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Subject and independence

Role: independent reviewer in a fresh context, commissioned for refutation and given only the assignment. The reviewer took no part in writing the page, had no contact with its author, and read no other review of it.

Frozen subject: wiki/research/erdos_1221/ko26b_lemma_4_2_reconstruction.md as it stood on 2026-09-28T05:03:27Z, read from the committed text. The path read is the page reviewed. The page is the Lemma 4.2 reconstruction.

Artifact: the PDF beside the library card Korsky 2026, resolution, arXiv:2609.07196v2, 16 pages; physical page $k$ carries printed page $k$. Read from the text layer, clause by clause against the page: pp. 7--8 (Section 4: the definition of $H_L$, Theorem 4.1, its derivation from Larcher's proof, Lemma 4.2 and its proof). Read from the text layer at ordinary depth: pp. 1--2 (notation and Theorem 1.1), pp. 4--6 (the definitions of $P_t$ and $N_t$, hypothesis (2.1), Lemma 2.1 and Proposition 3.1), p. 9 (Section 5) for the Role paragraph, and pp. 15--16 (acknowledgments and references [9] and [11]) for the citations of Schmidt and Larcher. Page images were rendered at 130 dpi for pp. 6--9; pp. 7 and 8 were read as images for every displayed formula. The canonical conversion beside the PDF was read for Section 4 and agrees with the PDF there; the PDF decided every reading.

Allowed material actually read: the card's provenance paragraph; the Statement sections of the sibling pages Lemma 2.1 (with its Definitions section, which the page imports), Theorem 1.1 and Proposition 3.1 (named in the page's Role paragraph), each from the committed text of the same state; the Statement paragraph of Problem 1221; the sections "Report contract" (which holds the whole-claim report rules) and "Audit checklist" of docs/verification.md, the section "Source fidelity" of docs/evidence.md, and docs/math_authoring.md.

Exposures, disclosed: the whole card index was printed, so its "Read status" and "Relation to Problem 1221" sections (standing and acceptance text) reached the reviewer; the Status paragraph of the problem page was printed together with its Statement; the Source and Standing paragraphs of the three sibling pages were printed with their Statement sections; and a listing of library folder names matching "larcher" or "schmidt" was taken to check the page's "not held" sentence (names only, no content; no Larcher folder exists, and a folder named schmidt_1972_irregularities_distribution exists). None of this bears on the mathematics checked below. No web search was made and no evidence folder was read.

Restatement

Let $(x_n)_{n\ge1}$ be a sequence of distinct points of $\mathbb T=\mathbb R/\mathbb Z$, $P_n=\{x_1,\ldots,x_n\}$, and $N_n(I)=\#(P_n\cap I)$ for an oriented half-open arc $I=(x,x+\lambda]$ with $\lambda<1$. For a finite list $z_1,\ldots,z_L\in[0,1)$ the maximum prefix counting error is

$$H_L(z_1,\ldots,z_L)=\max_{1\le j\le L}\ \sup_{0\le u\le1}\ \bigl|\#\{i\le j:\ z_i<u\}-ju\bigr| ,$$

with the strict convention $z_i<u$ and the supremum over the closed range $0\le u\le1$.

Imported input (Theorem 4.1 of the source). There is an absolute integer $L_0$ such that for every integer $L\ge L_0$ and every list $z_1,\ldots,z_L\in[0,1)$, $H_L(z_1,\ldots,z_L)\ge\frac1{16}\log L$. The page states it as used and does not verify it; the source derives it from a finite-list bound it attributes to Section 3 of Larcher's paper.

Claim (Lemma 4.2 of the source). Let $B\ge1$ and $S\ge2$ be real numbers, and suppose there is an integer $n_0$ such that for every integer $n\ge n_0$, every $x\in\mathbb T$ and every real $D$ with $0\le D\le S$,

$$\bigl|N_n\bigl((x,x+D/n]\bigr)-D\bigr|\le B .$$

If $\lfloor S\rfloor\ge L_0$, then $B\ge\frac1{16}\log\lfloor S\rfloor$. The conclusion involves neither $n_0$ nor the sequence; the hypothesis is used only at integer times $n\ge n_0$ and only on arcs of length less than $1$. The sequence must be infinite and its points distinct.

Checklist

Canonical failure modes:

• "Almost all" quietly upgraded to "all": absent. The hypothesis holds for $n\ge n_0$ and the proof applies it only at insertion times $n\ge n_0$; earlier times are handled by the separate early-prefix case, which uses $B\ge1$ instead.
• Induction that presupposes termination: inapplicable; there is no induction.
• Probabilistic or averaging heuristics presented as proofs: absent. The one averaging step (an $L$-span of length at most the mean $L/N$) is a finite pigeonhole with the mean computed exactly.
• Circular use of a statement equivalent to the claim: absent. The only input is Theorem 4.1, which concerns lists, not point sequences on the circle.
• Exceptional sets dropped from density arguments: inapplicable; no density argument.
• Finite verification cited as more than base-case coverage: absent. The only computation is the arithmetic of $c_{7/2}$, which the page labels as arithmetic.
• Convergence of a relaxed or averaged system standing in for the actual objects: inapplicable.

