---

Subject and independence

The reviewer is an independent reviewer working in a fresh context from the commissioning assignment alone; the reviewer took no part in writing the connections page, the reconstruction pages, the shared lemmas or any problem page, and read no earlier review or grade of any of them.

Subject: path wiki/research/erdos_18/connections.md as it stood at 2026-09-28T13:50:48Z, the H2 section "Other divisor problems examined", lines 736–778 (the next H2, "Leads created", starts at line 779). The section names no lead page; the two lead folders named under "Leads created" belong to the Problem 304/293 and Problem 18(a) sections and were not read.

Read, as of the time named above: the section in full; the page's opening conventions paragraph (lines 19–45) and the Shared lemmas section (lines 47–243) in full, as inputs whose proofs are another reviewer's subject; the reconstruction pages Corollary 3 (in full) and Lemma 4 (definitions, statement, consequence and the factorial ratio proof), as premises; the Statement paragraphs of the fourteen problem pages the section names, E0144, E0886, E0887, E1100, E0468, E0469, E0470, E1054, E0350 (with its Formulation paragraph, which sits inside the statement block), E0882, E0446, E0448, E0449 and E0450; the provenance paragraph of the library card Hughes (2026) and its result page Theorem 2 (quoted) in full; the text layer of the held Hughes PDF, searched for every mention of Berend–Harmse and of a Theorem 3 or Theorem 4; the commissioned sections of docs/verification.md, docs/approach_vetting.md, docs/evidence.md "Source fidelity" and docs/math_authoring.md. The Berend–Harmse 1993 paper is not held and was not read.

Exposures: while locating the Statement paragraph of E0144, its Status line (status, prize and tags) was displayed; while reading the provenance paragraph of the Hughes card, its "Read status" and "Bears on" paragraphs and the first lines of its Overview were displayed, including one sentence comparing the paper's Theorem 1 with a site-accepted proof. File names (not contents) of the folder's evidence/verify/ directory and of untracked files in the worktree were displayed by directory listings. Nothing from any review, grade, Current assessment or Known results section reached the reviewer. Small exact computations (divisor enumeration and subset-sum tables below $3000$, and the two width comparisons at $n=2^{16}$ and $n=10^6$) were run as sanity checks of derivations that are written out below; they are base-case coverage only and carry no weight in the verdicts.

Restatement

Conventions of the page: $h(N)$, $\tau(N)$, $v_p$, $L_x$, $\varepsilon_j$ and the windows $[\sqrt{(j-1)!},\sqrt{j!}]$ are as in the page's conventions paragraph and the Corollary 3 page; a number $N$ has the ratio-two property when consecutive divisors of $N$ have ratio at most $2$. Lemma A (shared) states: if $v_2(N)\ge\lfloor\log_2p\rfloor$ for every odd prime $p\mid N$, then $N$ has the ratio-two property; the hypothesis holds for $N=n!$ and $N=L_x$.

