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Subject and independence

The reviewer is an independent reader commissioned for this one page in a fresh context, given only the assignment, who took no part in writing the page or any page of its folder and had no access to the author's working notes. The charge was refutation.

Frozen subject: wiki/research/erdos_1221/ko26b_lemma_6_2_reconstruction.md as it stood on 2026-09-28T05:03:27Z, read from the committed text.

Artifact: the retained PDF of S. Korsky, A resolution of the de Bruijn--Erdős consecutive-gap problem, arXiv:2609.07196v2 (16 pages; the physical page numbers equal the printed ones), held under Korsky 2026, resolution. Physical pages 10--12 were read in full, in the text layer and in page images rendered at 150 dots per inch; every display on pages 10--11 that the subject uses ((6.1)--(6.6), the definitions of $\Delta_t$ and $Z_t$, the equation for $s(u)$, the injection $T_u$, the lift identity and the four unnumbered displays of the proof of Lemma 6.2) was checked against the image. Physical pages 4--5 were read in the text layer, and page 4 also in its page image, for the Section 2 conventions and the upper-bound half of the proof of Lemma 2.1, which the subject's proof borrows. Page images were rendered for pages 4, 5, 10, 11 and 12; those of pages 4, 10, 11 and 12 were read. The canonical conversion beside the PDF was read for its Section 6 block from the definition of $\Delta_t$ through the end of the proof of Lemma 6.2 and agrees with the PDF at every display read; the PDF decided.

Allowed material actually read: the subject page; the reconstruction pages of Lemma 6.1 and Lemma 2.1 in the same folder and the same state, in full, because the deduction of (6.6) applies Lemma 6.1 at real times (which needs the floor handling in its proof) and the subject delegates the range of $s(u)$ and the exchange of integrations to the Lemma 2.1 proof; the card's provenance paragraph; the Statement paragraph of Problem 1221; the "Whole-claim report" and "Audit checklist" sections of docs/verification.md, the "Source fidelity" section of docs/evidence.md, and docs/math_authoring.md. The subject's Source paragraph links no result page of the card, so none was read.

Exposures, none bearing on the mathematics: the card's _index.md was displayed whole, so its Read status paragraph, Overview and "Relation to Problem 1221" section, which carry standing and acceptance wording, were seen; a structural listing of the problem page showed the first line of its Status, Claim, Supported status and Remaining gaps paragraphs; a directory listing of evidence/verify/ showed the file names of eight other review files, whose contents were not opened; the Lemma 6.1 and Lemma 2.1 pages were read whole, including their Standing paragraphs; a search of the canonical conversion returned one sentence from the proof of Proposition 6.4 (source text). Nothing among the private working files, no other review, no evidence code and no web search was consulted.

Restatement

Setting (source pp. 4 and 10). $(x_n)_{n\ge1}$ are distinct points of $\mathbb T=\mathbb R/\mathbb Z$; for real $t\ge1$, $P_t=\{x_1,\dots,x_{\lfloor t\rfloor}\}$ and $N_t(I)=|P_t\cap I|$, so the sets are nested. Arcs are oriented half-open $(x,x+\lambda]$ with $\lambda<1$, and lifts to $\mathbb R$ measure displacements. The integers $r$ and $k\ge1$ are fixed and $t$ is large enough that $kr<|P_t|$. For $p\in P_t$, $L_{t,k}(p)$ is the clockwise distance from $p$ to the point $kr$ places after it in the cyclic order of $P_t$; $F_t$ moves every point of $P_t$ forward by $kr$ places and $B_t=F_t^{-1}$. Hypothesis (6.1): a number $A\ge1$ is fixed, and one fixed alternative, $nM_n^{(r)}-r\le A$ or $r-nm_n^{(r)}\le A$, holds for every sufficiently large integer $n$. For $D\ge0$,

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

Claim (Lemma 6.2, p. 11). Under (6.1), for any real $D,E>0$ and integer $k\ge1$ with $q=E/(kr)<1$, and for every real $t$ beyond a threshold that may depend on $r$, $k$, $A$, $D$, $E$ and the sequence,

$$Z_t(D)\ \le\ \frac{(1+q)D}{E}\,Z_{(1+q)t}(E)+qD+8kA+\frac{4kr}{t} .$$

Convention: $Z_{(1+q)t}(E)$ is taken at the real time $t_+=(1+q)t$, so it counts the set $P_{\lfloor t_+\rfloor}$ in arcs of length $E/t_+$. The source's statement displays no time threshold; the section's standing conventions (the eventual hypothesis, $kr<|P_t|$, arcs shorter than one) supply one, and the page states it explicitly (F1). The hypothesis $q<1$ is carried from the source but is not used by the argument: $s(u)$ is defined and lies in $[t,t_+]$ for every $q>0$.

