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Subject and independence

This report is by an independent reviewer working in a fresh context from the commissioning assignment alone. The reviewer took no part in writing the page under review, its sibling reconstruction pages, the library card or the problem page, and had no contact with the folder before this assignment. The charge is refutation; the review assigns no tier.

Frozen subject: path wiki/research/erdos_354/yu_chen_lemma_2_3_reconstruction.md as it stood at 2026-09-28T05:03:27Z, read in full as of that time.

Artifact: the seventeen-page PDF held by the library card Yu and Chen (2026), the folder-name PDF in the card folder. Physical pp. 3 and 4 were read in full, once from the layout text extraction and once from page images rendered at 130 dpi; the hypotheses of Lemma 2.3, the display (2.3) and the eight sentences of its proof were checked on the image of p. 4, and the definitions of $h$, $\operatorname{span}$ and $\operatorname{gap}$ on the image of p. 3. The top of physical p. 1 was read from the text extraction for the title, the author line and the date. No other page of the artifact was read.

Allowed material read: the sibling pages wiki/research/erdos_354/yu_chen_lemma_2_1_reconstruction.md and wiki/research/erdos_354/yu_chen_lemma_2_2_reconstruction.md as of the same time (printed whole; their Definitions and Statement sections were used and their Proof sections were not relied on); the frontmatter and the provenance paragraph of the library card; the Statement paragraph of wiki/problems/additive_bases/E0354/_index.md; the sections "Audit checklist -- the canonical failure modes", "Whole-claim report" and "Audit checklist" of docs/verification.md; the section "Source fidelity" of docs/evidence.md; and docs/math_authoring.md in full. The card's result page theorem.md is not linked from the page's Source paragraph and was not read.

Exposures, disclosed in full: the card's frontmatter desc, printed with the provenance paragraph, characterizes the manuscript's standing (unrefereed, listed as a proof claim, not reviewed in the corpus); a heading search over the card printed its "Standing" heading and the first line each of its "Read status" and "Bears on" paragraphs; a heading search over the problem page printed the first line of its "Status" paragraph; and a directory listing of wiki/research/erdos_354/evidence (file names only) was taken to see whether the verify directory existed. Nothing under any evidence/ folder was opened, and none of the exposed fragments concerns Lemma 2.3 or influenced the verdict.

Restatement

Convention. $W$ is a finite set of integers with at least two elements, listed as $w_0<w_1<\cdots<w_n$; $\operatorname{span}(W)=w_n-w_0$ and $\operatorname{gap}(W)=\max_j(w_{j+1}-w_j)$, so $\operatorname{gap}(W)\ge1$. For an integer $m\ge1$, $W\bmod m$ is the image of $W$ in $\mathbb Z/m\mathbb Z$, a nonempty set. For a nonempty $X\subseteq\mathbb Z/m\mathbb Z$, $h(X)$ is the largest $r$ such that some $r$ cyclically consecutive residues all lie outside $X$, and $h(X)=0$ when $X$ is the whole group; nonemptiness gives $h(X)\le m-1$.

Statement. For every such $W$, every integer $m$ with $1\le m\le\operatorname{span}(W)$, and every $k$ with $\operatorname{gap}(W)\le k$,

$$h(W\bmod m)\le k-1.$$

The statement is universal in $W$, $m$ and $k$; it has no exceptional set, no asymptotic clause and no constant. The source (p. 4) and the page leave the type of $k$ implicit; the argument proves the bound for every real $k\ge\operatorname{gap}(W)$, and the manuscript applies it with the integer $k=N$ (p. 3, display (1.1)). The hypothesis $\operatorname{span}(W)\ge m$ is necessary: $W=\{0,2\}$ and $m=5$ give $\operatorname{gap}(W)=2$ and $h(W\bmod5)=2$.