• S0 (preamble). A search of the problems, research and theory trees for the objects of the proofs surfaced the pages listed, and each is a resemblance without a transfer. This is a process statement about a search and a summary of S1–S8; it is not a mathematical claim.
• S1 (Problem 144). For every $n$ and every $x$, the numbers $n!$ and $L_x$ have two divisors $d_1<d_2<2d_1$, and this is said to follow as "the trivial instance" of Lemma A. The target, that the set of integers with two such divisors has natural density $1$, is a density-one statement over all integers, which no statement about the sparse families $n!$ and $L_x$ reaches.
• S2 (Problems 886 and 887, widths). For $N=n!$ and fixed $\eta>0$, $C>0$, the interval $(\sqrt N,\sqrt N+N^{1/2-\eta})$ is a multiplicative window of relative width $N^{-\eta}$ and $(\sqrt N,\sqrt N+CN^{1/4})$ one of relative width $CN^{-1/4}$; both are exponentially narrower, as $n\to\infty$, than the precision $\varepsilon_n=\exp(-(\log n)^2/(2\log2)\,(1-o(1)))$ of the imported Berend–Harmse estimate.
• S3 (Problems 886 and 887, no transfer). The reconstructed estimate (Corollary 3: for $j\ge2^{16}$, $n\ge j$ and consecutive divisors $a<b$ of $n!$ with $\sqrt{ab}$ in the window of index $j$, $\log(b/a)\le3\varepsilon_j$) gives no information on the number of divisors of $n!$ in windows of relative width $N^{-\eta}$ or $CN^{-1/4}$; in particular nothing toward either problem's requested uniform upper bound ($O_\eta(1)$ for all large $n$, or an absolute $K$ valid for every $C$).
• S4 (Problems 886 and 887, parenthetical). Theorems 3 and 4 of the Berend–Harmse 1993 paper concern the most balanced factorization of $n!$ and are not reconstructed in the folder. This is a characterization of a source that is not held.
• S5 (Problem 1100). With $\tau_\perp(N)$ the number of indices $i$ with $\gcd(d_i,d_{i+1})=1$ among the sorted divisors of $N$, the pairs $(d,d+1)$ with $1\le d\le n-1$ give $\tau_\perp(n!)\ge n-1$ for every $n\ge1$; this lower bound is far below the threshold $\exp((\log n!)^{o(1)})$ of the problem's second question (whether $\tau_\perp(N)<\exp((\log N)^{o(1)})$ for all $N$); the folder's mechanisms bound divisor gaps and say nothing about coprimality of consecutive divisors; no transfer.
• S6 (Problems 468, 469, 470 and 1054). The four problems concern whether a number $n$ itself, or a given $n$, has one representation of a prescribed shape (an initial partial sum of the divisor chain of some $m$; a sum of distinct proper divisors; the failure of that for abundant $n$; a sum of the $k$ smallest divisors of some $m$), not whether every $m\le N$ has a short representation; the practical-number mechanism (Lemma 3.1 extensions, greedy expansions, the ratio-two property) does not address them; no transfer.
• S7 (Problems 350 and 882). Problem 350 concerns finite sets with pairwise distinct subset sums and Problem 882 sets whose nonempty subset sums contain no two distinct elements one dividing the other; these are "the opposite of the many-representations property that practicality asserts"; no transfer.
• S8 (Problems 446, 448, 449 and 450). The four problems are density statements about the divisors of typical integers in short multiplicative intervals, and the families studied in the folder ($n!$, $L_x$, the extension chains, practical numbers) have natural density zero; no transfer.

Checklist

• Quantifiers and scope. Defect found in S1: Lemma A's conclusion is the non-strict ratio bound $b\le2d$, while the Problem 144 property is the strict $d_2<2d_1$; the boundary cases $n\le2$ and $x\le2$ fail the property (F1). S5's comparison is loosely phrased, since $\exp((\log N)^{o(1)})$ is a growth class and $n-1$ lies inside it (F6). The almost-all targets of S1 and S8 are correctly kept apart from the density-zero families. Elsewhere clean.
• Circularity. Inapplicable: no bullet derives a target from itself; the only deductions are the width computations, the trivial coprime pairs and the appeal to Lemma A.
• Model and convention changes. The section's $\tau_\perp$ agrees with the E1100 definition; the window intervals agree with E0886 and E0887; the page's $h$ convention is not used in the section. The word "window" is used for the precision $\varepsilon_n$, whereas the Corollary 3 page reserves it for $[\sqrt{(j-1)!},\sqrt{j!}]$ (F5, note). The reason given in S6 substitutes a shape mismatch for the actual relation, which is an entailment (practical and abundant implies pseudoperfect; F3), and the reason in S7 asserts an opposition that the divisor set of $2^k$ refutes (F4); neither affects the no-transfer conclusions.
• Finite and statistical overreach. Clean: no finite case is cited as more than an example; the reviewer's own finite checks are not used as evidence.
• Uniformity. Clean: the width comparison in S2 holds for every fixed $\eta>0$ and $C>0$ with the explicit rate rederived below; the $(1-o(1))$ in $\varepsilon_n$ is the explicit factor $1-2\log(\log n/\log2)/\log n$ of the Corollary 3 page. The dropped constant $C$ in "relative width $(n!)^{-1/4}$" is harmless (F5).
• Extremal conclusions. Inapplicable: no bullet asserts sharpness or an attained extremum.
• Consequences and composition. S1 composes Lemma A's conclusion with the strict target without the bridge (a pair of consecutive divisors with ratio strictly below $2$), which exists for $n,x\ge3$ but is not supplied by Lemma A (F1). S6's and S7's "hence no transfer" survive separately from their reasons (F3, F4). S3's "says nothing" is an honest negative: an upper bound on consecutive gaps yields neither an upper bound on divisor counts nor a lower bound in windows narrower than the gap bound.
• Computation. Inapplicable to the section (no computation is claimed); the arithmetic of S2 and S5 was rederived by hand below.
• Reproduction. Inapplicable: the section states no rerun command or coverage claim.
• Source and verdict fidelity. The fourteen problem descriptions match the Statement paragraphs (checked clause by clause; the section's status labels were not checked, status text being outside the read set). The Berend–Harmse import is used at the form the Theorem 2 page and the Corollary 3 page print. S4 characterizes theorems of a paper that is not held; the held Hughes PDF does not mention them, and the page's own "Sources read" paragraph says text-layer statements are marked, yet this one is not marked (F2).

