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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 under review, the library card, the result pages or the neighboring reconstructions, and had not seen any of them before this review.

Subject: path wiki/research/erdos_18/hughes_remark_6_reconstruction.md as it stood at 2026-09-28T05:03:27Z (the reconstruction page), read in full as of that time.

Artifact: the held PDF of Hughes, Sums of distinct divisors of factorials, arXiv:2609.10902v1, five pages, under the library card (hughes_2026_sums_distinct_divisors_factorials.pdf). Physical pages 4 and 5 (Remark 6 in full, with Remarks 5 and 7 and the references around it) were read clause by clause, first in the layout text extraction and then on page images rendered at 150 dpi, every displayed formula being read on the image. Physical page 1 (the definition of $h(N)$, the sentence crediting Erdős with $h(n!)\le n$, and the logarithm convention) was read the same way. Page images rendered: pages 1, 4 and 5. The canonical conversion beside the PDF was read for Remark 6 only; it agrees with the page images, and the PDF decided.

Allowed material actually read: the Statement section of the Theorem 1 reconstruction page in the same folder as of the same time, with its Definitions section, which the page under review cites for $h(N)$; the provenance paragraph of the library card; the Statement section of the Remark 6 result page; the Statement paragraph of Problem 18; the "Whole-claim report" and "Audit checklist" sections of docs/verification.md; the "Source fidelity" section of docs/evidence.md; all of docs/math_authoring.md.

Exposures, each by over-wide extraction, none used in the verdict: the library card's Overview, Read status and Bears on sections and the result page's Proof sketch, Reconstruction and Bears on sections arrived with the provenance paragraph and the Statement; the Theorem 1 reconstruction's Source and Standing paragraphs arrived with its Statement, and its Standing paragraph carries a supersession sentence; the Problem 18 page has no Statement heading, so its Status, Provenance, Source and References paragraphs arrived with the Statement paragraph, and the Status paragraph is status text; the "Durable reports and current standing" and "Audit checklist — the canonical failure modes" sections of docs/verification.md arrived with the two commissioned sections. Nothing under any evidence/ folder, the research folder's _index.md, other reviews, or the web was read.

Restatement

Conventions. $\log$ is the natural logarithm. An integer $N\ge1$ is practical when every integer $1\le m\le N$ is a sum of distinct positive divisors of $N$; for practical $N$, $h(N)$ is the least $k$ such that every integer $1\le m\le N$ is a sum of at most $k$ distinct positive divisors of $N$, the set of divisors being chosen afresh for each $m$. $\tau(N)$ is the number of positive divisors of $N$, $v_p$ the $p$-adic valuation, $\pi(x)$ the number of primes not exceeding the real number $x$, and Chebyshev's bound is taken as $\pi(x)\le Cx/\log x$ for every real $x\ge2$ with one absolute constant $C$.

Claim. There are an absolute constant $c>0$ and an integer $n_0$ such that for every integer $n\ge n_0$,

$$h(n!)\ge c\,(\log n)^2 .$$

The claim presupposes that $h(n!)$ is defined, that is, that $n!$ is practical; the source takes this from Erdős's $h(n!)\le n$ (p. 1). The bound is for every large $n$, not almost every; the constant does not depend on $n$; no sharpness is claimed. The page's proof gives more than the claim: an explicit absolute $c$ valid for every $n\ge4$. The page adds in its Qualifications the consequence that a bound $h(n!)\le(\log n)^A$ holding for all large $n$ forces $A\ge2$.

