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Subject and independence

Role: independent reviewer in a fresh context, commissioned for refutation, who took no part in writing the reviewed page or any page in its folder and received only the assignment, the frozen subject and the read set below.

Frozen subject: path wiki/research/erdos_18/doorn_theorem_1_1_reconstruction.md as it stood at 2026-09-28T05:03:27Z, read in full as of that time.

Artifact: the note Practical numbers and Egyptian fractions (author line as printed: Wouter van Doorn and GPT-6 Astra Pro), the seven-page folder-name PDF held under the library card; no canonical conversion sits beside it as of that time. Its text was extracted in layout mode for all seven pages, and page images were rendered at 150 dots per inch for all seven pages. Pages read on the images: p. 1 clause by clause (the definitions of practical and of $h$, and Theorem 1.1); p. 5 clause by clause (the Section 4 opening sentence and the statement of Proposition 4.1), with the proof text on pp. 5–6 read once and not checked; p. 6 (the end of that proof and the Section 5 remark that fixes $p_*=3$); p. 2 (the convention that logarithms are natural and the AI-usage section). Pages 3, 4 and 7 were seen only in the text extraction. Printed and physical page numbers coincide.

Allowed material actually read, all as of 2026-09-28T05:03:27Z:

• the Proposition 4.1 reconstruction, the one input the page cites: read in full (Statement, Proof and Qualifications), although the deduction under check needs only its Statement;
• the provenance paragraph of the library card and the Statement of the theorem result page;
• the Statement paragraph of Problem 18;
• docs/verification.md "Whole-claim report" and both "Audit checklist" sections, docs/evidence.md "Source fidelity", and docs/math_authoring.md.

Exposures. The whole library card was printed while locating its provenance paragraph, so its Read status, Bears on, Overview and Lean sections were seen; the whole theorem result page was printed, so its Proof pointer, Lean, Standing and Bears on sections were seen; the frontmatter description of the problem page, which summarizes status, was seen while locating its Statement. None of this entered a verdict: every verdict below rests on the page, the artifact and the problem statement. No evidence folder, folder index, Current assessment, Known results, other review or web page was read.

Restatement

Conventions: logarithms are natural (artifact p. 2); $c_0=14/\log2$, about $20.198$. A positive integer $n$ is practical when every integer $m$ with $1\le m\le n$ is a sum of distinct positive divisors of $n$; for practical $n$, $h(n)$ is the least integer $k$ such that every such $m$ is a sum of at most $k$ distinct positive divisors of $n$ (artifact p. 1). The claim (Theorem 1.1, p. 1): the set of practical $n$ with

$$h(n)\le c_0(\log\log n)^2$$

is infinite. There is no rate, no uniformity clause and no statement about factorials. The deduction on the page consumes exactly the statement of Proposition 4.1 (p. 5): there exist an integer $E\ge4$ and a real $x_0>e^e$ such that for every real $x\ge x_0$ and every odd prime $p_*$ there is a practical $n$ with $x\le n<x^2$, $2^E\parallel n$, $p_*\nmid n$ and $h(n)\le c_0(\log\log x)^2-1<c_0(\log\log n)^2$; only the practicality, the interval and the last two inequalities are used.

Checklist

• Quantifiers and scope. Pass. "Infinitely many" is obtained from an injective sequence, not from an unbounded-set argument; the boundary $x=x_0$ is inside the proposition's range $x\ge x_0$; the derived bound is strict and the theorem claims only $\le$, so nothing is strengthened.
• Circularity. Pass. The deduction uses the proposition and nothing equivalent to the theorem; the proposition's own proof is outside this review's remit and is not consumed here.
• Model and convention changes. Pass with one note (F3). The page's $h$ is the note's $h$ (range $m\le n$); the page's comparison with the site's $h$ (range $m<n$) is proved correct below; the logarithm convention is not stated on the page.
• Finite and statistical overreach. Inapplicable. The deduction contains no finite verification and no averaging; the averaging inside Lemma 3.3 belongs to the imported proposition.
• Uniformity. Pass. $E$ and $x_0$ are fixed once by the proposition; the page fixes $p_*=3$ before choosing any $x$, so the deduction does not even need $x_0$ to be uniform in $p_*$; $c_0$ is explicit.
• Extremal conclusions. Inapplicable. No infimum, supremum or sharpness is claimed; the approximation $c_0\approx20.2$ was rechecked ($14/0.6931\ldots=20.198$).
• Consequences and composition. Pass with one suggested label (F1). Each "hence" was rederived; the consumed clause is used at exactly the proposition's strength; the consequence sentences in "What it would settle" were checked separately against the problem statement (F5 notes an omission of context, not an error).
• Computation. Inapplicable. Nothing is computed beyond the decimal value of $c_0$.
• Reproduction. Inapplicable. The page states no rerun commands; the locators (seven pages, Theorem 1.1 on p. 1, Proposition 4.1 on pp. 5–6, the Section 4 opening) were rechecked on the page images.
• Source and verdict fidelity. Pass with notes (F2, F4, F5). The Statement matches p. 1 clause by clause; the Section 4 opening is paraphrased ("somewhat stronger" became "the stronger"), which changes nothing; the standing sentences claim no tier and label the page's reading as a reading; one causal attribution, one undated site observation and one omitted piece of the note's own framing are noted.