Weakest steps

W1 (S1): from the ratio-two property to two divisors $d_1<d_2<2d_1$. Lemma A gives, for every divisor $d<N$, a divisor in $(d,2d]$. For $N=2^k$ the hypothesis of Lemma A holds vacuously and every pair of consecutive divisors has ratio exactly $2$, so $N=2^k$ has the ratio-two property and no pair $d_1<d_2<2d_1$; hence the strict property is not a consequence of Lemma A's conclusion. For $N=n!$ with $n\ge3$ and for $N=L_x$ with $x\ge3$ the pair $(2,3)$ divides $N$ and $3<2\cdot2$, so the property holds directly; for $n\le2$ and $x\le2$ the divisor sets are $\{1\}$ or $\{1,2\}$ and it fails. So the bullet's content is right for $n,x\ge3$, by a witness rather than by Lemma A.

W2 (S2): the width comparison. The relative width of $(\sqrt N,\sqrt N+N^{1/2-\eta})$ is $N^{1/2-\eta}/N^{1/2}=N^{-\eta}$ and that of $(\sqrt N,\sqrt N+CN^{1/4})$ is $CN^{-1/4}$. With $N=n!$, $\log(1/N^{-\eta})=\eta\log n!\ge\eta(n\log n-n)$, while the Corollary 3 page's display (1) gives $\log(1/\varepsilon_n)\le(\log n)^2/(2\log2)$. Hence

$$\frac{N^{-\eta}}{\varepsilon_n} \le\exp\Bigl(\frac{(\log n)^2}{2\log2}-\eta(n\log n-n)\Bigr)\to0$$

at a rate exponential in $n\log n$, and the same holds with $C N^{-1/4}$ in place of $N^{-\eta}$. At $n=2^{16}$ the two logarithms are about $6.6\cdot10^5\eta$ against $44.4$.

W3 (S6): the relation between practicality and pseudoperfection. Let $n$ be practical (every $1\le m\le n$, or every $m<n$, is a sum of distinct divisors of $n$) with sorted divisors $1=d_1<\cdots<d_\tau=n$. For $i<\tau$ the target $m=d_{i+1}-1<n$ is a sum of distinct divisors, each at most $m<d_{i+1}$, hence among $d_1,\dots,d_i$; so $d_{i+1}\le1+d_1+\cdots+d_i$ for every $i<\tau$. The standard completeness lemma (if $a_1=1$ and $a_{i+1}\le1+a_1+\cdots+a_i$ for $i<k$, then every integer in $[0,a_1+\cdots+a_k]$ is a sum of distinct $a_i$; induction on $k$, subtracting $a_k$ from targets above $a_1+\cdots+a_{k-1}$) applied to the proper divisors $d_1,\dots,d_{\tau-1}$ shows that every integer in $[0,\sigma(n)-n]$ is a sum of distinct proper divisors of $n$. Two consequences: $\sigma(n)\ge2n-1$ for every practical $n$ (take $i=\tau-1$), and if $\sigma(n)\ge2n$ then $n$ itself is such a sum, so $n$ is pseudoperfect and, in the terms of Problem 470, no weird number is practical. So the "shape" reason of S6 is wrong for Problems 469 and 470: the every-target property entails the one-representation property whenever $n$ is abundant or perfect. The no-transfer conclusion still holds: odd $n>1$ is never practical (the target $2$ is not a sum of divisors of an odd $n$), so nothing is learned about odd weird numbers; primitive weird numbers are non-practical by the above, so the mechanism cannot produce or count them; and the convergence of $\sum1/n$ over primitive pseudoperfect numbers is untouched, since the mechanism says nothing about which pseudoperfect numbers are primitive.