Checklist

• Quantifiers and scope. Pass. The statement quantifies over all sufficiently large $n$ with one absolute constant; the proof covers every $n\ge4$; there is no almost-all, limit inferior or exceptional set. The source's $k\gg(\log n)^2$ (p. 5) carries the same meaning. The excluded $n\le3$ would also satisfy the bound with a smaller $c$, since $h(n!)\ge1$.
• Circularity. Pass. The target is not assumed, no statement equivalent to it is used, and there is no induction.
• Model and convention changes. Pass. The $h$ used is the source's own (p. 1: least $k$, fresh set for each $m$), and the objects counted are the actual subsets of the actual divisor set of $n!$; Chebyshev's bound is applied to the actual prime counts of dyadic ranges.
• Finite and statistical overreach. Inapplicable: no finite check, average or heuristic is used anywhere on the page.
• Uniformity. Pass. Every constant is absolute: $C$ from Chebyshev, $\sum_{r\ge0}(r+1)2^{-r}=4$, $\log(n+1)\le2\log n$ for $n\ge2$, $(\log n)^2\le(16/e^2)\sqrt n$ and $(\log n)^2\le(4/e^2)\,n$ for all $n\ge1$; the $r$-sum is bounded independently of $n$ because its terms are positive and the full series converges.
• Extremal conclusions. Pass for the single extremal sentence, the Qualifications' "exponent at least $2$": if $h(n!)\le(\log n)^A$ for all large $n$ then $c(\log n)^2\le(\log n)^A$ for all large $n$, so $A\ge2$, checked in the claim's own units.
• Consequences and composition. Pass with one undischarged trivial hypothesis (F1). Each "hence" and "so" was re-derived separately (Weakest steps below). The composition of the binomial-tail bound with the counting inequality needs $1\le k\le T$; the page discharges $k\le T$ only, and $k\ge1$ holds for every $n$.
• Computation. Inapplicable: the page runs no computation. The numeric facts used here (the value $4$ of the $r$-series, the extremes of $(\log n)^2/\sqrt n$ and $(\log n)^2/n$) were derived by hand in this report.
• Reproduction. Inapplicable: the page states no rerun command and no coverage claim.
• Source and verdict fidelity. Pass. The statement, the locator (Remark 6, physical pp. 4–5, which are also the printed pages, in a five-page arXiv v1 PDF), the citation of Erdős and Graham pp. 37–38, the case split at $T/2$, the dyadic count and the $k>T/2$ clause all match the page images. The Standing paragraph claims author-recorded standing only. The labeling of supplied steps and of the Problem 18 questions is the subject of F2 and F3.

Weakest steps

1. The binomial tail and its composition with the counting inequality. For $0<x\le1$ and $0\le i\le k$, $x^{i-k}\ge1$, so $\binom Ti\le x^{-k}\binom Ti x^i$; summing over $i\le k$ and enlarging the sum to $i\le T$ (all terms are nonnegative) gives

$$\sum_{i\le k}\binom Ti\le x^{-k}(1+x)^T\le x^{-k}e^{xT},$$

by $1+x\le e^x$. For $1\le k\le T$ the choice $x=k/T\in(0,1]$ is admissible and gives $(T/k)^ke^k=(eT/k)^k$. Composition: for $k=h(n!)$, every integer $1\le m\le n!$ has a set $S_m$ of at most $k$ distinct divisors of $n!$ with sum $m$; $m\mapsto S_m$ is injective because a set determines its sum; when $k\le T$ the sets of at most $k$ of the $T$ divisors number exactly $\sum_{i\le k}\binom Ti$. Hence $n!\le(eT/k)^k$ and $\log n!\le k(1+\log T-\log k)\le k(1+\log T)$, the last step because $k\ge1$. If $k>T$ the tail bound is unavailable, but then $k>T/2$ and the page's second case applies, so the split is exhaustive. The step is the weakest only because the page verifies $k\le T$ and not $k\ge1$ before using it (F1); $k\ge1$ holds since $m=1$ is not the empty sum.

2. The large primes. For a prime $\sqrt n<p\le n$, $p^2>n$, so Legendre's formula $v_p(n!)=\sum_{j\ge1}\lfloor n/p^j\rfloor$ leaves only $v_p(n!)=\lfloor n/p\rfloor$. Let $r\ge0$ be the unique integer with $n/2^{r+1}<p\le n/2^r$ (the ranges partition $(0,n]$). Then $n/p<2^{r+1}$, so $\lfloor n/p\rfloor\le2^{r+1}-1$ and $\log(v_p(n!)+1)\le(r+1)\log2$; and $n/2^r\ge p>\sqrt n$. For $n\ge4$, $\sqrt n\ge2$, so Chebyshev's bound applies at $x=n/2^r$: the primes of the $r$-th range number at most $\pi(n/2^r)\le Cn/(2^r\log(n/2^r))$, and $\log(n/2^r)>\tfrac12\log n$ turns this into $2Cn/(2^r\log n)$. Summing the weight $(r+1)\log2$ over the ranges, and over all $r\ge0$ since the terms are positive,

$$\sum_{\sqrt n<p\le n}\log\bigl(v_p(n!)+1\bigr) \le\frac{2Cn\log2}{\log n}\sum_{r\ge0}\frac{r+1}{2^r} =\frac{8C\log2\cdot n}{\log n},$$

using $\sum_{r\ge0}(r+1)y^r=(1-y)^{-2}$ at $y=\tfrac12$. This is the page's display with the constant made explicit. The hypothesis $x\ge2$ is what the page's "let $n\ge4$" buys; for $n=3$ the range $r=0$ has $n/2^r=3\ge2$ but $\sqrt3<2$ leaves no margin, and the page rightly does not claim $n\le3$.