Named patterns:

• Model-class transport instead of entailment: inapplicable; no axiom system or certificate class.
• Uniformity over an infinite family asserted from finitely many instances: absent. $L_0$ is stated absolute; in the derivation of Theorem 4.1 the additive $O_a(1)$ is bounded by $\log(2a)$ for $L\ge2a$ (re-derived below) and is independent of the list, as the page says.
• Extremal claims audited in the claim's own units: inapplicable; no sharpness or attainment sentence.
• Consequence sentences are claim surfaces: checked. "Hence $H_L\le B$, and Theorem 4.1 ... gives $B\ge\frac1{16}\log L$" holds for $L=\lfloor S\rfloor\ge L_0$; the Role sentence matches p. 9 of the source; the remark's "that form suffices for $r(\mu_r-1)\to\infty$" is correct given the rest of the source's Section 5 (with $A=\kappa\log r$, $B=3\kappa\log r+O(1)$ against $\frac c2\log r-O(\log\log r)$ contradicts for $\kappa<c/6$).
• Carry hypotheses actually used: checked. $B\ge1$ is stated and used in the one-point prefix; distinctness is stated on the imported definitions page and used for $\delta>0$ and for the exact count $L$; $S\ge2$ is stated and idle beyond $L\ge2$.
• A composition inherits its unproved premises: the page discloses that Theorem 4.1 is unverified and that Larcher's paper is not held; the conclusion is not presented as unconditional. Finding F2 asks the same disclosure for the remark's use of Schmidt's theorem.
• Reproducibility notes are claims: inapplicable; no rerun line.
• Verifier quotations are claims: inapplicable; the page quotes no verifier.
• Verdict words spelled in full: inapplicable to the page; this report carries no such verdict.
• Certified-bracket functions fail loudly: inapplicable; no numerics.
• A harness leg with no failing input is decoration: inapplicable; no harness.
• A gate that reads caches instead of re-running is defective: inapplicable; no gate.

Weakest steps

W1. The window $J$. $P_N$ has $N>L$ distinct points. The $L$-span from a point is the clockwise distance to the point $L$ places later. A gap between consecutive points lies in the spans starting at the $L$ points before its right end, which are distinct because $L<N$, so the $N$ spans sum to $L$ and one, from $p$ say, has length $\ell\le L/N<1$. The points after $p$ sit at clockwise distances $0<d_1<\cdots<d_L=\ell<d_{L+1}$ (with $d_{L+1}=1$ when $N=L+1$), so $(p,p+\ell]$ holds exactly $L$ points and $\ell>0$. For $0<\varepsilon<\min(d_1,\,d_{L+1}-\ell)$ the arc $J=(p+\varepsilon,p+\varepsilon+\ell]$ holds the same $L$ points, no other, and neither endpoint is a point. Two points of $P_{n_0}$ inside $J$ would be at circular distance at most $\ell<\delta$, so $|J\cap P_{n_0}|\le1$. This composes with the rest by supplying the list, the bound $N\ell\le L$, and the early-prefix count.

W2. Prefixes and the early case. With the points of $J$ listed as $x_{m_1},\ldots,x_{m_L}$, $m_1<\cdots<m_L\le N$, and $n=m_j$: since $P_n\subseteq P_N$, $P_n\cap J=\{x_{m_i}:m_i\le m_j\}$ is exactly the first $j$ listed points, so $N_n(J)=j$ and $\#\{i\le j:z_i\le u\}=N_n((a,a+u\ell])$ for $0\le u\le1$, the case $u=0$ giving $0=0$ because every $z_i>0$. If $n<n_0$ then all $m_i\le m_j<n_0$, so the first $j$ points lie in $P_{n_0}\cap J$ and $j\le1$; a one-point list has error $\sup_u|\mathbf 1[z_1<u]-u|\le\max(u,1-u)\le1\le B$. This is the only place $B\ge1$ is used.

