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Subject and independence

Role. The reviewer is an independent reviewer in a fresh context, given only the commissioning assignment and charged with refutation. The reviewer took no part in writing the page, any other page in its folder, or the library card it cites, and read no other review.

Subject. Path wiki/research/erdos_18/hughes_corollary_3_reconstruction.md as it stood at 2026-09-28T05:03:27Z, read whole as of that time: the reconstruction page.

Artifact. The folder-name PDF hughes_2026_sums_distinct_divisors_factorials.pdf under library/divisors/hughes_2026_sums_distinct_divisors_factorials/ (arXiv:2609.10902v1, five pages, 271,095 bytes, held by the Hughes card). Physical pages 1–3 were extracted as layout text. Page images rendered: pages 1–3 at 150 dpi, page 1 at 110 dpi, and a 300 dpi crop of page 2 covering Theorem 2 through the end of the proof of Corollary 3. Physical page 2 (printed page 2) was read clause by clause on the 150 dpi image, and every displayed formula on it was read again on the 300 dpi crop; page 1 was read on the 110 dpi image for the logarithm convention and Theorem 1; page 3 was read in extracted text only, for the "window" vocabulary of its lower-range paragraph. The canonical conversion beside the PDF was read for Sections 1 and 2 and consulted for Section 3; the PDF decided, and the two did not differ on anything checked.

Allowed material read. The provenance paragraph of the Hughes card; the "Statement (as quoted)" section of the Theorem 2 page; in docs/verification.md the Erdos-specific "Whole-claim report" and "Audit checklist" subsections; docs/evidence.md "Source fidelity"; docs/math_authoring.md whole; the Statement paragraph of wiki/problems/divisors/E0018/_index.md. The page cites the Theorem 1 reconstruction as its consumer, not as an input, so it was not read; no other reconstruction page and not the folder index were read; the two other files under evidence/verify/ were seen by file name only. The 1993 Berend–Harmse paper is not held and was not read.

Exposures. Four, none used in any verdict and none bearing on the mathematics checked: (1) the card's _index.md printed whole, so its "Read status", "Bears on" (which names a record that supersedes Theorem 1) and "Overview" sections were seen beyond the provenance paragraph; (2) the Theorem 2 page printed whole, so its "Corollary 3 (the paper's own deduction)", "Reconstruction" and "Bears on" sections were seen beyond the statement; (3) the Problem 18 page's "Status" and "Provenance of the proof file" paragraphs, which follow the Statement paragraph under the same heading, were seen; (4) in docs/verification.md the shared "Audit checklist — the canonical failure modes" section and the opening of "Durable reports and current standing" printed alongside the two named subsections.

Restatement

Convention: $\log$ is the natural logarithm and $\lg t=\log_2t$ (source p. 1, the line before Section 1). For real $x\ge2^{16}$ the page sets $\varepsilon_x=x^{-(\lg x/2-\lg\lg x)}$, so that $\log(1/\varepsilon_x)=(\lg x/2-\lg\lg x)\log x$; the source defines the same quantity at integers $j$ only.

Imported theorem, in the form the preprint quotes from Berend–Harmse (1993), Theorem 2, and not checked against the 1993 paper: for every integer $n\ge2^{16}$ and every real $D$ with $\sqrt{(n-1)!}\le D\le\sqrt{n!}$ there exists a positive divisor $x$ of $n!$ with $|x/D-1|\le\varepsilon_n$. The preprint's display first gives the sharper bound $5\cdot10^7(\lg n/n)^{(\lg n-\lg\lg n+1)/2+\lg e}$ and then compares it to $\varepsilon_n$; the page consumes only the outer bound.

Result (Corollary 3): for all integers $j\ge2^{16}$ and $n\ge j$, and for every pair $a<b$ of consecutive positive divisors of $n!$ (no divisor of $n!$ lies strictly between them) whose geometric mean satisfies $\sqrt{(j-1)!}\le\sqrt{ab}\le\sqrt{j!}$, both endpoints included,

$$\log\frac ba\le3\varepsilon_j .$$

The constant $3$ is absolute and $\varepsilon_j$ depends on the window index $j$ alone, not on $n$, $a$ or $b$. Two auxiliary facts carry the proof and are stated for the consumer: $(\varepsilon_j)_{j\ge2^{16}}$ is strictly decreasing, and $\varepsilon_j\le\varepsilon_{2^{16}}=2^{-64}$ on that range.