3. The small primes, the factorial bound, the division, and the other case. For $p\le\sqrt n$, Legendre gives $v_p(n!)\le\sum_{j\ge1}n/p^j=n/(p-1)\le n$, so $\log(v_p(n!)+1)\le\log(n+1)\le\log(2n)\le2\log n$ for $n\ge2$; there are at most $\sqrt n$ such primes, so they contribute at most $2\sqrt n\log n$. Since $(\log n)^2/\sqrt n$ has its maximum $16/e^2<2.2$ at $n=e^4$, $2\sqrt n\log n\le4.4\,n/\log n$; altogether $\log T\le C_1\,n/\log n$ with $C_1=8C\log2+4.4$ for $n\ge4$. For the factorial, $\log n!\ge\sum_{n/2<j\le n}\log j$; there are $\lceil n/2\rceil\ge n/2$ such $j$, each exceeding $n/2$, so the sum is at least $\tfrac n2\log\tfrac n2=\tfrac n2(\log n-\log2)\ge\tfrac n4\log n$ once $\log n\ge2\log2$, that is, for $n\ge4$. With $1\le n/\log n$ for $n\ge2$,

$$\frac n4\log n\le\log n!\le k\,(1+\log T)\le k\,(1+C_1)\frac n{\log n},$$

so $k\ge(\log n)^2/(4(1+C_1))$. In the other case, $1,\dots,n$ are $n$ distinct divisors of $n!$, so $T\ge n$ and $k>T/2\ge n/2$; as $(\log n)^2/n$ has its maximum $4/e^2$ at $n=e^2$, $n/2\ge(e^2/8)(\log n)^2$. Both cases give an explicit absolute constant, as the page states.

Strongest attack

The attack aimed at the one place where an absolute constant could fail: the dyadic ranges nearest $\sqrt n$. There $n/2^r$ is barely above $\sqrt n$, the divisor $\log(n/2^r)$ is only about half of $\log n$, and Chebyshev's bound needs $x\ge2$. The attempt was to make those ranges contribute more than $n/\log n$ to $\log T$, or to find an $n$ for which a range meets $x<2$. It failed on both counts: every range that contains a prime $p>\sqrt n$ has $n/2^r\ge p>\sqrt n\ge2$ for $n\ge4$, so Chebyshev's bound applies with the same $C$ in every range; the loss from $\log(n/2^r)>\tfrac12\log n$ is exactly the factor $2$ the page writes; and the weights $(r+1)\log2$ grow linearly while the counts halve, so the sum is $8C\log2\cdot n/\log n$ regardless of how many ranges occur. A second attempt, to inflate the small-prime contribution through $p=2$ (where $v_2(n!)$ is nearly $n$), is absorbed by $\log(n+1)$ per prime and at most $\sqrt n$ primes, which is below $4.4\,n/\log n$. A third attempt, to break the counting inequality by a mismatch between the divisor sets and the $T$-element set, failed because the divisors of $n!$ form exactly a $T$-element set and a set determines its sum. A fourth, to make the case $k>T/2$ weak by a small $T$, failed on $T\ge n$. The reconstruction survives.

Premises

• Chebyshev's bound. Interface: $\pi(x)\le Cx/\log x$ for every real $x\ge2$, $C$ absolute. Not held in the library; the source (p. 4) names it as "Chebyshev's bound $\pi(x)\ll x/\log x$" with no reference; it is a standard textbook theorem and the page names it as imported. Reading depth: none beyond the source's sentence. Used once, at $x=n/2^r\ge2$.
• Legendre's formula. Interface: $v_p(n!)=\sum_{j\ge1}\lfloor n/p^j\rfloor$. Standard; used by the page, and by the source in the same way, without being named (F4).
• Divisor count. Interface: $\tau(n!)=\prod_{p\le n}(v_p(n!)+1)$. Standard; used without being named (F4).
• Definition of $h(N)$. Taken by the page from the Theorem 1 reconstruction's Definitions; checked here against the source's p. 1: identical, including the fresh choice of divisors for each $m$.
• $n!$ is practical. Implicit in writing $h(n!)$; the source (p. 1) credits Erdős with $h(n!)\le n$, which implies it. Not proved on the page or in the source.
• Elementary facts derived in this report. $\sum_{r\ge0}(r+1)2^{-r}=4$; $\log(n+1)\le2\log n$ for $n\ge2$; $(\log n)^2\le(16/e^2)\sqrt n$ and $(\log n)^2\le(4/e^2)\,n$ for $n\ge1$; $n/\log n\ge1$ for $n\ge2$.
• Explicit assumptions. Only $n\ge4$, which the page states.
• Consumed local claims. None; the page consumes no native L-claim.