W3. The convex combination and the convention. For $n\ge n_0$ and $0\le u\le1$, the arcs $(a,a+u\ell]$ and $(a+u\ell,a+\ell]$ are disjoint with union $J$ and have $n\times$length equal to $nu\ell$ and $n(1-u)\ell$, both at most $n\ell\le N\ell\le L\le S$; the hypothesis at time $n$ gives $|f(u)|\le B$ and $|(j-N_n((a,a+u\ell]))-n(1-u)\ell|=|f(1)-f(u)|\le B$. Then $N_n((a,a+u\ell])-ju=f(u)+n\ell u-ju=f(u)-uf(1)=(1-u)f(u)-u(f(1)-f(u))$, whose absolute value is at most $(1-u)B+uB=B$ because $u,1-u\ge0$. For the strict convention, $\#\{i\le j:z_i<u\}$ is the left limit at $u$ of $\#\{i\le j:z_i\le u'\}$ for $u>0$, both are $0$ at $u=0$, and both are $j$ at $u=1$ because every $z_i<1$; the function $ju$ is continuous, so the two suprema over $[0,1]$ coincide. Hence $H_L\le B$, and Theorem 4.1 at $L\ge L_0$ closes.

Strongest attack

The attack aimed at the range of the hypothesis. The transfer needs the counting hypothesis at time $n$ on every sub-arc of $J$, so it needs $n\cdot|J|\le S$ at every insertion time $n\le N$; the largest value is $N\ell$. Had the window been any arc holding $L$ points, $N\ell$ could exceed $S$ (a window with $L$ points and length $(L+1)/N$ already breaks it when $S=L$), and the prefix bound would fail at $u$ near $1$. The attack fails because the window is the shortest $L$-span, $\ell\le L/N$, so $N\ell\le L=\lfloor S\rfloor\le S$ with no slack needed; the choice $L=\lfloor S\rfloor$ rather than $\lceil S\rceil$ is exactly what makes the hypothesis available.

A second attack tried to make an insertion time $n=m_j$ with $n<n_0$ and $j=2$, which would need two of the listed points inside $P_{n_0}$; it fails because $\delta$ is fixed before $N$ and $\ell<\delta$ forbids two points of $P_{n_0}$ in $J$. A third tried a listed point on an endpoint of $J$, which would give $z_i\in\{0,1\}$ and break the count identity at $u=0$ or the convention switch at $u=1$; the forward shift by $\varepsilon$ excludes it.

On fidelity, every clause of the Statement, of Theorem 4.1 as used, of the derivation and of the proof was compared with pp. 7--8; the one defect found is the locator of the derivation (F1).

Premises

• Theorem 4.1 (finite-prefix discrepancy bound). Interface: for every integer $L\ge L_0$ and every list in $[0,1)$, $H_L\ge\frac1{16}\log L$, $L_0$ absolute. Held source: the Korsky PDF, p. 7 for the statement and p. 8 for the derivation, read clause by clause; the page states it verbatim and applies it with $L=\lfloor S\rfloor\ge L_0$ and $z_i\in(0,1)\subset[0,1)$, so its hypotheses are met. Its standing is named as imported and unverified. Explicit assumptions behind it, as the source states them: Larcher's Section 3 proves $H_N\ge c_a\log N$ for every list of length $N=\lfloor a^h\rfloor$, $3<a<4$, with $c_a=(a-2)(8a+3)/(16(1-2a)^2\log a)$. Not held; not checked. The derivation from that assumption was re-derived here: for $a=7/2$, $c_a=\tfrac32\cdot31/(576\log3.5)=31/(384\log3.5)=0.06444\ldots>1/16$; the largest $N=\lfloor a^h\rfloor\le L$ satisfies $a^{h+1}>L$, so $N>L/a-1\ge L/(2a)$ for $L\ge2a$ and $\log N\ge\log L-\log(2a)$; $H_L\ge H_N$ because $H_N$ is the same maximum restricted to the first $N$ prefixes; so $H_L\ge c_a\log L-c_a\log(2a)\ge\frac1{16}\log L$ once $(c_a-\tfrac1{16})\log L\ge c_a\log(2a)$, an absolute threshold. One consistency observation, from the reviewer's recollection and verified against no held source: the supremum of $c_a$ over $3<a<4$, computed here, is $0.0646363\ldots$ at $a\approx3.719$, which agrees with the constant the reviewer recalls from Larcher's abstract; this supports the transcription of the formula and says nothing about the finite-list form.
• Schmidt's planar theorem (1972). Used only in the page's authored remark. Interface as quoted on the page: every $N$-point set in $[0,1]^2$ has an origin-anchored box whose count differs from $Nuv$ by at least $c\log N$, $c>0$ absolute. Not read here; the source cites it as [9] without using it. The deduction on the page from that statement to $H_L\ge c\log L-1$ was re-derived: the set $\{(z_i,i/L)\}$ has count $\#\{i\le j:z_i<u\}$ in $[0,u)\times[0,v]$ with $j=\lfloor Lv\rfloor$, and $|ju-Luv|=u(Lv-\lfloor Lv\rfloor)<1$. See F2 and F5.
• Definitions of $P_n$, $N_n$ and distinctness. From the Lemma 2.1 page's Definitions section (read) and the source's Section 2 (p. 4, read); the page uses them at integer times only.
• Proposition 3.1 and Section 5 (source pp. 6 and 9, read; the Proposition 3.1 page's Statement section, read): consumed only by the Role paragraph, which reports them correctly ($B=3A+C_1A/\Lambda$, $S=\sqrt{Ar}/\Lambda^2$, and $\tfrac3{100}\log r$ against $\tfrac1{32}\log r$).

