---

Subject and independence

Role. Independent reviewer in a fresh context, given only the review assignment. The reviewer took no part in writing the page under review, the library card, the result page or any other page of the folder, and the charge was refutation. This is a focused review of one reconstruction page; it is not an acceptance record.

Frozen subject. wiki/research/erdos_1221/clst25_theorem_2_reconstruction.md as it stood on 2026-09-28T05:03:27Z, read whole (the working tree of the worktree was at the same state and clean).

Artifact. The retained PDF beside the library card, arXiv:2511.14637v1, 12 physical pages; the printed page numbers coincide with the physical ones. Read clause by clause, in the text layer and on page images rendered at 160 dots per inch: p. 2 (Theorem 2, Theorem 3 and their displayed formulas), p. 3 (the remark after Theorem 3) and p. 9 (the paragraph "Theorem 3 implies Theorem 2"). Read on the page image at 110 dots per inch: p. 1 (abstract and Section 1.1, for the piece count). Skimmed in the text layer only, for section boundaries and the convention sentences of Section 2: pp. 4, 8 and 10--12. The proofs of Theorem 3 (Sections 2--3, pp. 3--11) were not read. Page images were rendered for pp. 1--11 at 110 dots per inch and for pp. 2, 3 and 9 at 160 dots per inch; the ones read are the ones listed.

Allowed material actually read. The page; the Statement section of the ko26b Theorem 1.1 reconstruction (its lines 64--84 in the frozen state); the provenance paragraph of the library card and the Statement section of the Theorem 2 result page; the statement paragraph of Problem 1221; the sections "Whole-claim report" and "Audit checklist" of the verification rules (both the general list of canonical failure modes and the repository-specific list of ten items), the section "Source fidelity" of the evidence rules, and the math authoring page in full, all in the frozen state. The canonical conversion beside the PDF was not opened; the PDF decided every reading.

Exposures. Three, none of which influenced a finding. (1) The library card and the Theorem 2 result page were printed whole, so their overview, proof-pointer and bears-on paragraphs were seen beyond the provenance paragraph and the Statement section. (2) The problem page has no "Statement" heading; the block read to reach its statement paragraph also contained its "Formulation", "Status", "Source", "References" and "Formalization" paragraphs, and the "Status" paragraph is status text. (3) A directory listing showed seven sibling review files by name; none was opened. Nothing among the private working files, no evidence folder, no current-assessment or standing text of any claim, and no web search was used. A finite computation was run as a private sanity check of one step's conclusion; it is not evidence and no finding rests on it.

Restatement

Convention. The circle $S^1=\mathbb R/\mathbb Z$ has length $1$. For a sequence whose terms are distinct points of the circle and for $n\ge3$, list the first $n$ terms in cyclic order as $y_1,\ldots,y_n$ and read indices modulo $n$. For $1\le r\le n-2$ and each $i$, the $r$-span at $i$ is the closed arc from $y_i$ forward to $y_{i+r}$; its length is the sum of the $r$ consecutive gaps between $y_i$ and $y_{i+r}$, and it contains exactly the $r+1$ terms $y_i,\ldots,y_{i+r}$ and no other of the first $n$ terms. Write $M_n^r$ and $m_n^r$ for the largest and the smallest $r$-span. The point $0$ is not a break point: the page uses $n$ points and $n$ arcs, the convention of Problem 1221, and says so.

Imported input (Theorem 3, p. 2, in the circle form of the remark on p. 3, read for $R\ge2$). For each of the two sequences, the base-$2$ van der Corput sequence and the Kronecker sequence $x_k=\{k\varphi\}$ with $\varphi=(1+\sqrt5)/2$, there is a constant $c>0$, independent of $R$ and $n$, such that for every integer $R\ge2$ there is a threshold $n_0(R)$ with the following property: for every $n\ge n_0(R)$ and every closed arc $I$ of the circle of length $R/n$,

$$\bigl|\#\{1\le k\le n:\ x_k\in I\}-R\bigr|\le c\log R .$$

The page reads the source's "for all $r\in\mathbb N$" as $r\ge2$, because the instance $R=1$ is false for both sequences (its Source notes), and the review adopts that reading.