Checklist

• Quantifiers and scope. Pass, with F1. Hypotheses, the ranges of $D$, $E$ and $k$, the definition of $q$, the constants and the time arguments match p. 11. The page adds "for all sufficiently large $t$", the reading forced by the eventual hypothesis and the p. 10 convention, without marking it. The boundary cases $u=0$ (where $s=t$, $T_0$ is the identity and $\eta_0=0$) and $u=\ell$ (where $s=t_+$) are covered by the argument. No almost-all versus all issue arises.
• Circularity. Pass. The proof consumes Lemma 6.1, elementary measure theory and the source's Section 2 constructions; (6.5) is not invoked at another scale inside its own proof.
• Model and convention changes. Pass. The average over $u$ is the source's own device, not a substitution; real times, half-open arcs and lifts are the source's conventions; the page's explicit $R_u$ is one function satisfying exactly the two properties the source asserts of its $R_u$, and the page declares the expansion in its Standing paragraph.
• Finite and statistical overreach. Inapplicable: no finite verification, sampling or heuristic enters the argument.
• Uniformity. Pass. (6.6) is uniform in $u\in[0,\ell]$ because $s(u)\in[t,t_+]$ and Lemma 6.1's bound $2kA+kr/s$ decreases in $s$; one threshold on $t$ serves every $s(u)$ because $\lfloor s(u)\rfloor\ge\lfloor t\rfloor$ and (6.1) is eventual; the page claims no uniformity in $D$, and neither does the source's lemma.
• Extremal conclusions. Inapplicable: the claim bounds an integral; no infimum, supremum, attained value or sharpness is asserted.
• Consequences and composition. Pass. The interfaces consumed (Lemma 6.1 at $t$ and at each $s(u)$; the range of $s(u)$ and the exchange of integrations from the Lemma 2.1 page) are supplied at the strength used and were re-derived here. The Role section's consequence, that (6.4) turns a positive-mass bound into an $L^1$ bound at integer times, follows from the zero-mean identity re-derived under F4.
• Computation. Inapplicable: the page carries no code or numerics.
• Reproduction. Inapplicable: the page states no rerun commands or coverage claims.
• Source and verdict fidelity. Pass, with F3 and F4. Statement, constants and the labels (6.4)--(6.6) match p. 11; the Standing paragraph claims only an author-recorded reconstruction of an unrefereed preprint; the Source paragraph's page range over-includes p. 12, and the justification of (6.4) is a supplied expansion not marked as such.

Weakest steps

1. The transport bound (6.6). Fix $u\in[0,\ell]$ and let $s=s(u)=kr/(kr/t-u)$. Since $u\le\ell=E/((1+q)t)$, one has $ut/(kr)\le E/((1+q)kr)=q/(1+q)<1$, so $s$ is finite, and $s$ increases from $t$ at $u=0$ to $t/(1-q/(1+q))=(1+q)t=t_+$ at $u=\ell$; this uses only $q>0$. For $p\in P_t$ put $p'=F_t(p)\in P_t\subseteq P_s$ and $p''=B_s(p')\in P_s\subseteq P_{t_+}$, so $T_u(p)=p''$. By the definition of $L$, the clockwise distance from $p$ to $p'$ is $L_{t,k}(p)$, and, because $F_s(p'')=p'$, the clockwise distance from $p''$ to $p'$ is $L_{s,k}(p'')$; both lie in $(0,1)$ because $kr<|P_t|\le|P_s|$ and the points are distinct. Lifting $p$ to $\tilde p$ and taking the lift $\tilde p+L_{t,k}(p)-L_{s,k}(p'')$ of $T_u(p)$, the displacement is exact, and with $u=kr/t-kr/s$,

$$\eta_u(p)=\Bigl(L_{t,k}(p)-\frac{kr}{t}\Bigr) -\Bigl(L_{s,k}(p'')-\frac{kr}{s}\Bigr)$$

as an identity of real numbers. Summing over $p\in P_t$, the first bracket contributes at most $2kA+kr/t$ in absolute value by Lemma 6.1 at $t$. The map $p\mapsto p''$ is the composition of the bijection $F_t$ of $P_t$, the inclusion $P_t\subseteq P_s$ and the bijection $B_s$ of $P_s$, hence injective, so the $p''$ are distinct elements of $P_s$ and