Weakest steps

1. Infinitely many distinct $n$ from the iteration. Put $x_i=x_0^{2^i}$ for $i\ge0$. Since $x_0>e^e>1$, $x_i\ge x_0$, so the proposition applies to each $x_i$ and returns a practical $n_i$ with $x_i\le n_i<x_i^2=x_{i+1}$. The half-open intervals $[x_i,x_{i+1})$ are pairwise disjoint (consecutive ones share only the endpoint $x_{i+1}$, which belongs to the later one), so $i\mapsto n_i$ is injective and the set $\{n_i\}$ is infinite. This is the whole content of the theorem once the bound is transferred, and the transfer is step 2. A shorter route, taking $x=\max(x_0,N)$ for each $N$, gives the same conclusion; the page's route is correct as written.

2. The bound in terms of $n$. The proposition's (4.1) already contains $c_0(\log\log x)^2-1<c_0(\log\log n)^2$; rederived: $n\ge x>e^e$ gives $\log\log n\ge\log\log x>1>0$, so $(\log\log n)^2\ge(\log\log x)^2$ and $c_0(\log\log n)^2\ge c_0(\log\log x)^2>c_0(\log\log x)^2-1\ge h(n)$. Hence every $n_i$ satisfies $h(n_i)<c_0(\log\log n_i)^2$, which implies the theorem's non-strict inequality. The positivity of $\log\log x$ is what makes squaring monotone, and $x_0>e^e$ supplies it.

3. The two definitions of $h$ in "What it would settle". The site's $h(m)$ ranges over $1\le m'<m$, the note's over $1\le m'\le m$. For $m\ge2$ the site's value is at least $1$ (the integer $1$ needs one summand), and the extra integer $m'=m$ is the single divisor $m$, so the note's value is $\max(\text{site},1)=\text{site}$. For $m=1$ the site's condition is vacuous (value $0$) while the note's is $1$. So the note's $h$ is at least the site's, with equality for $m\ge2$, exactly as the page says; the note's bound therefore implies the site's, which is the direction the page needs.

Strongest attack

The attack tried to break the deduction at the quantifier structure of the proposition. If $x_0$ depended on $p_*$, a deduction that varied $p_*$ with $x$ would fail; the page fixes $p_*=3$ before choosing any $x$, so a single $x_0$ for that prime suffices and the argument survives even a failure of the uniformity clause. A second attempt tried to make two of the produced numbers coincide, so that the sequence would be finite; the intervals $[x_0^{2^i},x_0^{2^{i+1}})$ are disjoint, which rules it out. A third attempt looked for a hidden hypothesis in Theorem 1.1 on p. 1 (a size condition, a restriction on $n$, or a different $h$); there is none, and the definition of $h$ on p. 1 is the one the page states. A fourth attempt checked whether the note in fact claims the theorem follows from the proposition; p. 5 calls the proposition "somewhat stronger" than the theorem and prints no deduction, so the page's proof is a supplied completion (F1), but it is a correct one. No defect in the mathematics was found.

