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Subject and independence

Role. Independent reviewer commissioned in a fresh context with only the assignment, charged with refutation. The reviewer took no part in writing the page, had not seen it or any review of it before this commission, and read nothing outside the commissioned set listed below. The page's author is a different role (the reconstruction's author); no result of the page was consumed by the reviewer for any other purpose.

Subject. Path wiki/research/erdos_354/fan_remark_4_2_reconstruction.md as it stood at 2026-09-28T05:03:27Z (the page), read in full, clause by clause, as of that time.

Artifact. The v5 PDF under the source card Fan (2026) (arXiv:2607.14071v5, 16 September 2026, 36 physical pages; the file on disk matches the card's recorded SHA-256), physical pages 2, 3, 4, 19 and 20, read in the text layer and as page images rendered at 130 dots per inch: Remark 4.2 (p. 20) clause by clause; the definitions (1.1) (p. 2), (1.5) (p. 3), (1.8) and (1.9) (p. 4), the dyadic-equivalence and dyadic-rational conventions and Hegyvári's conjecture (p. 4), the observation that strongly complete sets satisfy (1.5) with its five-line proof (p. 3), Theorem 1.1 (pp. 3--4) and Corollary 1.2 (p. 4), all clause by clause; Remark 4.1 (pp. 19--20) at statement level for the value $M_2^*\ge2$; the surrounding text of pp. 19--20 (the end of the proof of Theorem 1.1, Proposition 4.2) only to fix the remark's boundaries. The v4 PDF, physical page 19, text layer and page image, for the label map: its Remark 4.1 was compared word by word with v5's Remark 4.2. The canonical conversion beside the v5 PDF, which the page says it read instead of the PDF: the remark's paragraph and the p. 4 pointer to Remark 4.2 were compared with the PDF and agree; the PDF decided every check below.

Other allowed material read. The source card's provenance paragraph and label map; the Source and Statement sections of the result page Remark 4.2; the Statement section only of the Remark 4.1 reconstruction, to check the page's cross-link; the Statement of Problem 354; in docs/verification.md the sections "Whole-claim report" and "Audit checklist" and the shared "Audit checklist" of canonical failure modes; in docs/evidence.md the section "Source fidelity"; docs/math_authoring.md in full for the record's mechanics.

Exposures. Four, none used: (1) printing the source card and the result page printed their whole text, including the card's Overview, Bears-on and Results paragraphs and the result page's Dependencies and Bears-on sections, which carry status context; (2) printing the head of the problem page to locate its Statement also printed its Formulation paragraph and the opening of its Status paragraph; (3) the search of the conversion for the remark printed neighboring source text (Theorem 1.3, Propositions 4.2 and 4.3), which is source material; (4) the page images of pp. 19--20 carry the end of the proof of Theorem 1.1 and Proposition 4.2, source material read only to place the remark. Nothing under any evidence/ folder, no other review, nothing under the private working files or outside the repository, and no web search.

Restatement

Conventions. $\mathbb N=\{1,2,\ldots\}$. For $A\subseteq\mathbb N$, $\operatorname{FS}(A)$ is the set of sums over nonempty finite subsets of $A$; $A$ is complete when $\mathbb N\setminus\operatorname{FS}(A)$ is finite and strongly complete when $A\setminus B$ is complete for every finite $B\subseteq A$. $\|x\|$ is the distance from $x$ to the nearest integer. Condition (1.5) for $A$: $\sum_{a\in A}\|a\theta\|=\infty$ for every real $\theta\notin\mathbb Z$ (the source writes $\theta\in\mathbb T\setminus\{0\}$ with $\mathbb T=\mathbb R/\mathbb Z$, the same set of $\theta$ since $a$ is an integer). $M_2^*$ is the least positive integer $M$ such that every $A\subseteq\mathbb N$ with (1.5) and $|A\cap(2^k,2^{k+1}]|\ge M$ for all sufficiently large $k$ is strongly complete. For reals $\alpha,\beta>0$, $A_{\alpha,\beta}$ is the set (not multiset) of nonzero values $\lfloor2^k\alpha\rfloor$ and $\lfloor2^k\beta\rfloor$, $k\ge0$; $\alpha\sim\beta$ when $\alpha/\beta$ is an integer power of $2$ (positive, zero or negative exponent); $\alpha$ is a dyadic rational when $\alpha\sim n$ for a nonzero integer $n$, that is, $\alpha=2^mn$ with $m\in\mathbb Z$.