$$\sum_{p\in P_t}\Bigl|L_{s,k}(p'')-\frac{kr}{s}\Bigr| \le\sum_{y\in P_s}\Bigl|L_{s,k}(y)-\frac{kr}{s}\Bigr|\le2kA+\frac{kr}{s} \le2kA+\frac{kr}{t}$$

by Lemma 6.1 at $s$ and $s\ge t$. The triangle inequality gives $\sum_p|\eta_u(p)|\le4kA+2kr/t$ with a right side independent of $u$. Check at $u=0$: $s=t$, $p''=B_tF_tp=p$ and $\eta_0=0$. Lemma 6.1 at $s$ needs (6.1) at $\lfloor s\rfloor$ and $kr<\lfloor s\rfloor$; both follow from the same conditions at $\lfloor t\rfloor\le\lfloor s\rfloor$ because (6.1) is eventual. This bound feeds step 2 and, doubled, bounds $\int R_u$.

2. Moving one atom and the pointwise inequality. For a point $y$ and fixed $u$, $y\in I_x+u=(x+u,x+u+D/t]$ holds exactly when $x\in[y-u-D/t,\,y-u)$ modulo $1$, an arc of length $D/t$ (this needs $D/t<1$). For $y=p+u$ the arc is $J_p=[p-D/t,p)$, whose indicator in $x$ is $\mathbf 1[p\in I_x]$; for $y=T_u(p)=p+u+\eta_u(p)$ the arc is $J_p+\eta_u(p)$. For an arc of length $\lambda$ and its rotation by a real number $\eta$ the symmetric difference has measure at most $2\min(\|\eta\|,\lambda)\le2|\eta|$, where $\|\eta\|$ is the circular distance; no smallness of $\eta$ is needed, which matters because under the second alternative of (6.1) a single $kr$-span need not be small. Hence $\int_{\mathbb T}|\mathbf 1[p\in I_x]-\mathbf 1[T_u(p)\in I_x+u]|\,dx\le2|\eta_u(p)|$, and summing over $p$ gives $\int_{\mathbb T}R_u\le2\sum_p|\eta_u(p)|\le8kA+4kr/t$. Pointwise, $N_t(I_x)=\sum_p\mathbf 1[p\in I_x]\le\sum_p\mathbf 1[T_u(p)\in I_x+u]+R_u(x)$, and the first sum is at most $N_{t_+}(I_x+u)$ because $T_u$ is injective into $P_{t_+}$, so the $T_u(p)$ are distinct points of $P_{t_+}$. This is the page's inequality $N_t(I_x)\le N_{t_+}(I_x+u)+R_u(x)$, which step 3 averages.

3. Averaging and positive parts. Put $R(x)=\ell^{-1}\int_0^\ell R_u(x)\,du$. As $u$ runs over $[0,\ell]$, $\lfloor s(u)\rfloor$ takes finitely many values, so $R_u(x)$ is a finite sum of indicators of sets that are arcs in $x$ on finitely many $u$-intervals; $R$ is measurable and Tonelli gives $\int_{\mathbb T}R=\ell^{-1}\int_0^\ell\int_{\mathbb T}R_u\le8kA+4kr/t$. For the exchange of integrations, fix $y\in P_{t_+}$: the set $\{u\in[0,\ell]:y\in I_x+u\}$ equals $\{u\in[0,\ell]:y-u\in I_x\}$; with $v=y-u$, the condition $u\in[0,\ell]$ reads $y\in[v,v+\ell]$, so the set has the measure of $\{v\in I_x:y\in(v,v+\ell]\}$ up to endpoints. Summing over $y$,

$$\int_0^\ell N_{t_+}(I_x+u)\,du=\int_{I_x}N_{t_+}\bigl((v,v+\ell]\bigr)\,dv =\int_{I_x}\bigl(\Delta_{t_+}(v,E)+E\bigr)\,dv ,$$

the last step because $\ell=E/t_+$. Dividing by $\ell$, the constant contributes $E|I_x|/\ell=E\,(D/t)\,(t_+/E)=(1+q)D$. Averaging the pointwise inequality of step 2 and subtracting $D$,