Reconstructed claim (Theorem 2, p. 2, for $r\ge2$). For either sequence there is a constant $c'<\infty$, depending only on $c$, such that for every integer $r\ge2$ there is a threshold $n_1(r)$, depending on $r$ and $c$, with

$$\frac{M_n^r}{m_n^r}\le1+\frac{c'\log r}{r}\qquad\text{for every }n\ge n_1(r).$$

Consequence recorded on the page. With $\mu_r$ the infimum over all sequences of $\limsup_n M_n^r/m_n^r$, either witness gives $\mu_r\le1+c'\log r/r$, that is $r(\mu_r-1)\le c'\log r$, for every $r\ge2$; the same bound holds over the family of distinct-point sequences, since both witnesses have distinct terms.

Checklist

• Quantifiers and scope. The page makes every quantifier of the source explicit ($c$, then $r$, then the threshold, then $n$, then the arc), and labels its restriction to $r\ge2$ and the reason. Fails in one place: the lower-bound step applies the imported input at $R^-=1$, outside the range $R\ge2$ the page itself declares, for every $r<2+c\log2$, which always includes $r=2$ (F1). Passes elsewhere.
• Circularity. None. Theorem 3 is a different statement from Theorem 2, imported as such; nothing equivalent to the claim is assumed.
• Model and convention changes. One convention change, disclosed: the page works with $n$ points and $n$ arcs, while the source counts $n+1$ pieces. The transfer is not the issue (the derivation is carried out entirely in the page's convention with the circle form of Theorem 3), but the page's description of the source's convention is inaccurate, and the page does not say that the source's version follows by the same two steps (F2, suggested).
• Finite and statistical overreach. None. The two boundary witnesses are exact and refute only the $r=1$ instances they name; no finite case is used as a proof of anything infinite.
• Uniformity. Passes for $r\ge2+c\log2$: $c'$ depends only on $c$, the threshold $n_1(r)$ depends on $r$ and $c$ as the source allows, and the case split at $r_1(c)$ keeps one constant for all $r$. The constant for the finite range $2\le r<2+c\log2$ rests on the defective step (F1).
• Extremal conclusions. Inapplicable beyond the ratio itself: the page claims an upper bound, not sharpness or attainment, and the strict inequality $M_n^r/m_n^r<R^+/R^-$ is derived in the claim's own units (lengths, then their ratio).
• Consequences and composition. The passage from the two span bounds to the ratio, the asymptotics of $R^\pm$, the choice of $c'$ and the consequence $\mu_r\le1+c'\log r/r$ were each re-derived and hold (Weakest steps). The composition inherits the standing of Theorem 3, which the page names as an unreconstructed same-paper import. Fails in one consumed clause: the lower-bound step consumes Theorem 3 at $R=1$ at a strength the import does not have (F1). The sentence "Both fail for both sequences" is short one line for the Kronecker sequence (F3, note).
• Computation. Inapplicable: the page runs no computation and cites none.
• Reproduction. Inapplicable: the page states no rerun commands and no coverage claims.
• Source and verdict fidelity. Theorems 2 and 3 were checked clause by clause against p. 2, the circle-form remark against p. 3, and the derivation paragraph and its two label slips against p. 9; all locators (p. 2, p. 3, p. 9, Section 2 on pp. 3--9, Section 3 on pp. 9--11) are right. The standing sentence claims author-recorded work only. The one inaccurate characterization of the source is the piece-count parenthesis (F2); the citation of the Korsky lower bound drops its "for all sufficiently large $r$" (F5, note).