Findings

F1. Severity: suggested. Location: "the case $k\le T/2$ (so $k\le T$)". Defect: the page's binomial-tail bound is stated for $1\le k\le T$ and its use needs $k\ge1$ (the choice $x=k/T$ must be positive, and $(eT/k)^k$ and $\log(eT/k)$ are undefined at $k=0$); the page discharges $k\le T$ only. Witness: the page's paragraph "The binomial tail" opens "If $1\le k\le T$"; the source (p. 4) writes the bound under "if $k\le T/2$" and is silent on $k\ge1$ as well. The hypothesis holds: $k=h(n!)$ is a positive integer, since $n!$ is practical and $m=1$ is not the empty sum. Replacement: "the case $k\le T/2$ (so $1\le k\le T$, as $k\ge1$ because $m=1$ is not an empty sum)".

F2. Severity: suggested. Location: Qualifications, "The source splits at $k\le T/2$; the binomial-tail bound holds for all $k\le T$". Defect: the steps the page supplies are not marked as supplied. Witness: on p. 4 the source asserts, without proof or a range for $n$, the bound $\sum_{i\le k}\binom Ti\le(eT/k)^k$, the contribution $O(\sqrt n\log n)$ of the primes $p\le\sqrt n$, the count $O(n/(2^r\log n))$ of primes in a dyadic range, and (p. 5) $\log(n!)\asymp n\log n$; the page proves each, fixes the constants, and adds "let $n\ge4$", nowhere saying that these are the page's additions. Replacement, appended to Qualifications: "The source states without proof the binomial-tail bound, the contribution $O(\sqrt n\log n)$ of the primes $p\le\sqrt n$, the count $O(n/(2^r\log n))$ and $\log n!\asymp n\log n$; their proofs, the explicit constants and the range $n\ge4$, which makes $n/2^r>\sqrt n\ge2$ so that Chebyshev's bound applies, are supplied here. The binomial-tail bound needs $1\le k\le T$."

F3. Severity: suggested. Location: Qualifications, "questions (b) and (c) of Problem 18". Defect: the Problem 18 Statement asks its three questions in prose and letters none of them; the two questions the source restates are its second and third. Witness: the Problem 18 Statement, "Is it true that $h(n!)<n^{o(1)}$? Or perhaps even $h(n!)<(\log n)^{O(1)}$?", and the source, p. 5, "Erdős asked whether $h(n!)<n^{o(1)}$, or even $h(n!)<(\log n)^{O(1)}$ [2, pp. 37–38]". Replacement: "which are the second and third questions of Problem 18; the remark shows that any bound $h(n!)\le(\log n)^A$ valid for all large $n$ has $A\ge2$."

F4. Severity: note. Location: Standing, "The only imported input is Chebyshev's bound", and Definitions. Defect: the proof also rests on Legendre's formula (for $v_p(n!)\le n/(p-1)$ and for $v_p(n!)=\lfloor n/p\rfloor$ when $p^2>n$) and on $\tau(n!)=\prod_{p\le n}(v_p(n!)+1)$, neither stated. Witness: the page's paragraph "The divisor count", first three sentences. Both facts are elementary and the source (p. 4) uses them the same way, so the standing is unaffected. Replacement: in Definitions add "By Legendre's formula $v_p(n!)=\sum_{j\ge1}\lfloor n/p^j\rfloor$, and $\tau(n!)=\prod_{p\le n}(v_p(n!)+1)$."; in Standing write "The only imported input beyond these elementary formulas is Chebyshev's bound".

F5. Severity: note. Location: frontmatter desc, "from the Chebyshev estimate log tau(n!) << n/log n". Defect: the estimate $\log\tau(n!)\ll n/\log n$ is derived on the page from Chebyshev's bound $\pi(x)\ll x/\log x$; it is not itself the Chebyshev estimate. Witness: the source, p. 4, "the estimate $\log T\ll n/\log n$, which follows from Chebyshev's bound". Replacement: "from the bound log tau(n!) << n/log n that Chebyshev's estimate gives."

Verdict

Source fidelity: faithful. The statement, its quantifiers, the convention for $h$, the case split, the dyadic count, the $k>T/2$ clause and every locator match physical pp. 4–5 of the held arXiv v1 PDF, and the Standing sentence claims nothing beyond author-recorded standing.

The argument as reconstructed: sound. Every deduction was re-derived above with explicit constants; the one hypothesis the page uses without discharging, $k\ge1$ (F1), holds for every $n$. No required correction; the three suggested corrections (F1–F3) and two notes (F4, F5) concern labeling, a cross-reference and wording.

Limitations: this review is noncomputational; Chebyshev's bound was accepted as a standard imported theorem, no source for it being held; the Erdős and Graham pages 37–38 are not held and were not read; the definition of $h$ was checked against the source's p. 1 and the Theorem 1 reconstruction's Definitions only; the practicality of $n!$ was accepted from the source's citation of Erdős and not re-proved.

This focused review assigns no tier and changes no status.