Proposition (the page's Remark 4.2). Let $\alpha,\beta>0$ with $\alpha\not\sim\beta$, and suppose at least one of $\alpha,\beta$ is not a dyadic rational. Then: (i) there is $k_0$ such that for every $k\ge k_0$ the interval $(2^k,2^{k+1}]$ contains at least two elements of $A_{\alpha,\beta}$; (ii) $A_{\alpha,\beta}$ satisfies (1.5). Consequently, if $M_2^*=2$, then $A_{\alpha,\beta}$ is strongly complete; since a strongly complete set is complete, $M_2^*=2$ implies Hegyvári's conjecture, which asserts completeness of $A_{\alpha,\beta}$ under exactly these hypotheses. The hypothesis $M_2^*=2$ is not proved anywhere; the page uses it only as the antecedent of the implication. Auxiliary proposition (the page's Observation): every strongly complete $A\subseteq\mathbb N$ satisfies (1.5); it is proved on the page but not used in the deduction of the remark.

Checklist

• Quantifiers and scope. Pass, with one boundary slip filed as F1. The statement's quantifiers match the source (all $\alpha,\beta>0$ with $\alpha\not\sim\beta$ and one not a dyadic rational; "for every sufficiently large $k$" in (1.8); (1.5) for every $\theta\notin\mathbb Z$). In Step 1 the displayed inclusion into $A_{\alpha,\beta}$ is asserted for $k_0\ge\max(s,t,0)$ but holds only for $k_0\ge1$, which the page itself imposes two sentences later; the argument never uses $k_0=0$.
• Circularity. Pass. $M_2^*=2$ is the remark's antecedent, not assumed to prove itself; Steps 1--3 use only the definitions and the hypotheses on $\alpha,\beta$.
• Model and convention changes. Pass. $\theta\in\mathbb T\setminus\{0\}$ is rendered as $\theta\in\mathbb R\setminus\mathbb Z$, the form the source's abstract itself uses; $A_{\alpha,\beta}$ is the source's set; the source's closing phrase $H_1(A)=\{0\}$ is read as (1.5), which is what (1.8) requires and what Step 3 proves, so no spectrum definition is consumed (F4).
• Finite and statistical overreach. Inapplicable: no finite case check or heuristic stands in for a proof; the witness in F1 is a counterexample to one displayed sentence, not evidence for the argument.
• Uniformity. Pass. Every "for $k$ large" threshold depends only on the fixed pair $\alpha',\beta'$: the disjointness threshold, $k_0\ge\max(s,t,1)$, and $\min(\alpha',\beta')\ge1/2+2^{-k_0-1}$; nothing is claimed uniform in $\alpha,\beta$, and the source's "$k_0$ can be arbitrarily large" is matched.
• Extremal conclusions. Inapplicable to the deduction. The Scope paragraph's $2\le M_2^*\le5$ is quoted from the source (Corollary 1.2 and Remark 4.1 at $\rho=2$, $u_2=2$) as author-recorded context and matches pp. 4 and 19.
• Consequences and composition. Pass. Each "hence" was re-derived (see Weakest steps); Step 4 composes (i) and (ii) with the definition (1.8) at $\rho=2$ exactly; the "in particular" clause uses strongly complete implies complete (p. 2).
• Computation. Inapplicable: the page runs no computation and cites no evidence driver.
• Reproduction. Inapplicable: the page states no rerun command and no coverage claim.
• Source and verdict fidelity. Pass with notes. Every quoted statement, label and page number matches the v5 PDF; the v4 label map is right (F3 refines its wording); the Standing paragraph claims author-recorded standing only; the page's admission that it read the conversion and not the PDF is honest, and the conversion agrees with the PDF at the remark (F5).