Strongest attack

• S1. Attack: find $N$ satisfying Lemma A's hypothesis without the strict property. Witness $N=2^k$ (and $N=1!,2!,L_1,L_2$ within the named families). The attack succeeds against the sentence's appeal to Lemma A and against the boundary cases, and fails against the content for $n,x\ge3$ (witness $2<3<4$) and against the no-transfer conclusion (a density-one statement over all integers is untouched by two sparse families). Recorded as F1.
• S2. Attack: seek $\eta$ or $n$ for which $N^{-\eta}$ is not below $\varepsilon_n$. Fails: W2 gives the explicit inequality for every fixed $\eta>0$ once $\eta(n\log n-n)>(\log n)^2/(2\log2)$, which holds for all large $n$ and, for instance, for every $\eta\ge10^{-3}$ at $n\ge2^{16}$.
• S3. Attack: extract from Corollary 3 either an upper bound on the number of divisors in a window or a lower bound in a window narrower than $3\varepsilon_n$. Fails: the estimate is an upper bound on the ratio of consecutive divisors only; it implies at least one divisor in every multiplicative window of relative width above the gap bound and nothing else. The problem windows lie just above $\sqrt{n!}$, outside the range $\sqrt{ab}\le\sqrt{n!}$ of the statement, but the map $d\mapsto n!/d$ mirrors the divisor set, so this changes nothing.
• S4. Attack: check the characterization against a held source. Not possible: the 1993 paper is not held, and the held Hughes PDF cites only its Theorem 2 and Remark 2. Recorded as F2.
• S5. Attack: find a pair $(d,d+1)$, $1\le d\le n-1$, that is not a coprime pair of consecutive divisors of $n!$, or two such pairs sharing an index. Fails: $d,d+1\le n$ both divide $n!$, no integer lies strictly between them, $\gcd(d,d+1)=1$, and distinct $d$ give distinct indices; so $\tau_\perp(n!)\ge n-1$ (with equality of the count of witnesses, and the true value larger from $n=5$ on). The comparison with the problem's threshold is loose (F6) but not false: for every fixed $c>0$, $n-1=\exp(\log(n-1))$ is far below $\exp((\log n!)^c)$.
• S6. Attack: exhibit a transfer from the practical-number mechanism to one of the four problems. W3 exhibits an entailment (practical and abundant implies pseudoperfect) that the bullet's reason denies, but no transfer to the questions asked: Problems 469 and 470 ask about primitive pseudoperfect numbers, odd weird numbers and primitive weird numbers, all outside the reach of a statement about practical numbers (W3); Problems 468 and 1054 ask about initial-segment sums of a divisor chain, a shape the mechanism never produces. The conclusion survives; the reason is corrected in F3.
• S7. Attack: exhibit a set that is at once dissociated and the divisor set of a practical number. Witness: the divisors $\{1,2,\dots,2^k\}$ of the practical number $2^k$, which is also the extremal set of Problem 350. So the properties are compatible, and practicality asserts a covering property (every target hit at least once), not "many representations". The attack succeeds against the word "opposite" and fails against the no-transfer conclusion: nothing in the folder bounds $\sum1/a$ over dissociated sets or the size of a divisibility-free subset-sum set; for Problem 882 the divisor set of any $n\ge2$ contains $1\mid2$ and is never admissible. Recorded as F4.
• S8. Attack: find one of the four problems for which a density-zero family could bear on the question. Fails for 446, 448 and 449, which are statements about natural density or almost all $n$, unchanged by any density-zero set; for 450, a uniform statement over all short intervals $(x,x+y)$, the reason applies only indirectly (a sparse family gives no upper bound valid for every interval), and there is still no transfer (F7, note). That $n!$, $L_x$ and the extension chains have density zero is immediate from their growth; that practical numbers do is a known theorem not established in the folder, and the bullet does not depend on it.