Premises

• Proposition 4.1 of the note (held source, statement on p. 5, read clause by clause on the page image; proof on pp. 5–6 read once and not checked). Interface consumed: there exist an integer $E\ge4$ and a real $x_0>e^e$ such that for every real $x\ge x_0$ and every odd prime $p_*$ there is a practical $n$ with $x\le n<x^2$ and $h(n)\le c_0(\log\log x)^2-1<c_0(\log\log n)^2$; the clauses $2^E\parallel n$ and $p_*\nmid n$ are not used. Standing: claimed; the page names it as a reconstructed claimed input, and the folder's reconstruction of it is author-recorded, not reviewed. This review does not examine its proof.
• Definitions of practical and of $h$ (p. 1) and the convention that all logarithms are natural (p. 2), held source, read on the page images.
• The site's statement of Problem 18 (the corpus's problem page, Statement paragraph), used only for the consequence sentences.
• Provenance facts in the page's Standing paragraph (a partial proof claim on the site's proof-claims tab, mostly AI-generated by the note's own account, not refereed, not on arXiv, an author-side Lean file not built here): checked against the card's provenance paragraph and the note's p. 2; the site's OPEN status and the absence of acceptance were not checked, as web reading is excluded.
• Explicit assumptions: none beyond the proposition.

Findings

F1. Severity: suggested. Location: the "Proof" section, "Taking $x=x_0,x_0^2,x_0^4,\dots$". Defect: the deduction is supplied by the page and is not labeled as such. Witness: p. 5 opens Section 4 with the sentence that the section proves a "somewhat stronger version of Theorem 1.1", and the proof of Proposition 4.1 ends on p. 6 followed directly by Section 5; no deduction of Theorem 1.1 is printed anywhere in the note, and the choice $p_*=3$ is borrowed from the Section 5 remark about Theorem 1.2 (p. 6). The Source paragraph's parenthetical hints at this, but the Proof section reads as the note's own argument. Proposed replacement, as the first sentence of the Proof section: "The note prints no deduction: Section 4 opens by calling the proposition a somewhat stronger version of the theorem (p. 5). The argument below is supplied."

F2. Severity: note. Location: Standing, "registered as partial because it answers only the first of the three questions". Defect: the causal clause attributes a reason to the site registration that the held provenance does not record; the card's provenance paragraph says only that the note was registered as a partial proof claim. Witness: the card's provenance paragraph; the note's Theorem 1.1 (p. 1) addresses only the first of Problem 18's three questions, which is the fact the clause should state. Proposed replacement: "registered as partial; the note addresses only the first of the three questions".

F3. Severity: note. Location: Statement, "Let $c_0=14/\log2$". Defect: the logarithm convention is not stated, although the base changes both $c_0$ and the bound; the page fixes it only implicitly through "$c_0\approx20.2$" in the last section. Witness: p. 2, "all logarithms are natural". Proposed replacement: "Let $c_0=14/\log2$, all logarithms being natural."

F4. Severity: note. Location: Standing, "the site shows OPEN and no independent acceptance is documented". Defect: the observation is undated, while the card's provenance paragraph dates its own checks. Witness: the card's provenance paragraph ("as of 2026-09-27"). Proposed replacement: "the site showed OPEN and no independent acceptance was documented as of 2026-09-27" (or the date on which the page author made the observation).

F5. Severity: note. Location: "What it would settle", "improving the $(\log m)^{1/2}$ of Vose's construction". Defect: the sentence is true, but it presents exponent $2$ as the note's advance over the last proved bound without the note's own framing, which is that the exponent was already claimed and the note makes it "explicit" and "simplified". Witness: p. 1, the abstract ("This makes a recent bound posted by Liam Price explicit") and the sentence before Theorem 1.1 ("Here we record a simplified and explicit version of this result"). Proposed replacement: "improving the $(\log m)^{1/2}$ of Vose's construction; the note presents the exponent $2$ as already claimed by the posted preprint it cites and its own contribution as the explicit constant (p. 1)".

Verdict

Source fidelity: faithful. The Statement reproduces Theorem 1.1 (p. 1) with its constant and the note's definition of $h$; the locators (seven pages, p. 1, pp. 5–6, the Section 4 opening) are correct; there are no required corrections.

The argument as reconstructed: sound. The deduction from the statement of Proposition 4.1 is complete and was rederived above; the consequence sentences for Problem 18 are correct, including the comparison of the two definitions of $h$.

Limitations: this review checks only the deduction of Theorem 1.1 from the statement of Proposition 4.1 and the page's statement, labels and standing sentences. It does not examine the proof of Proposition 4.1 or the lemmas behind it, so the theorem's truth rests entirely on that claimed input; the site's status and acceptance were not checked. The exposures listed above did not enter any verdict.

This focused review assigns no tier and changes no status.