Weakest steps

1. Disjointness of the rescaled rays (Step 1). Re-derived: with $\alpha',\beta'\in(1/2,1]$ distinct, for $k\ge1$ the value $\lfloor2^k\alpha'\rfloor$ lies in $[2^{k-1},2^k]$ because $2^k\alpha'\in(2^{k-1},2^k]$, and likewise for $\beta'$. Two such values with indices $k,j\ge1$ can agree only if $[2^{k-1},2^k]$ meets $[2^{j-1},2^j]$, so $|k-j|\le1$. Choose $K\ge1$ with $2^K|\alpha'-\beta'|\ge1$ and $\min(\alpha',\beta')\ge1/2+2^{-K}$. For $k\ge K$: the case $j=k$ is impossible since $|2^k\alpha'-2^k\beta'|\ge1$ forces different floors; the case $j=k+1$ forces the common value $2^k=\lfloor2^{k+1}\beta'\rfloor$, that is $\beta'<1/2+2^{-k-1}<1/2+2^{-K}$, false; the case $j=k-1$ forces $2^{k-1}=\lfloor2^k\alpha'\rfloor$, that is $\alpha'<1/2+2^{-k}\le1/2+2^{-K}$, false. A coincidence inside $U_K(\alpha')\cap U_K(\beta')$ needs $k\ge K$, so the intersection is empty, and the same holds for every $k_0\ge K$. The page's version uses the threshold $2^k|\alpha'-\beta'|\ge2$ and "for $k$ large" in each case; both are correct. Composition: this supplies the distinctness in Step 2 and makes $U_{k_0}(\alpha')$, $U_{k_0}(\beta')$, $B$ a partition; the case $\beta'=1$ ($\beta$ a dyadic rational, allowed) was checked and causes no exception.

2. Divergence of the distance sums (Step 3). Re-derived: write $c_k=\lfloor2^k\alpha'\rfloor$ and $f_k=2^k\alpha'-c_k\in[0,1)$. Then $c_{k+1}-2c_k=\lfloor2f_k\rfloor\in\{0,1\}$. If this were $0$ for all $k\ge k_1$, then $f_{k+1}=2f_k<1$ for all $k\ge k_1$, so $2^mf_{k_1}<1$ for every $m\ge0$, forcing $f_{k_1}=0$; then $2^{k_1}\alpha'=c_{k_1}$ is an integer, positive because $\alpha'>1/2$, and $\alpha=2^s\alpha'=2^{s-k_1}c_{k_1}\sim c_{k_1}$ is a dyadic rational, against the hypothesis. So $c_{k+1}=2c_k+1$ for infinitely many $k$. Now let $\theta\notin\mathbb Z$ with $\sum_{a\in A_{\alpha,\beta}}\|a\theta\|<\infty$. The values $c_k$, $k\ge k_0$, are distinct elements of $A_{\alpha,\beta}$ ($c_{k+1}\ge2c_k>c_k$ as $c_k\ge1$), so $\sum_{k\ge k_0}\|c_k\theta\|$ is bounded by the full sum and $\|c_k\theta\|\to0$. On the infinite set of $k\ge k_0$ with $c_{k+1}-2c_k=1$,

$$\|\theta\|=\|(c_{k+1}-2c_k)\theta\| \le\|c_{k+1}\theta\|+\|2c_k\theta\| \le\|c_{k+1}\theta\|+2\|c_k\theta\|\to0,$$

so $\|\theta\|=0$ and $\theta\in\mathbb Z$, a contradiction. The page's argument is the same with the limit $2^{-k}c_k\to\alpha'$ in place of the fractional-part doubling. Composition: this is hypothesis (1.5) in the definition (1.8); only the $\alpha$-ray is used, which is legitimate since the sum runs over all of $A_{\alpha,\beta}$.

3. Two elements in every large dyadic interval (Step 2). Re-derived: for $k\ge0$, $2^{k+1}\alpha'\in(2^k,2^{k+1}]$, so $\lfloor2^{k+1}\alpha'\rfloor\in[2^k,2^{k+1}]$, with the value $2^k$ exactly when $2^{k+1}\alpha'<2^k+1$, that is $\alpha'<1/2+2^{-k-1}$; this fails as soon as $2^{-k-1}\le\alpha'-1/2$, and likewise for $\beta'$. For $k\ge k_0$ with $k_0\ge\max(s,t,1)$ past these thresholds and past the disjointness threshold of step 1, the two values lie in $(2^k,2^{k+1}]$, are distinct, and belong to $A_{\alpha,\beta}$ (they are $\lfloor2^{k+1-s}\alpha\rfloor$ and $\lfloor2^{k+1-t}\beta\rfloor$ with nonnegative exponents, and are positive). Composition: this is the counting hypothesis of (1.8) at $\rho=2$ with the value $2$; Step 4 then reads $M_2^*=2$ as "every $A\subseteq\mathbb N$ with (1.5) and at least two elements in every large dyadic interval is strongly complete" and applies it to $A_{\alpha,\beta}\subseteq\mathbb N$.