Premises

• Lemma A (shared, connections page lines 52–78): hypothesis $v_2(N)\ge\lfloor\log_2p\rfloor$ for every odd prime $p\mid N$; conclusion the ratio-two property (non-strict), practicality, distinct greedy divisors and $h(N)\le\log_2N+1$; the hypothesis is stated to hold for $n!$ and $L_x$. Author-recorded; its proof is another reviewer's subject. Used in S1 only for the non-strict ratio bound, within its hypothesis; the interface confirmed at the use, with the strictness gap noted in F1.
• Corollary 3 reconstruction (folder page, read in full): defines $\varepsilon_x=(1/x)^{\lg x/2-\lg\lg x}$ for $x\ge2^{16}$ with $\log(1/\varepsilon_x)=(\log x)^2/(2\log2)\,(1-2\log(\log x/\log2)/\log x)$, and states $\log(b/a)\le3\varepsilon_j$ for consecutive divisors of $n!$ with geometric mean in the window of index $j\le n$, $j\ge2^{16}$. Independently reviewed as of 2026-09-28T05:03:27Z per the assignment; consumed at that standing. Used in S2 (form of $\varepsilon_n$; rederived the expansion of display (1)) and S3 (what the estimate says).
• Lemma 4 reconstruction (folder page; definitions, statement, consequence and the factorial ratio proof read): not used by the section directly; it is the source of Lemma A's consequence clause.
• Hughes (2026) library card provenance paragraph and Theorem 2 page: the Berend–Harmse estimate is quoted second-hand from the held preprint (arXiv:2609.10902v1, p. 2); the 1993 paper is not held. Held PDF: text layer searched for the cited theorems (only Theorem 2 and Remark 2 of Berend–Harmse appear). Reading depth: the Theorem 2 page in full, the card to its Overview.
• Problem pages: the Statement paragraphs of the fourteen pages listed above, read in full and compared clause by clause with the section's descriptions; nothing else on those pages was read except the exposures disclosed.
• Standard facts used by the reviewer: the completeness lemma for sequences with $a_{i+1}\le1+a_1+\cdots+a_i$ (proved inline in W3); $\log n!\ge n\log n-n$; the mirror symmetry $d\mapsto n!/d$ of the divisor set.