$$\Delta_t(x,D)\le qD+\frac1\ell\int_{I_x}\Delta_{t_+}(v,E)\,dv+R(x).$$

Positive parts: $z\le w$ implies $z_+\le w_+$; $(a+b+c)_+\le a_++b_++c_+$; $qD\ge0$ and $R\ge0$ are their own positive parts; and $(\int f)_+\le\int f_+$ because $\int f\le\int f_+$ and the right side is nonnegative. Integrating over $x\in\mathbb T$, which has measure $1$: $\int qD=qD$; the middle term is $\ell^{-1}\int_{\mathbb T}(\Delta_{t_+}(v,E))_+\,|\{x:v\in I_x\}|\,dv$, and $v\in(x,x+D/t]$ exactly when $x\in[v-D/t,v)$, of measure $D/t$, so the middle term equals $(D/t)(t_+/E)\,Z_{t_+}(E)=\frac{(1+q)D}{E}Z_{t_+}(E)$; and $\int R\le8kA+4kr/t$. This is (6.5) exactly as displayed on p. 11.

Strongest attack

The strongest attempt aimed at the transport bound. The quantity $\eta_u(p)$ mixes a $kr$-span at time $t$ with a $kr$-span at a different time $s$ whose point set is larger, Lemma 6.1 controls real deviations from $kr/\tau$, and the source's one-atom sentence speaks of a circular distance. If the lift of $T_u(p)$ implicit in $T_u(p)=p+u+\eta_u(p)$ could differ from the one used in the decomposition by an integer, or if the one-atom bound needed a small displacement, then (6.6) would not control $\int R_u$; and under the second alternative of (6.1) a single $kr$-span can be of size comparable to $A$, so $\eta$ is not small in general. The attack fails: the decomposition of $\eta_u(p)$ in step 1 is an identity of real numbers for the explicit lift $\tilde p+L_{t,k}(p)-L_{s,k}(p'')$, the page's one-atom paragraph uses that same $\eta$, and the symmetric difference of an arc and its rotation by a real $\eta$ is at most $2\|\eta\|\le2|\eta|$ with no smallness assumption. A second attempt targeted the second use of Lemma 6.1: if $p\mapsto B_s(F_t(p))$ had collisions, the sub-sum over $P_t$ could exceed the full sum over $P_s$; it has none, being two bijections around an inclusion. A third looked for a degenerate endpoint: at $u=\ell$, $s=t_+$ and $T_\ell=B_{t_+}\circ F_t$, still an injection, and at $u=0$ the inequality reduces to $N_t(I_x)\le N_{t_+}(I_x)$, true by nesting. A fourth asked whether the threshold on $t$ can be independent of $u$; it can, since $\lfloor s(u)\rfloor\ge\lfloor t\rfloor$ and (6.1) is eventual. A fifth checked whether dropping the unused hypothesis $q<1$ could hide a use; it does not, since $s(u)$ stays in $[t,t_+]$ for every $q>0$. The page survives.

Premises

• Lemma 6.1 (source p. 10, display (6.2); held; read in the text layer and the page image; the folder's reconstruction page read in full, including its proof, which was found consistent with the source at that depth but not independently reviewed). Interface used: for every sufficiently large real $\tau$, $\sum_{p\in P_\tau}|L_{\tau,k}(p)-kr/\tau|\le2kA+kr/\tau$, applied at $\tau=t$ and at $\tau=s(u)$ for every $u\in[0,\ell]$; the page's version matches the source's. Its standing is author-recorded per its own Standing paragraph; the subject names it as imported ("The span input is Lemma 6.1").
• Proof of Lemma 2.1, upper-bound half (source pp. 4--5; held; text layer read, p. 4 image read; the folder's reconstruction page read in full). Interfaces borrowed by the subject: $t\le s(u)\le t_+$, the injectivity of $T_u$, and the exchange $\int_0^\ell N_{t_+}(I_x+u)\,du=\int_{I_x}N_{t_+}((v,v+\ell])\,dv$. All three were re-derived above, so the dependence is not load-bearing. The subject does not link this page (F2).
• Hypothesis (6.1) with $A\ge1$ (source p. 10), consumed only through Lemma 6.1.
• Section conventions (source p. 4): real times with $P_t=P_{\lfloor t\rfloor}$, nested point sets, oriented half-open arcs of length less than one, lifts to $\mathbb R$ for displacements.
• Standard facts, not held and elementary: Tonelli's theorem for nonnegative measurable functions on $[0,\ell]\times\mathbb T$; subadditivity of the positive part; $(\int f)_+\le\int f_+$; the symmetric-difference bound for an arc and its rotation; rotation invariance of Lebesgue measure on $\mathbb T$.
• Explicit assumptions on $t$: (6.1) holds, in the fixed alternative, at every integer $n\ge\lfloor t\rfloor$; $kr<\lfloor t\rfloor$; $D/t<1$; $E/t_+<1$. No batch acceptance order applies.