Weakest steps

W1, the lower bound "every $r$-span is longer than $R^-/n$". Re-derived: let $R$ be an integer with $R+c\log R\le r$, and let $n\ge n_0(R)$ with $n>r+1$; then $R\le r<n$, so $R/n<1$. Suppose an $r$-span $J$ has $|J|\le R/n$. Extend $J$ to a closed arc $I\supseteq J$ of length exactly $R/n$. $I$ contains the $r+1$ terms of $J$, so its count is at least $r+1$; Theorem 3 at $R$ bounds the count by $R+c\log R\le r$. Contradiction, so $|J|>R/n$. The step is valid exactly when Theorem 3 at $R$ is available, that is for $R\ge2$ under the page's reading. The page takes $R=R^-=\max\{R\in\mathbb N:R+c\log R\le r\}$ and notes that $R=1$ qualifies. $R^-\ge2$ holds if and only if $2+c\log2\le r$; for every $r<2+c\log2$, and in particular for $r=2$ whatever $c>0$ is, the page's $R^-$ is $1$ and the step invokes the false instance $R=1$. Composition: the step supplies the denominator of the ratio bound, and its failure on the finite range $2\le r<2+c\log2$ leaves the constant $c'$ unsupported there (F1).

W2, the upper bound "every $r$-span is shorter than $R^+/n$". Re-derived: $R^+=\min\{R\in\mathbb N:R-c\log R\ge r+2\}$ exists since $R-c\log R\to\infty$, and $R^+\ge r+2\ge4$, so Theorem 3 at $R^+$ is inside the reading. Take $n\ge\max(n_0(R^+),R^++1)$; then $R^+/n<1$ and $n\ge r+3$, so $r\le n-2$ and every $r$-span omits at least one term and is a proper arc. If an $r$-span $J$ has $|J|\ge R^+/n$, it contains a closed sub-arc $I$ of length exactly $R^+/n$ (the arc of that length starting at the left end of $J$); Theorem 3 at $R^+$ gives $I$ at least $R^+-c\log R^+\ge r+2$ terms, but $I\subseteq J$ and $J$ holds exactly $r+1$. Contradiction. The circle form of Theorem 3 is needed here, since $J$ or $I$ may pass through $0$; the page imports that form explicitly. Composition: supplies the numerator; sound.

W3, the asymptotics and the universal constant. Re-derived: with $x_0=\lceil r+2+2c\log r\rceil$ and $r$ large enough that $r+3+2c\log r\le r^2$, one has $x_0\le r^2$, so $c\log x_0\le2c\log r$ and $x_0-c\log x_0\ge r+2$; minimality gives $R^+\le x_0\le r+3+2c\log r$. With $x_1=\lfloor r-c\log r\rfloor\ge1$, $\log x_1\le\log r$ gives $x_1+c\log x_1\le r$, so $R^-\ge x_1\ge r-c\log r-1$. When also $r-c\log r-1\ge r/2$,

$$\frac{R^+}{R^-}\le\frac{r+3+2c\log r}{r-c\log r-1} =1+\frac{4+3c\log r}{r-c\log r-1}\le1+\frac{8+6c\log r}{r} \le1+\frac{(6c+12)\log r}{r},$$

the last step because $8\le12\log2$ ($12\log2=8.32$). For $2\le r<r_1(c)$ the ratio is below $R^+(r)/R^-(r)\le R^+(r)\le R^+(r_1)$, since $R^+$ is nondecreasing in $r$ (the defining set shrinks as $r$ grows), while $\log r/r\ge\log2/r_1$; so any $c'\ge\max(6c+12,\ R^+(r_1)r_1/\log2)$ works for all $r\ge2$. The arithmetic is correct. Composition: the three thresholds folded into $r_1(c)$ are all eventually satisfied; the small-$r$ branch uses $R^-\ge1$, which is where F1 enters.