Findings

• F1 (suggested; S1, lines 744–746: "Lemma A is the trivial instance for $n!$ and $L_x$"). Defect: Lemma A concludes the non-strict bound $b\le2d$, and the Problem 144 property needs $d_2<2d_1$; the two differ, and the named families fail the property at $n\le2$ and $x\le2$. Witness: $N=2^k$ satisfies Lemma A's hypothesis and has no divisors $d_1<d_2<2d_1$; $2!$ and $L_2$ likewise. Replacement: "For $n,x\ge3$ the divisors $2<3<4$ of $n!$ and $L_x$ give the property outright (Lemma A's non-strict bound alone does not, as $N=2^k$ shows); the target is a density-one statement over all $n$."
• F2 (suggested; S4, lines 753–755: "Berend and Harmse's Theorems 3 and 4 concern the most balanced factorization of $n!$ and are not reconstructed here"). Defect: a characterization of a source that is not held, taken (per the page's "Sources read" paragraph) from a text layer with garbled displays, and not marked as such although that paragraph says such statements are marked; the held Hughes PDF cannot confirm it. Witness: no held source mentions those theorems. Replacement: either drop the parenthetical or mark it, e.g. "(Berend and Harmse's Theorems 3 and 4, read in the Numdam text layer and not verified on the page images, concern the most balanced factorization of $n!$ and are not reconstructed here.)"
• F3 (suggested; S6, lines 765–767: "Each concerns whether $n$ itself, or a given $n$, has one representation of a prescribed shape, not whether every $m\le N$ has a short one; the practical-number mechanism does not address them"). Defect: for Problems 469 and 470 the two properties are not disjoint shapes but related by entailment: every practical $n$ with $\sigma(n)\ge2n$ is pseudoperfect, so no weird number is practical (W3). Witness: the derivation W3; for instance every $2^ap$ with $p$ an odd prime below $2^{a+1}$ is practical, abundant or perfect, and pseudoperfect. The no-transfer conclusion stands for the questions asked. Replacement: "Problems 468 and 1054 concern initial-segment sums of a divisor chain, a shape the mechanism never produces. For Problems 469 and 470, practicality is stronger than pseudoperfection on abundant numbers (every practical $n$ with $\sigma(n)\ge2n$ is a sum of distinct proper divisors, so weird numbers are never practical), but the questions asked, about odd weird numbers (odd $n>1$ is never practical), primitive weird numbers and the reciprocal sum of primitive pseudoperfect numbers, lie outside what a statement about practical numbers can reach."
• F4 (suggested; S7, lines 770–771: "the opposite of the many-representations property that practicality asserts"). Defect: practicality asserts that every target $m\le n$ is hit at least once, not that it has many representations, and dissociation is compatible with it. Witness: the divisor set $\{1,2,\dots,2^k\}$ of the practical number $2^k$ has pairwise distinct subset sums and is the extremal set of Problem 350. Replacement: "Problem 350 asks for the extremal reciprocal sum under distinct subset sums and Problem 882 for the extremal size under divisibility-free subset sums; practicality asks only that every target be hit, which is compatible with distinct subset sums (the divisors of $2^k$), and none of the mechanisms here bounds either extremal quantity."
• F5 (note; S2, lines 750–752). The phrase "the Berend–Harmse window $\varepsilon_n$" uses "window" for the precision, whereas the Corollary 3 page reserves "window" for $[\sqrt{(j-1)!},\sqrt{j!}]$; and the second relative width is $C(n!)^{-1/4}$, the constant being dropped. Neither affects the comparison. Replacement: "the Berend–Harmse precision $\varepsilon_n$" and "relative width $(n!)^{-\eta}$ or $C(n!)^{-1/4}$".
• F6 (note; S5, line 758: "far below the question's $\exp((\log n!)^{o(1)})$"). $\exp((\log N)^{o(1)})$ is a growth class, and $n-1=\exp(\log(n-1))$ belongs to it at $N=n!$; the intended comparison is with $\exp((\log n!)^c)$ for fixed $c>0$, below which $n-1$ indeed lies. Replacement: "well inside the bound the second question conjectures".
• F7 (note; S8, lines 775–777). Problem 450 is a uniform statement over all short intervals $(x,x+y)$ rather than a density statement, so the density-zero reason applies to it only indirectly; the no-transfer conclusion is unaffected.

Verdict

• S0: not a mathematical claim; no verdict.
• S1: refutation-failed for the content restricted to $n,x\ge3$, by the witness $2<3<4$ and the density-one nature of the target; the appeal to Lemma A does not supply the strict inequality, and the boundary cases fail (F1, suggested).
• S2: refutation-failed.
• S3: refutation-failed.
• S4: no mathematical verdict; a characterization of a source outside the read set, unverifiable here and unmarked (F2, suggested).
• S5: refutation-failed (F6, note).
• S6: refutation-failed for the no-transfer conclusion; the stated reason is inaccurate for Problems 469 and 470 and is replaced in F3 (suggested).
• S7: refutation-failed for the no-transfer conclusion; the word "opposite" is false as a mathematical characterization, witness the divisor set of $2^k$, and is replaced in F4 (suggested).
• S8: refutation-failed (F7, note).

Limitations: the section's status labels for the fourteen problems were not checked, status text being outside the read set; the Berend–Harmse 1993 paper is not held, so S4 and the second-hand import behind $\varepsilon_n$ were taken at the standing the Corollary 3 page records; the proofs of the shared lemmas and of the reconstruction pages were not re-examined beyond their interfaces. Required corrections: none. Suggested: F1–F4. Notes: F5–F7. This focused review assigns no tier and changes no status.