Checklist

• Quantifiers and scope: pass. The page keeps the source's universal quantification over $W$, $m$ and $k$, treats $m=1$ separately as the source does, and the boundary cases $W\cap[0,m)=\{0\}$ and $w_-=m-1$ are covered by its argument (re-derived under Weakest steps).
• Circularity: pass. The proof uses only the definitions and the two hypotheses; neither the conclusion nor an equivalent is assumed, and there is no induction.
• Model and convention changes: pass. The translation to $\min W=0$ is a proved transfer (rotation of residues, invariance of $h$, span and gap); the definitions of span, gap and $h$ on the page agree with the source's p. 3 definitions clause by clause.
• Finite and statistical overreach: inapplicable. The proof is a complete deductive argument; no finite case list or averaging stands in for a proof.
• Uniformity: pass. There are no constants or error terms; the bound $k-1$ is stated and proved for every admissible triple $(W,m,k)$ with no hidden dependence.
• Extremal conclusions: pass. The lemma claims an upper bound only, and neither the source nor the page claims sharpness; the bound happens to be attained, for example by $W=\{0,3,5\}$, $m=5$, $k=3$.
• Consequences and composition: pass. The page states no consequence beyond the display; the only composition is with the definitions imported from the sibling pages, whose interfaces (Definitions sections) match the source.
• Computation: inapplicable. The page carries no computation and no evidence driver; the finite examples in this report were checked by hand.
• Reproduction: inapplicable. No rerun commands or coverage claims are made.
• Source and verdict fidelity: pass. The hypotheses, the display (2.3) and the locators (Lemma 2.3, display (2.3), physical p. 4, seventeen pages, manuscript dated 13 September 2026, authors as on p. 1) were checked against the page image; the Standing paragraph claims author-recorded status only. Two labeling points are filed as suggestions (F1, F2), not as fidelity failures.

Weakest steps

Step 1: the wraparound bound. After the translation, $0=\min W$ and $\max W=\operatorname{span}(W)\ge m\ge2$. Let $w_-=\max(W\cap[0,m))$, which exists since $0\in W\cap[0,m)$, and $w_+=\min\{w\in W:w\ge m\}$, which exists since $\max W\ge m$. Then $w_-<m\le w_+$. If $x\in W$ satisfied $w_-<x<w_+$, then either $x<m$, contradicting the maximality of $w_-$ in $[0,m)$, or $x\ge m$, contradicting the minimality of $w_+$. So $w_-$ and $w_+$ are adjacent in the increasing listing of $W$, and $w_+-w_-\le\operatorname{gap}(W)\le k$. Since $w_+\ge m$,

$$m-w_-\le w_+-w_-\le k,\qquad\text{so}\qquad m-1-w_-\le k-1.$$

This is the only place the span hypothesis enters, and the example $W=\{0,2\}$, $m=5$ in the Restatement shows that without it the wraparound run can exceed $k-1$. The step composes with Step 2 by bounding the one missing run that is not bounded by two integer points of $W\cap[0,m)$.

Step 2: the maximal missing runs of the truncated set. Let $S=W\cap[0,m)$, viewed inside $\mathbb Z/m\mathbb Z$ (each element is its own residue), so $0\in S$ and $w_-=\max S$. List $S$ as $0=s_0<s_1<\cdots<s_t=w_-$. Consecutive elements $s_i<s_{i+1}$ of $S$ are adjacent in $W$: an element of $W$ strictly between them would lie in $[0,m)$, hence in $S$. So $s_{i+1}-s_i\le k$, and the residues strictly between them, namely $s_i+1,\ldots,s_{i+1}-1$, number $s_{i+1}-s_i-1\le k-1$. The residues of $\mathbb Z/m\mathbb Z$ outside $S$ and not strictly between two consecutive elements of $S$ are exactly $w_-+1,\ldots,m-1$, which number $m-1-w_-\le k-1$ by Step 1 and may be none. Cyclically, the residue after $m-1$ is $0\in S$, and the residue before $w_-+1$ is $w_-\in S$. Hence every maximal missing run of $S$ is either a block $\{s_i+1,\ldots,s_{i+1}-1\}$ or the block $\{w_-+1,\ldots,m-1\}$, each of length at most $k-1$, and $h(S)\le k-1$. The boundary cases behave: when $S=\{0\}$ there are no consecutive pairs and the single missing block $\{1,\ldots,m-1\}$ has length $m-1\le k-1$ by Step 1; when $w_-=m-1$ the wraparound block is empty; when $S$ is all of $[0,m)$, $h(S)=0$. The step composes with Step 3, which passes from $S$ to $W\bmod m$.