Strongest attack

The attack that succeeded targets W1 at small $r$. Fix $r=2$. For every $c>0$ the inequality $R+c\log R\le2$ fails at $R=2$ (it reads $2+c\log2\le2$), so the page's $R^-$ is $1$, and the lower-bound step reads: every closed arc of length $1/n$ contains at most $1+c\log1=1$ of the first $n$ terms. This is Theorem 3 at $R=1$, which the page's Source notes declare false and outside the reading. Witness (the page's own, checked here): for the van der Corput sequence with $n=2^k-1$ the first $n$ terms are the points $j/2^k$, $1\le j\le2^k-1$, because bit reversal permutes $\{1,\ldots,2^k-1\}$, and the closed arc $[2^{-k},2^{-k}+1/n]$ contains $2^{-k}$ and $2\cdot2^{-k}$ since $1/n>2^{-k}$ (p. 2 defines the sequence). For the Kronecker sequence and any $n\ge2$ the gaps are not all equal (equal spacing would put $\varphi=\{2\varphi\}-\{\varphi\}$ modulo $1$ in $\tfrac1n\mathbb Z$), so some gap is shorter than $1/n$ and the closed arc of length $1/n$ starting at its left endpoint contains two terms. The same defect occurs for every $r<2+c\log2$, and the source gives no value of $c$.

The gap cannot be closed from the imported input alone. Take $n-r$ equally spaced points and $r$ further points within a distance $\varepsilon$ of one of them. A closed arc of length $R/n$ holds between $R-1$ and $R+1$ grid points once $n\ge Rr$ (its length is $R-Rr/n$ grid spacings, and a closed arc of length $L$ holds $\lfloor L/s\rfloor$ or $\lfloor L/s\rfloor+1$ points of a grid of spacing $s$), plus at most $r$ cluster points, so the inequality of Theorem 3 holds at every $R\ge2$ whenever $r+1\le c\log2$, while one $r$-span has length at most $2\varepsilon$. Hence no argument that uses only the Theorem 3 inequalities for $R\ge2$, with $c$ unspecified, can bound the smallest $r$-span below for such $r$. Tightening the page's condition to the integer count, $\lfloor R+c\log R\rfloor\le r$ with $R\ge2$, rescues $r=2$ only when $c<1/\log2$, which the source does not provide.

What survives: the step's conclusion is nonetheless true for the van der Corput sequence, by its gap structure and not by Theorem 3. For $2^k\le n<2^{k+1}$ the terms $x_1,\ldots,x_{2^k-1}$ are the points $j/2^k$ and each later term $x_m$, $2^k\le m\le n$, is an odd multiple of $2^{-k-1}$ (its top bit lands in the $2^{-k-1}$ place), so every gap is $2^{-k-1}$ or $2^{-k}$ except the one through $0$, which is at least $2^{-k}$; hence every gap is at least $2^{-k-1}\ge1/(2n)$ and every $r$-span is at least $r/(2n)$. The source states the two gap lengths on p. 4 without proof. For the Kronecker sequence the corresponding bound rests on the three-gap structure of $\{k\varphi\}$, cited by the source on p. 9 to its references [18, 19, 22] and not verified here. Either way the range $2\le r<2+c\log2$ needs an input that the page does not import.

Attacks that failed: (a) an $r$-span passing through $0$ (the circle form of Theorem 3 covers it, and the page imports that form); (b) the count "exactly $r+1$" (both sequences have distinct terms: bit reversal is injective, and $\{k\varphi\}$ repeats only if $\varphi$ is rational); (c) the ratio's strictness and the monotonicity of $R^+$ (both hold); (d) dependence of $c'$ on the sequence (the claim is existential in the sequence, so one constant per witness suffices, and the page's $c'$ depends on $c$ alone).

