giving $C\theta h<22$. A third attack tried the descent at $D$ near $S$, where the term $C''D\theta\Lambda$ is largest; it equals $C''A/\Lambda<0.73\,C''A$ there, inside the constant. The mathematics survives; the first attack exposes an inaccurate sentence on the page (F1).

Premises

• Lemma 6.2 (imported from the same folder's reconstruction page; source held, p. 11, statement and proof read from the page image and the text layer, the statement compared clause by clause with the page). Exact interface: under (6.1), for fixed $D,E>0$, integer $k\ge1$ and $q=E/(kr)<1$, for all sufficiently large $t$, $Z_t(D)\le\frac{(1+q)D}E Z_{(1+q)t}(E)+qD+8kA+4kr/t$; by the page's proof, "sufficiently large" means (6.1) at the integer parts of the times in $[t,(1+q)t]$, $kr<\lfloor t\rfloor$, and arcs of lengths $D/t$ and $E/((1+q)t)$ shorter than $1$. The page names it as an input; its proof was not verified here.
• Lemma 6.3 (imported from the same folder's reconstruction page; source held, p. 12, statement and proof read from the image). Exact interface: under (6.1), $Z_t(r)\le A$ for all sufficiently large $t$. Not verified here.
• Identity (6.4) (source p. 11; stated on the Lemma 6.2 page). Exact interface: at an integer time $n$ and for $0\le D\le n$, $\int_{\mathbb T}|\Delta_n(x,D)|\,dx=2Z_n(D)$. The restriction $D\le n$ is implicit in the source and supplied here.
• Hypothesis (6.1) (source p. 10), with $A\ge1$ and the section's standing convention that $t$ is large enough for the spans used to exist.
• Explicit assumptions. The points are distinct and $r$ is a positive integer; $C_2$ is large enough that $\theta<1/2$ (hence $\Lambda=-2\log\theta>2\log2>1$) and $\theta\Lambda\le1$; all implied constants are absolute. No batch acceptance order applies; this is a single focused review.

Findings

F1. Severity: required. Location: "Integer times", the sentence "from the largeness needed by the finitely many comparisons, none of which depends on $D$". Defect: the descent step is one Lemma 6.2 instance for each $D\in(0,S]$, not finitely many, and its largeness requirement does depend on $D$ (the arc of length $D/t$ must be shorter than $1$), as does identity (6.4) at integer time ($D\le n$); both are uniform over $0\le D\le S$ once $n>S$, but the page does not say so, and the sentence as written is inaccurate at the statement's clause "not on $D$". Witness: source p. 13, whose reason is "The transport error above is independent of $D$, so one late time works uniformly for $0<D\le S$"; Lemma 6.2 page in the same state, Statement ("Fix $D,E>0$ ... for all sufficiently large $t$") and the closing sentence of its Proof. Proposed replacement: "The threshold on $n$ comes from (6.9) at the time $(1+\theta)n$, from $4r/n\le A$, from the finitely many chain comparisons behind (6.9), and from the descent comparison, whose transport error $8A+4r/t$ is free of $D$ and whose only $D$-dependent largeness requirement (Lemma 6.2 page, end of proof) is that the arc of length $D/n$ be shorter than $1$; identity (6.4) needs the same. Since $D\le S$, any $n>S$ meets both at once, so one late time serves every $0\le D\le S$."

F2. Severity: suggested. Location: "Descent to short intervals", "this is at most $8A+C''D\theta\Lambda+4r/t$ with $C''$ absolute". Defect: the source's display writes this line with the constant $C'$ of (6.9), which is not literally valid, since $(1+\theta)C'D\theta\Lambda+\theta D$ exceeds $C'D\theta\Lambda$; the page's $C''$ is the correct repair ($C''=2C'+1$ serves, using $\theta\le1$ and $\theta\le\theta\Lambda$), but the departure from the source is not recorded, and the Standing sentence on constants covers constants the reconstruction names, not a constant the source reuses. Witness: source p. 13, second line of the descent display, "$\le 8A+C'D\theta\Lambda+4r/t$". Proposed replacement: after "with $C''$ absolute" add "(the source's display writes $C'$ here, reusing the constant of (6.9); absorbing $\theta D$ and the factor $1+\theta$ needs a larger constant, and $C''=2C'+1$ serves)".

F3. Severity: note. Location: "Proof", "take $C_2$ large enough that $\theta$ and $\theta\Lambda$ are small". The proof also uses $2\theta<1$ (for $q<1$ in the chain) and $\Lambda\ge1$ (in "$h=O(\Lambda)$", "$\theta\le\theta\Lambda$" and "$A/\Lambda\le A$"); both follow from $\theta<1/2$ because $\Lambda=-2\log\theta$, but the page does not say so. Witness: source p. 13, "taking $C_2$ large enough that $\theta\Lambda$ is small", equally silent. Proposed replacement: "take $C_2$ large enough that $2\theta<1$ (so $q<1$ below and $\Lambda=-2\log\theta>1$) and $\theta\Lambda\le1$; both hold once $r/A$ is large."

F4. Severity: note. Location: frontmatter desc, "intervals holding at most S points". The intervals are arcs of length $D/n$ with $D\le S$; $D$ is their mean count over $x$, not a bound on their count. Witness: source p. 12, (6.7), whose integrand is $N_n((x,x+D/n])-D$ with $0\le D\le S$. Proposed replacement: "intervals of length at most S over n".

Verdict

Source fidelity: faithful with corrections. The Statement section reproduces Proposition 6.4 with its hypotheses, quantifiers, constants, the definition of $S$ and the clause on the threshold exactly as on p. 12 of the artifact, and every locator and label is correct; the corrections are F1, to the page's own justification of the threshold's independence from $D$, and F2, the unrecorded repair of a constant the source reuses.

The argument as reconstructed: sound. Every deduction was re-derived and holds; the closing step's clause "not on $D$" is true, but its stated reason does not cover the descent comparison, and F1 supplies the missing observation. No step is defective.

Limitations: Lemma 6.2 and Lemma 6.3 were consumed at their stated interfaces and their proofs were not verified; the review covers pp. 10--13 of the source and says nothing about the rest of the paper or its main theorem; the source is an unrefereed preprint; no computation was involved. This focused review assigns no tier and changes no status.