Strongest attack

The strongest attack was on the index bookkeeping of Step 1, where the page supplies the rescaling the source leaves to the reader: the reviewer tried to make one of $U_{k_0}(\alpha')$, $U_{k_0}(\beta')$ leave $A_{\alpha,\beta}$, or the complement $B$ infinite, by choosing $\alpha,\beta$ with negative or mixed exponents $s,t$ and by pushing $k_0$ to the bottom of its stated range. The complement is always finite (an element $\lfloor2^k\alpha\rfloor$ misses $U_{k_0}(\alpha')$ only when $k<k_0-s$), and the inclusion holds for every $k_0\ge\max(s,t,1)$. It fails at the one boundary value $k_0=0$, reachable when $\alpha,\beta\le1$: for $\alpha=7/10$, $\beta=9/10$ one has $s=t=0$, $\lfloor\alpha'\rfloor=0\in U_0(\alpha')$ and $0\notin A_{\alpha,\beta}$. This refutes the displayed sentence at its stated range $k_0\ge\max(s,t,0)$ but not the argument, because the page's own parenthetical justifies positivity only for $k_0\ge1$ and its next paragraph fixes $k_0\ge\max(s,t,1)$ before anything is deduced (F1). The disjointness case analysis was then attacked with $\beta'=1$ (a dyadic-rational $\beta$ is allowed) and with $\alpha'$ close to $1/2$; each of the three cases $|k-j|\le1$ still closes for $k$ large. Step 3 was attacked by asking whether coincidences between the two rays could shrink the $\alpha$-ray's contribution to the sum; they cannot, since the sum is over the set $A_{\alpha,\beta}$ and the $c_k$ are distinct members of it. The statement was attacked on its quantifiers: the source's remark concerns the set $A_{\alpha,\beta}$ for all $\alpha,\beta>0$ under Hegyvári's condition, and the page's statement, "in particular" clause and Scope paragraph claim exactly that and nothing about the problem page's multiset reading or other bases. No attack succeeded against the mathematics.

Premises

• Definition (1.8) of $M_2^*$ (v5, p. 4). Held; read clause by clause in the PDF. Interface used: the antecedent $M_2^*=2$ means that every $A\subseteq\mathbb N$ satisfying (1.5) with at least two elements in $(2^k,2^{k+1}]$ for every sufficiently large $k$ is strongly complete; the minimality in "least" is not used. This is the only imported statement the deduction consumes.
• Definitions (1.1), (1.5), (1.9), dyadic equivalence, dyadic rational, complete, strongly complete (v5, pp. 2--4). Held; clause by clause; the page's renderings match, with $\theta\in\mathbb R\setminus\mathbb Z$ for $\theta\in\mathbb T\setminus\{0\}$.
• Hegyvári's conjecture as the source records it (v5, p. 4). Held at statement level; the page cites it only through the source, and Hegyvári's paper is not held and not needed.
• Theorem 1.1 (v5, pp. 3--4) and Corollary 1.2 (p. 4). Held; read at statement level; standing: statements of a preprint, imported and named as such by the page. They are used only in the Scope paragraph for $M_2^*\le5$, not in the deduction. The page's restatement omits the source's "let $M\ge M_\rho$ be an integer"; harmless for a statement used as context, since the real-$M$ form follows from the integer form applied to $\lceil M\rceil$.
• Remark 4.1 of v5 (pp. 19--20), base-two case. Held; read at statement level; supplies $M_2^*\ge u_2=2$ for the Scope paragraph only; the page's cross-link to its reconstruction resolves and that page's Statement section states the same fact.
• Elementary facts, supplied without citation. $\lfloor2x\rfloor-2\lfloor x\rfloor\in\{0,1\}$; $\|x+y\|\le\|x\|+\|y\|$ and $\|2x\|\le2\|x\|$ on $\mathbb R/\mathbb Z$; $\|n\theta\|$ depends only on $\theta$ modulo $1$ for integer $n$; $\sim$ is an equivalence relation; terms of a convergent series of nonnegative reals tend to zero. All checked.
• Explicit assumptions. Only the remark's hypotheses ($\alpha,\beta>0$, $\alpha\not\sim\beta$, one of them not a dyadic rational) and its antecedent $M_2^*=2$; the convention $\mathbb N=\{1,2,\ldots\}$, which the source's usage supports (its Remark 4.1 counts $|A\cap[1,2^{k+1}]|=k$ for $A=\{2^k+1:k\in\mathbb N\}$, which needs $1\in\mathbb N$ and $0\notin\mathbb N$). No batch acceptance order.