Checklist

• Quantifiers and scope: pass. Every variable is universally quantified over the stated integer ranges; the window endpoints are inclusive on the page and in the source (p. 2, Corollary 3); the boundary cases $x=a$ and $x=b$ are covered by the weak inequalities of the two cases; no exceptional set appears.
• Circularity: pass. The conclusion is derived from the imported bound, $j!\mid n!$, and two elementary logarithm inequalities; nothing equivalent to the corollary is assumed.
• Model and convention changes: pass, with a note. The logarithm convention matches p. 1. The page extends $\varepsilon$ to a real argument for its monotonicity argument, where the source keeps $\varepsilon_j$ at integers and writes the exponent as a function of real $x$ only in its monotonicity sentence; the extension agrees with the source at every integer and changes no statement (F3).
• Finite and statistical overreach: pass. The only evaluated instances are $\varepsilon_{2^{16}}=2^{-64}$ and the bound comparison at $n=2^{16}$, and each comes with a monotonicity argument covering every larger argument; no finite check stands for a universal claim.
• Uniformity: pass. The constant $3$ is absolute; the threshold $u\le\tfrac14$ of the logarithm inequality is met uniformly through $\varepsilon_j\le2^{-64}$; no limit, sum or error term is exchanged.
• Extremal conclusions: inapplicable. No infimum, supremum, attained value or sharpness is claimed; the intermediate $\tfrac83\varepsilon_j$ is a bound, not an extremum.
• Consequences and composition: pass. Each "so" was rederived below. The imported interface is applied with $n=j$ and $D=\sqrt{ab}$, both inside its hypotheses, and used at exactly its quoted strength; the bridge from $j!$ to $n!$ is the divisibility $j!\mid n!$, which the page states.
• Computation: pass. The page attaches no code; its three numerical claims (the value about $-5.7$, the identity for the base-two logarithm of the bound ratio, and $16\log2-\log16-1>0$) were recomputed by hand and in floating point; the identity agrees with direct evaluation to $10^{-13}$ at sampled $n$ from $2^{16}$ to $10^{30}$. No enclosure or limit is involved.
• Reproduction: inapplicable. The page states no rerun command and retains no evidence directory; all derivations are inline and were rederived here.
• Source and verdict fidelity: pass with one required correction. The hypotheses, conclusion, both bounds of the quoted theorem, display (1), the citation of the 1993 paper, and the locators (physical p. 2, Theorem 2, display (1), Corollary 3 with proof) match the PDF; the standing sentence claims author-recorded only. The monotonicity derivation is supplied by the page and not marked as supplied (F1).

Weakest steps

1. The case $x\le a$ and the logarithm inequality. Rederived: with $D=\sqrt{ab}$ and $a<b$, $a/D=\sqrt{a/b}<1$. If $x\le a$ then $x/D\le a/D$, and the imported bound gives $x/D\ge1-\varepsilon_j$, so $1-\varepsilon_j\le\sqrt{a/b}$. Since $0<1-\varepsilon_j$ (as $\varepsilon_j\le2^{-64}$), squaring preserves the order: $(1-\varepsilon_j)^2\le a/b$, that is $b/a\le(1-\varepsilon_j)^{-2}$ and $\log(b/a)\le-2\log(1-\varepsilon_j)$. For $0\le u\le\tfrac14$, $1/(1-u)=1+u/(1-u)$, so $-\log(1-u)=\log(1+u/(1-u))\le u/(1-u)$ by $\log(1+t)\le t$, and $u/(1-u)\le\tfrac43u$ since $1-u\ge\tfrac34$. With $u=\varepsilon_j$ this gives $\log(b/a)\le\tfrac83\varepsilon_j<3\varepsilon_j$. Composition: the step needs only $\varepsilon_j\le\tfrac14$, which the size fact supplies with room to spare; the other case gives $\log(b/a)\le2\log(1+\varepsilon_j)\le2\varepsilon_j$, so $3$ covers both.