Findings

F1. Severity: required. Location: Source paragraph, "Theorem 4.1 (p. 7, with its derivation from Larcher's proof)". Defect: the locator places the derivation on p. 7; the statement of Theorem 4.1 is the last item on p. 7 and the paragraph "Derivation from Larcher's proof" opens p. 8 (physical and printed), above Lemma 4.2. Witness: PDF p. 7 ends with the display $H_L(z_1,\ldots,z_L)\ge\frac1{16}\log L$ and p. 8 begins "Derivation from Larcher's proof. Section 3 of [11] ...". The subheading "The source's derivation, as stated" carries no locator, so nothing corrects the reader. Proposed replacement: "Section 4: Theorem 4.1 (p. 7), its derivation from Larcher's proof (p. 8) and Lemma 4.2 (p. 8) of the retained PDF", and "The source's derivation, as stated (p. 8).".

F2. Severity: suggested. Location: the authored remark, "follows from Schmidt's theorem for planar point sets (W. M. Schmidt, ...)". Defect: the remark imports a theorem whose standing is not named; the page says neither whether Schmidt's paper is held nor that only the deduction, not the quoted statement, is what "checked here" covers, while the Standing paragraph names only Larcher's paper as not held. Witness: the page's Standing paragraph and the remark; the source (p. 16, [9]) cites the paper without stating its theorem. Proposed replacement: after the citation, "quoted from the literature and not read here; what is checked is the deduction from that statement", or, if the library holds the paper, a link to its card with the read status.

F3. Severity: suggested. Location: "The source's derivation, as stated": "restrict to its prefix of the largest such length $N\le L$; then $H_L\ge H_N$ (a maximum over fewer prefixes)". Defect: under a heading that promises the derivation as stated, "its prefix" is a reading of the source's "restrict it to the largest such $N\le L$" and the parenthetical is a supplied justification, neither marked. Both are correct (a prefix is the only restriction for which $H_L\ge H_N$ holds as written). The Proof section likewise supplies, unmarked, the sentence "(each gap lies in exactly $L$ of them)", the $\varepsilon$-shift details, and "the points of $P_n$ in $J$ are exactly the first $j$ listed points"; all three were re-derived above and hold. Witness: PDF p. 8, lines "Given an arbitrary list of length $L$, restrict it to the largest such $N\le L$" and "$H_L\ge H_N\ge c_a\log L-O_a(1)$". Proposed replacement: "restrict it to the largest such $N\le L$ (read here as the prefix of that length; then $H_L\ge H_N$, the same maximum over fewer prefixes, a justification supplied here)", and one sentence at the head of the Proof: "Parenthetical justifications and the $\varepsilon$-shift details are supplied here."

F4. Severity: note. Location: frontmatter desc, "intervals holding at most $S$ points". Defect: (4.1) constrains arcs whose expected count $D$ is at most $S$; such an arc may hold up to $S+B$ points. The body states (4.1) correctly. Witness: PDF p. 8, display (4.1), "$0\le D\le S$". Proposed replacement: "intervals of expected count at most $S$".

F5. Severity: note. Location: the authored remark, "has a box anchored at the origin ... $[0,u)\times[0,v]$". Defect: the box is half-open in $u$ and closed in $v$, a mixed convention; Schmidt's theorem in any one convention yields the same supremum in every other by one-sided limits, so the remark's conclusion $H_L\ge c\log L-1$ stands as a supremum statement, but the page does not say why the convention may be mixed. Witness: the remark's own text. Proposed replacement: add "(the supremum is the same for every endpoint convention, by one-sided limits)".

Verdict

Source fidelity: faithful with corrections. One correction is required (F1, a locator); the statement, Theorem 4.1 as used, the derivation and the proof match pp. 7--8 clause by clause, and the page neither strengthens nor silently alters what the source proves.

The argument as reconstructed: sound, given Theorem 4.1 as an imported input. Every deduction from the hypothesis (4.1) to $H_L\le B$ was re-derived and holds; the derivation of Theorem 4.1 from Larcher's finite-list bound is valid as an implication.

Limitations: Theorem 4.1 rests on a finite-list bound attributed to Larcher's Section 3, which is not held and was not checked, as the page discloses; Schmidt's paper, cited only in a remark, was not read; the reviewer's consistency observation on $c_a$ is a recollection and no warrant. The exposures listed above did not touch the mathematics checked.

This focused review assigns no tier and changes no status.