Step 3: the reductions at both ends. Translation: for an integer $c$, $(W+c)\bmod m=(W\bmod m)+(c\bmod m)$, a rotation of the circle, and a set of $r$ cyclically consecutive residues avoids $X$ exactly when its rotation avoids $X+c$, so $h$ is unchanged; span and gap are differences of elements and are unchanged. Monotonicity: $S\subseteq W\bmod m$ as subsets of $\mathbb Z/m\mathbb Z$, and any set of cyclically consecutive residues avoiding $W\bmod m$ avoids $S$, so $h(W\bmod m)\le h(S)\le k-1$. The case $m=1$: $\mathbb Z/1\mathbb Z$ has one residue, so $W\bmod1$ is full and $h=0$; since $W$ has two distinct elements, $\operatorname{gap}(W)\ge1$, so $k\ge1$ and $0\le k-1$. These reductions compose with Steps 1 and 2 to give the display for every admissible $(W,m,k)$.

Strongest attack

The attack sought a triple $(W,m,k)$ with $\operatorname{span}(W)\ge m\ge1$, $\operatorname{gap}(W)\le k$ and $h(W\bmod m)\ge k$. Every missing run of $W\bmod m$ is a missing run of $S=W\cap[0,m)$, and Step 2 shows that each maximal missing run of $S$ is either bounded by two integer points of $S$ adjacent in $W$, which forces length at most $k-1$ directly from the gap hypothesis, or is the wraparound block $\{w_-+1,\ldots,m-1\}$. So a counterexample must have a wraparound block of length at least $k$, that is $m-w_-\ge k+1$. But the next element of $W$ after $w_-$ is at least $m$, so the gap of $W$ at $w_-$ is at least $m-w_-\ge k+1>k$, contradicting the hypothesis. The attack fails because the span hypothesis supplies the next element $w_+\ge m$; the same computation with $\operatorname{span}(W)<m$ produces the counterexample $W=\{0,2\}$, $m=5$, $h=2>1$, which confirms that the page invokes the span hypothesis at exactly the step that needs it (the existence of $w_+$).

Secondary attacks: a non-integer $k$ (the bound still holds, since $w_+-w_-$ is an integer at most $k$); $m=\operatorname{span}(W)$ (then $w_+=\max W$ and $W\bmod m$ identifies $0$ with $\max W$, consistent with the argument); $m=2$ and $m=1$; and $W\cap[0,m)=\{0\}$. None breaks the argument. The statement was also attacked for fidelity by reading the hypotheses and display (2.3) on the image of p. 4 against the page's Statement section; they agree symbol for symbol.

Premises

• Definitions of $\operatorname{span}$ and $\operatorname{gap}$: source p. 3, first paragraph, held and read on the page image; the page imports them from the Definitions section of the mesh lemma page (Lemma 2.2 page), which states them for a finite integer set with at least two elements and agrees with the source. Interface used: $\operatorname{span}(W)$ is $\max W-\min W$, and $\operatorname{gap}(W)$ is the largest difference of adjacent elements, hence at least $1$.
• Definition of $h$: source p. 3, first paragraph, held and read on the page image; the page imports it from the Definitions section of the erosion lemma page (Lemma 2.1 page), which defines missing runs, maximal missing runs and $h$ for a nonempty subset of $\mathbb Z/d\mathbb Z$ and agrees with the source. Interface used: $h(X)$ is the largest length of a set of cyclically consecutive residues outside $X$, and $0$ for the full group.
• No theorem is imported; the lemma is self-contained and the sibling pages are consumed for definitions only. Both sibling pages record themselves as author-recorded reconstructions, and this review relies on nothing from their Proof sections.
• Explicit assumptions: $W$ is finite with at least two elements (needed for $\operatorname{span}$ and $\operatorname{gap}$ to be defined), $m$ is a positive integer (a modulus), and $k$ is any real number with $\operatorname{gap}(W)\le k$; the source and the page both leave the first two implicit in the lemma's sentence and fix them in their definitions.