2. Monotonicity of the exponent. The source asserts it in one sentence; the page proves it. Rederived: with $L=\log x$, $(\lg x/2)\log x=L^2/(2\log2)$ and $\lg(\lg x)\log x=L\log(L/\log2)/\log2$, so

$$h(L):=\log\frac1{\varepsilon_x} =\frac1{\log2}\Bigl(\frac{L^2}2-L\log\frac L{\log2}\Bigr),\qquad h'(L)=\frac1{\log2}\Bigl(L-\log\frac L{\log2}-1\Bigr),\qquad h''(L)=\frac1{\log2}\Bigl(1-\frac1L\Bigr).$$

$h''>0$ for $L>1$, and $L\ge16\log2\approx11.09$ on the range, so $h'$ is increasing there; $h'(16\log2)=(16\log2-\log16-1)/\log2$ with $16\log2-\log16-1\approx7.32>0$. Hence $h'>0$ and $h$ is strictly increasing on $[16\log2,\infty)$, so $\varepsilon_x=e^{-h(\log x)}$ is strictly decreasing for real $x\ge2^{16}$, in particular along the integers. This is the source's assertion exactly (same function, same range). Composition: it yields $\varepsilon_j\le\varepsilon_{2^{16}}=2^{-64}$, the input of step 1; at $j=2^{16}$ the exponent of $1/j$ is $16/2-\lg16=8-4=4$ and $(2^{-16})^4=2^{-64}$, as the source prints.

3. The comparison of the two quoted bounds. Not consumed, but stated as checked, so a claim surface. Rederived with $m=\lg n$ and $t=\lg m$: the left bound $A$ has $\lg A=\lg(5\cdot10^7)+\bigl(\tfrac{m-t+1}2+\lg e\bigr)(t-m)$ and the right bound $B$ has $\lg B=-\tfrac12m^2+mt$. Expanding $(m-t+1)(t-m)=-(m-t)^2-(m-t)$ and collecting,

$$\lg\frac AB =\lg(5\cdot10^7)-\tfrac12t^2-\bigl(\tfrac12+\lg e\bigr)m +\bigl(\tfrac12+\lg e\bigr)t ,$$

the page's expression. At $m=16$, $t=4$: with $\lg(5\cdot10^7)\approx25.575$ and $\tfrac12+\lg e\approx1.943$ this is $25.575-8-31.083+7.771\approx-5.74$. Its derivative in $m$ is $-(\tfrac12+\lg e)+(\tfrac12+\lg e-t)/(m\log2)$, and for $m\ge16$ one has $t\ge4>\tfrac12+\lg e$, so both terms are negative and the expression decreases in $n$. Thus $A<B$ for all $n\ge2^{16}$. Composition: none; the page says so, and consumes only the outer bound.

Strongest attack

The attack aimed at the two places where a hidden hypothesis could enter: the application of the imported theorem and the case split. (a) Placing $D=\sqrt{ab}$ outside the theorem's range fails: the corollary's window hypothesis is verbatim the theorem's $D$-range at $n=j$, endpoints included, and $j\ge2^{16}$ is the theorem's own threshold. (b) Placing the divisor $x$ strictly between $a$ and $b$ fails: $x\mid j!$ and $j!\mid n!$ because $j\le n$, and $a<b$ are consecutive divisors of $n!$. (c) Making the squaring in the first case or the $u\le\tfrac14$ threshold fail requires $\varepsilon_j\ge\tfrac14$, but $\varepsilon_j\le2^{-64}$ by step 2. (d) Breaking the monotonicity near the threshold fails: $h'$ is positive at $L=16\log2$ and increasing beyond. (e) Exhausting the constant $3$ fails: the worst case is $\tfrac83$. (f) The second-hand import cannot be attacked here: the page consumes the bound exactly as the preprint prints it and declares the 1993 paper unread, so any discrepancy between the preprint and the 1993 paper lies outside this page's claim. (g) The locators were checked on the page image: Theorem 2, display (1), Corollary 3 and its proof all sit on physical page 2, printed page 2, of the five-page PDF, and the 1993 citation matches the preprint's reference [1]. Every mathematical attack failed. What survived is a labeling defect: the monotonicity derivation has no counterpart in the source and is not marked as supplied (F1), unlike the neighboring numerical comparison, which the page marks as "checked here".