Findings

F1. Severity: suggested. Location: Statement, "then for all sufficiently large $t$". Defect: a reading not marked as such. Witness: the source's Lemma 6.2 (p. 11) reads "If $q<1$, then (6.5)" with no time quantifier; the quantifier is inherited from the eventual hypothesis (6.1) (p. 10, "for every sufficiently large integer $n$"), from the convention "$t$ is sufficiently large that $kr<|P_t|$" (p. 10) and from the Section 2 convention on arc lengths (p. 4); Lemma 6.3 (p. 12) displays its threshold, so the source is not uniform on the point. Proposed replacement: "If $q<1$, then, for all sufficiently large $t$ (a reading: the source's lemma displays no time threshold and inherits one from the eventual hypothesis (6.1) and the section's convention $kr<|P_t|$, p. 10; the threshold is spelled out at the end of the proof)," and, in the Standing paragraph, "The time threshold in the statement is a reading."

F2. Severity: suggested. Location: Definitions, "Notation as on the Lemma 2.1 and Lemma 6.1 pages", and the proof sentences "as on the Lemma 2.1 page, $t\le s\le t_+$", "the injection of the Lemma 2.1 proof" and "The same exchange of integrations as on the Lemma 2.1 page". Defect: a consumed input with no cross-link; the page relies on the Lemma 2.1 page for the notation $P_t$, $N_t$, $F_s$, $B_s$ and for three deductions, but links only the Lemma 6.1 page. Witness: the page in the frozen state contains no wikilink whose target is research/erdos_1221/ko26b_lemma_2_1_reconstruction. Proposed replacement for the opening of Definitions: "Notation as on the Lemma 2.1 page and the Lemma 6.1 page: $P_t$, $N_t(\cdot)$, ...".

F3. Severity: note. Location: Source paragraph, "displays (6.4)--(6.6) and Lemma 6.2 (pp. 10--12)". Defect: the page range over-includes a page. Witness: the definitions of $\Delta_t$ and $Z_t$ close p. 10; (6.4), Lemma 6.2, its proof, (6.5) and (6.6) all lie on p. 11; p. 12 holds Lemma 6.3 and Proposition 6.4, which only the Role section mentions. Proposed replacement: "(pp. 10--11)", or "(pp. 10--11; the iteration described under Role is on p. 12)".

F4. Severity: note. Location: Definitions, "Identity (6.4). At an integer time $n$, each of the $n$ points lies in $(x,x+D/n]$ for a set of $x$ of measure $D/n$, so ...". Defect: a correct supplied justification that is not marked as supplied and silently uses $D/n<1$. Witness: the source (p. 11) states "At integer times the mean of $\Delta_t(\,\cdot\,,D)$ is zero, and hence (6.4)" with no argument. Re-derivation: for $D/n<1$ and $y\in P_n$, $y\in(x,x+D/n]$ exactly when $x\in[y-D/n,y)$, of measure $D/n$, so $\int_{\mathbb T}N_n((x,x+D/n])\,dx=n\cdot D/n=D$, $\int\Delta_n=0$, the positive and negative parts have equal integrals and $\int|\Delta_n|=2Z_n(D)$. Proposed replacement: "Identity (6.4). (Justification supplied; the source asserts the zero mean.) At an integer time $n$ with $D/n<1$, each of the $n$ points ...".

Verdict

Source fidelity: faithful. The statement, its hypotheses, the constants and the labels (6.4)--(6.6) match the source at p. 11; the two suggested findings concern an unmarked reading and a missing cross-link, and the two notes a loose page range and an unmarked elementary expansion. No required correction was found.

The argument as reconstructed: sound. Every deduction was re-derived above; the transport bound, the one-atom bound and the averaging step compose into (6.5) exactly. The hypothesis $q<1$ is carried but unused, which is a fact about the source's statement and not a defect of the page.

Limitations: this is a focused, single-reviewer refutation review of one lemma; Lemma 6.1 was consumed at its stated interface and its proof was read only for consistency, not reviewed; the source is an unrefereed preprint; the review examined no evidence code because none exists for the page. This focused review assigns no tier and changes no status.