Findings

F1. Severity: suggested. Location: Step 1, "so for $k_0\ge\max(s,t,0)$, $U_{k_0}(\alpha')\cup U_{k_0}(\beta')\subseteq A_{\alpha,\beta}$". Defect: the inclusion is false at $k_0=0$, which the stated range allows whenever $s,t\le0$, because $U_0(\alpha')$ contains $\lfloor\alpha'\rfloor=0$ when $\alpha'<1$ while $A_{\alpha,\beta}$ excludes $0$ by (1.9); the parenthetical justification ("for $k_0\ge1$") and the later choice $k_0\ge\max(s,t,1)$ show the intended range. Witness: $\alpha=7/10$, $\beta=9/10$ (v5, p. 4 for (1.9)): $s=t=0$, $0\in U_0(\alpha')$, $0\notin A_{\alpha,\beta}$. The argument is unaffected. Replacement: "so for $k_0\ge\max(s,t,1)$," with the parenthetical unchanged.

F2. Severity: suggested. Location: Source paragraph, "The remark's rescaling, the finiteness of the intersection of the two rays and the "routine" triangle-inequality step are stated without detail in the source; they are written out below." Defect: the list of supplied steps is incomplete. The source (v5, p. 20) also states without proof that for $k_0$ large $\lfloor2^{k+1}\alpha\rfloor,\lfloor2^{k+1}\beta\rfloor\in(2^k,2^{k+1}]$ are distinct, and that a non-dyadic-rational $\alpha'$ has $\lfloor2^{k+1}\alpha'\rfloor=2\lfloor2^k\alpha'\rfloor+1$ for infinitely many $k$; the page supplies both derivations (Step 2 and the first paragraph of Step 3) without naming them as supplied. Replacement: "The remark's rescaling, the finiteness of the intersection of the two rays, the placement of the two floors in $(2^k,2^{k+1}]$, the infinitely many carries for a non-dyadic-rational $\alpha'$ and the "routine" triangle-inequality step are stated without detail in the source; they are written out below."

F3. Severity: note. Location: Source paragraph, "(Remark 4.1 of v4, p. 19, with the same content)". Defect: v4's Remark 4.1 (p. 19) is a longer remark whose first paragraph is the $\{2^k+1\}$ example (v5's Remark 4.1 at base two) and whose second paragraph is, word for word, v5's Remark 4.2; "the same content" holds for that paragraph, not for the whole v4 remark. Witness: v4, physical and printed p. 19, the two paragraphs of Remark 4.1. Replacement: "(the second paragraph of Remark 4.1 of v4, p. 19, word for word)".

F4. Severity: note. Location: the Source paragraph and Step 3. Defect: two readings of the source's notation are not marked. The source's last sentence (v5, p. 20) uses $\alpha'$ without defining it and writes $H_1(A)=\{0\}$ "for $A_{\alpha,\beta}$"; the page reads $\alpha'$ as the rescaled $\alpha$ and the conclusion as condition (1.5), which is what (1.8) requires and what Step 3 proves directly, so no spectrum definition is consumed. Both readings are right. Replacement: add to the Source paragraph "The source's $\alpha'$ is read as the rescaled $\alpha$, and its conclusion $H_1(A)=\{0\}$ as condition (1.5), the form the definition of $M_2^*$ uses."

F5. Severity: suggested. Location: Source paragraph, "Read in the canonical conversion beside the held v5 PDF, which was not itself opened for this page". Defect: the source-fidelity rule reads statements, formulas and proof details against the canonical PDF when one is held; the page records that it did not. The disclosure is honest, and this review compared the conversion's remark paragraph and the page's quoted statements, labels and page numbers with the v5 PDF at pp. 2--4 and 20 and found them to agree, so no content changes. Replacement: after the page's author reads the PDF at those pages, "Read against the held v5 PDF at these pages, with the canonical conversion beside it as the text layer."

Verdict

Source fidelity: faithful. The page's statement of Remark 4.2, its definitions, the observation, Theorem 1.1 and Corollary 1.2, and every locator (v5 physical and printed p. 20 for the remark; pp. 2--4 for the definitions and the introduction's results; v4 p. 19 for the earlier label) match the held PDF; nothing the source proves is altered or strengthened, and the Standing paragraph claims author-recorded standing only.

The argument as reconstructed: sound. Steps 1--4 and the observation were re-derived; the one false sentence (F1) sits at a boundary value of $k_0$ that the page excludes before deducing anything, and F2--F5 concern labeling and reading records, not mathematics.

Limitations. The proofs of Theorem 1.1, Corollary 1.2 and Remark 4.1 were not examined and are not the page's subject; the source's spectrum $H_1$ was not read, and the page does not depend on it; the source is a preprint; the hypothesis $M_2^*=2$ is unproved, and the page states so. This focused review assigns no tier and changes no status.