Premises

• Theorem 3 in circle form (same source). Interface: for either sequence, a constant $c>0$ and thresholds $n_0(R)$ such that for every $R\ge2$, $n\ge n_0(R)$ and closed arc $I$ of length $R/n$, $|\#\{k\le n:x_k\in I\}-R|\le c\log R$. Source held: the printed theorem on p. 2 takes $I=[x,x+R/n]$ with $0\le x\le1-R/n$ and "for all $r\in\mathbb N$"; the first sentence of p. 3 says the argument shows the bound for all intervals of length $r/n$ on $S^1$. Reading depth: statement and remark clause by clause on the page images; proofs not read. Standing: author-recorded import from an unrefereed preprint, the circle form resting on the authors' remark rather than a displayed statement; the page names it as an unreconstructed import. Explicit assumptions: $c$ is unspecified; the base of the logarithm is immaterial (it changes $c$ only); the instance $R=1$ is excluded.
• Distinctness of the terms. Used for "exactly $r+1$ points"; elementary and checked above; the page states it only implicitly through "cut the circle into $n$ arcs".
• Definitions of Problem 1221. $M_r$, $m_r$ as the largest and smallest sums of $r$ consecutive gaps on the circle, and $\mu_r=\inf_a\limsup_n M_r(a)/m_r(a)$ over all sequences; read from the problem page's statement paragraph, together with its sentence that the third expression is the same under both readings. Standing: the problem statement as filed.
• Korsky's Theorem 1.1, ratio part. Interface as read in the ko26b reconstruction's Statement section: $\limsup_n M_n^{(r)}/m_n^{(r)}\ge 1+\log r/(100r)$ for every sequence of distinct points and every $r\ge r_0$, hence $\mu_r-1\ge\log r/(100r)$ for all sufficiently large $r$. Standing: claimed and unreviewed; the page cites it only as the "matching claimed lower bound", which is the right register (F5 on the dropped range).

Findings

F1. Severity: required. Location: "$R^-$ exists because $R=1$ qualifies" and "Theorem 3 at $R^-$ gives $\#\{k\le n:x_k\in I\}\le R^-+c\log R^-\le r$". Defect: for every $r<2+c\log2$, always including $r=2$, the page's $R^-$ equals $1$ and the lower-bound step applies Theorem 3 at $R=1$, an instance outside the page's own reading ($r\ge2$) and false for both sequences; the small-$r$ branch of the constant ("the ratio is at most $R^+(r)/1$") inherits the gap. Witness: the page's boundary witness, van der Corput with $n=2^k-1$ and the arc $[2^{-k},2^{-k}+1/n]$ holding two terms (sequence defined on p. 2; Theorem 3's range "for all $r\in\mathbb N$" on p. 2); for the Kronecker sequence any gap shorter than $1/n$. The Strongest attack section shows the range cannot be covered by Theorem 3 at $R\ge2$ alone. Proposed replacement: define $R^-=\max\{R\in\mathbb N:\ R\ge2,\ R+c\log R\le r\}$, which exists exactly when $r\ge2+c\log2$, and state that the derivation from Theorem 3 covers every $r\ge2+c\log2$. Then add, as a labeled supplied input, the finite range: "For $2\le r<2+c\log2$ Theorem 3 gives no lower bound on the smallest $r$-span, since its inequalities at $R\ge2$ permit $r+1$ terms in an arbitrarily short arc when $r+1\le c\log2$. The bound there uses the smallest gap instead: for the van der Corput sequence every gap among the first $n$ terms is at least $1/(2n)$ (the two gap lengths $2^{-k}$ and $2^{-k-1}$ for $2^k\le n<2^{k+1}$, stated on p. 4 of the source and checked here), so every $r$-span is at least $r/(2n)$ and the ratio is at most $2R^+(r)/r$, a constant on the range; for the Kronecker sequence the same follows from a minimum-gap bound $\delta/n$ of the three-gap structure, an external import not held here." If the page prefers not to import the Kronecker bound, restrict the reconstructed Statement to $r\ge2+c\log2$ and record the finite range as not derived.