Findings

F1. Severity: suggested. Location: the Source paragraph, "Lemma 2.3 with its display (2.3), physical p. 4". Defect: the source proves the lemma in eight sentences (p. 4, the paragraph after display (2.3)); the page's proof writes them out, supplying the observation $k\ge1$ behind "The case $m=1$ is immediate", the existence of $w_-$, the argument that no element of $W$ lies strictly between $w_-$ and $w_+$, the enumeration of the maximal missing runs behind "This also bounds the wraparound interval to zero", and the monotonicity behind "Adding residues from other points only reduces gaps"; it also omits the source's aside that the translation "does not introduce negative original summands", which concerns the later application. None of this is marked, whereas the erosion lemma page marks its expansion in its Source paragraph. The route of the argument is the source's, so this is a labeling gap, not a fidelity failure. Witness: p. 4, the eight proof sentences from "Translate $W$ analytically" to "only reduces gaps". Proposed replacement: append to the Source paragraph "The source gives the proof eight sentences, which the proof below writes out; its aside that the translation introduces no negative summands concerns the sets to which the manuscript applies the lemma and is omitted here."

F2. Severity: suggested. Location: the Proof, "Any two consecutive ones differ by at most $k$". Defect: the gap hypothesis bounds differences of elements adjacent in $W$, and the sentence applies it to elements adjacent in $W\cap[0,m)$ without saying why these are adjacent in $W$ (an element of $W$ between them would itself lie in $[0,m)$). The source asserts the same without justification ("gaps among the points of $W\cap[0,m)$ are at most $k$", p. 4), so fidelity is unaffected, but the page's chain of deductions should carry the clause. Proposed replacement: "Any two consecutive ones are consecutive in $W$, since an element of $W$ between them would itself lie in $[0,m)$; so they differ by at most $k$, and between them at most $k-1$ residues are missing."

F3. Severity: note. Location: the frontmatter title, "projection to a smaller modulus". Defect: the source's heading reads "Lemma 2.3: projection to any smaller modulus" (p. 4), and the sibling pages reuse the source's headings verbatim ("erosion by one translate", "propagation of a finite integer mesh"). Proposed replacement: "Yu--Chen Lemma 2.3: projection to any smaller modulus".

F4. Severity: note. Location: the Proof, "a run of length $m-1-w_-$". Defect: when $w_-=m-1$ this "run" is empty, while the erosion lemma page's definition, which the page imports, gives every missing run length at least $1$; the inequality and the conclusion are unaffected. Proposed replacement: "The remaining residues are $w_-+1,\ldots,m-1$ (none when $w_-=m-1$), at most $m-1-w_-\le k-1$ of them in one block, and the block is followed cyclically by the residue $0$, which is present."

F5. Severity: note. Location: the Definitions, "as on the mesh lemma page". Defect: the mesh lemma page lists $W$ as $w_0<w_1<\cdots<w_m$, so its letter $m$ is the top index, while on this page $m$ is the modulus. The page never uses the index, so nothing is ambiguous, but a reader following the link meets the clash. Proposed replacement: "are as on the mesh lemma page (whose index letter $m$ is unrelated to the modulus $m$ here)".

Verdict

Source fidelity: faithful. The hypotheses, the conclusion, the display label, the physical page, the page count, the date and the author line match the held artifact, and the Standing paragraph claims no more than author-recorded status.

The argument as reconstructed: sound. Each step was re-derived above, the boundary cases $m=1$, $W\cap[0,m)=\{0\}$ and $w_-=m-1$ close, and the span hypothesis is used exactly where it is necessary.

Limitations: this is a focused review of one lemma against physical pp. 3 and 4 of the artifact; it does not examine how the manuscript applies the lemma (p. 3, Section 1.1), the sibling reconstructions beyond their definitions, or any other part of the manuscript. No computation was run; the finite examples were checked by hand. Required corrections: none. Suggested: F1, F2. Notes: F3, F4, F5.

This focused review assigns no tier and changes no status.