Premises

• Berend–Harmse (1993), Theorem 2, as quoted by the preprint (p. 2). Interface: for every integer $n\ge2^{16}$ and every real $D$ with $\sqrt{(n-1)!}\le D\le\sqrt{n!}$ there is a divisor $x$ of $n!$ with $|x/D-1|\le(1/n)^{\lg n/2-\lg(\lg n)}$. Held source: the preprint, read on the page image clause by clause, both bounds and their exponents checked symbol by symbol on the 300 dpi crop. The 1993 paper: not held, not read. Standing on the page: imported, second-hand, and named as such. Explicit assumption: the preprint's quotation is faithful to the 1993 paper; this review cannot check it and the page does not claim it.
• Elementary facts, supplied by the page without a source and verified here: $j\le n$ implies $j!\mid n!$; the definition of consecutive divisors; $\log(1+t)\le t$ for $t>-1$; one-variable calculus for the monotonicity.
• Local claims consumed: none; no L-claim and no other reconstruction page is an input. No batch acceptance order applies.

Findings

F1. Severity: required. Location: "Monotonicity. Write $L=\log x$. The exponent in (1) is ... whose derivative in $L$ is". Defect: this derivation is the page's own. The source (physical p. 2, the sentence between display (1) and Corollary 3) reads "Since $(\frac{\lg x}2-\lg(\lg x))\log x$ is increasing for $x\ge2^{16}$, the sequence $(\varepsilon_j)_{j\ge2^{16}}$ is decreasing" and gives no argument; the page does not mark its derivative computation as supplied, while it marks the neighboring bound comparison as "checked here", so a reader cannot tell which of the two facts the source proves. Witness: the quoted sentence at p. 2; nothing on pp. 1–5 computes a derivative. Proposed replacement: begin the paragraph with "Monotonicity. The source asserts, without proof, that $(\frac{\lg x}2-\lg(\lg x))\log x$ is increasing for $x\ge2^{16}$; the derivative argument below is supplied here." and, after "and it increases with $L$", add "(its derivative is $(1-1/L)/\log2>0$ for $L>1$)".

F2. Severity: note. Location: "The window of index $j\ge2$ is the interval". Defect: the definition is used nowhere on the page (the Statement writes its hypothesis out), and the vocabulary comes from the source's Section 3, physical p. 3 ("Each window $[\sqrt{(j-1)!},\sqrt{j!}]$ has logarithmic width $\frac12\log j$"), outside the "physical p. 2" locator; the index range $j\ge2$ is the page's own, the source using windows only at indices of at least $2^{16}$. Witness: p. 3, first paragraph of "Lower range". Proposed replacement: either delete the sentence, or append "(the vocabulary of the source's Section 3, p. 3; the index range is a convention of this page)".

F3. Severity: note. Location: "For real $x\ge2^{16}$ put". Defect: the source defines $\varepsilon_j$ at integers only ("the rightmost bound above at $n=j$", p. 2) and uses the real variable only inside the exponent of its monotonicity sentence; the page's real-variable definition is an unmarked reading. It agrees with the source at every integer and changes no statement. Proposed replacement: append "(the source defines $\varepsilon_j$ at integers $j$; the real argument serves only the monotonicity argument below)".

Verdict

Source fidelity: faithful with corrections (one required correction, F1, a missing supplied-step label; two notes). The argument as reconstructed: sound; every deduction was rederived and no step fails. Limitations: the imported theorem was checked only against the preprint's quotation, since the 1993 paper is not held; the review covers Corollary 3 and the two error-term facts and not their consumer; the numerical checks are floating-point confirmations of hand derivations, not certified enclosures, and nothing on the page depends on them beyond the sign of a comparison the page does not consume. This focused review assigns no tier and changes no status.