F2. Severity: suggested. Location: "the source's introduction counts $n+1$ pieces of a broken stick $[0,1]$". Defect: the source's abstract and Section 1.1 (p. 1) say the circular stick $S^1$ is broken into $n+1$ pieces, and Section 2 (p. 4) sets $x_0=0$ and reads indices cyclically; the source's count comes from treating $0$ as an extra break point on the circle, not from a stick $[0,1]$. The page's $n$-point convention is a disclosed reading, but the source is characterized inaccurately, and the page does not record that the source's version follows by the same argument. Witness: p. 1, abstract ("the 'circular stick' $S^1$ is broken into a total of $n+1$ pieces") and Section 1.1 ("we have $n+1$ intervals"); p. 4 ("It will be convenient to set $x_0=0$"). Proposed replacement: "(the source's abstract and introduction count $n+1$ pieces of the circular stick after $n$ breaks, and its Section 2 sets $x_0=0$, so the source treats $0$ as an extra break point; here $0$ is not a break point, the convention of Problem 1221. With $0$ added, an $r$-span among the $n+1$ points contains $r$ or $r+1$ of the first $n$ terms, and the two steps below go through with $R^-+c\log R^-\le r-1$ in place of $\le r$.)"

F3. Severity: note. Location: "Both fail for both sequences" and "for the Kronecker sequence the $n$ gaps are never all equal". Defect: the Kronecker witness as written refutes Theorem 2 at $r=1$ only; the refutation of Theorem 3 at $r=1$ for that sequence needs one more sentence. Witness: the gaps sum to $1$ and are not all equal, so one is shorter than $1/n$, and the closed arc of length $1/n$ starting at its left endpoint contains two terms. Proposed replacement: append "so some gap is shorter than $1/n$ and the closed arc of length $1/n$ from its left endpoint contains two terms" after "rational".

F4. Severity: note. Location: "the definitions of $R^\pm$ and the threshold $n_1(r)$ are the corpus's completion". Defect: the paragraph "Asymptotics of $R^\pm$", the case split at $r_1(c)$ and the choice of $c'$ are also supplied by the corpus (the source's paragraph on p. 9 ends at "This implies Theorem 2" with no passage from the two bounds to $1+c\log r/r$), but the label names only the definitions and the threshold. Proposed replacement: "the definitions of $R^\pm$, the threshold $n_1(r)$, the asymptotics of $R^\pm$ and the choice of $c'$ are the corpus's completion."

F5. Severity: note. Location: "the matching claimed lower bound $\mu_r-1\ge\log r/(100r)$ is the ratio part of". Defect: the cited statement holds for all sufficiently large $r$ (an unspecified threshold $r_0$), which the sentence drops. Witness: the Statement section of the ko26b Theorem 1.1 reconstruction ("for all sufficiently large $r$"). Proposed replacement: "the matching claimed lower bound $\mu_r-1\ge\log r/(100r)$ for all sufficiently large $r$ is the ratio part of".

Verdict

Source fidelity: faithful with corrections. The statements of Theorems 2 and 3, the circle-form remark, the derivation paragraph with its two label slips, and every page and section locator match the artifact; the one inaccurate characterization is the piece-count parenthesis (F2), and the standing sentence claims author-recorded work only.

The argument as reconstructed: defective at the step "Every $r$-span is longer than $R^-/n$", for every $r<2+c\log2$ (always including $r=2$), where it applies Theorem 3 at $R^-=1$, an instance the page itself records as false; sound for every $r\ge2+c\log2$, where both span bounds, the asymptotics of $R^\pm$ and the universal constant were re-derived and hold. The reconstructed statement is not refuted: on the defective range its conclusion follows for the van der Corput sequence from the gap structure derived above, and for the Kronecker sequence from a minimum-gap bound not verified here; neither is derived from Theorem 3, so the page must import or restrict (F1).

Limitations. Theorem 3 and its circle form were taken as the source states them and not reviewed; the Kronecker minimum-gap bound was not verified; the sibling reconstruction was read at its Statement section only; this review covers one page and its inputs in the frozen state. This focused review assigns no tier and changes no status.