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Subject and independence

Role: independent reviewer in a fresh context, commissioned for refutation of the Lemma 3.3 reconstruction and given only the commission. The reviewer took no part in writing that page or any page of its folder, had no exchange with the page's author, and saw no other review of it. The subject is path wiki/research/erdos_18/doorn_lemma_3_3_reconstruction.md as it stood at 2026-09-28T05:03:27Z, read as of that time.

Artifact. The seven-page PDF held by van Doorn (2026) (Wouter van Doorn and GPT-6 Astra Pro, the author line as printed, Practical numbers and Egyptian fractions); the on-disk file matches the LFS pointer recorded as of that time. Physical p. 4 (the definitions of $Q(k)$ and $t(k)$, Lemma 3.3 and its whole proof) was read in full, by text extraction and on page images rendered at 110 and 170 dots per inch, every display on the 170 dpi image. Physical p. 3 (the definition of $M_d(X)$, Lemma 3.2 with (3.1) and (3.2)) and p. 1 (the abstract and Theorem 1.1, where $c_0=14/\log2$ is defined) were read by text extraction and on 110 dpi page images; p. 2 (the definitions of $D(n)$ and $\omega(n)$) by text extraction; pp. 5–7 were skimmed by text extraction for the uses of Lemma 3.3 in Corollary 3.4 and Proposition 4.1 and were not otherwise used. Printed and physical page numbers coincide.

Allowed material read. The Definitions and Statement of the Lemma 3.2 reconstruction as of the same time; the provenance paragraph of the card named above; the statement paragraph of the Problem 18 page; the "Whole-claim report" and "Audit checklist" sections of docs/verification.md, the "Source fidelity" section of docs/evidence.md, and docs/math_authoring.md. The file names of the folder as of that time were listed, without reading, to check that the pages the Source paragraph links exist; they do.

Exposures. Four, none used below; every derivation in this report is the reviewer's own from the PDF. (1) The card's _index.md came into view whole, so its read-status, bears-on, overview and Lean paragraphs were seen, including site status wording and a one-paragraph summary of Lemma 3.3's strategy. (2) The Lemma 3.2 reconstruction came into view whole, so its Standing paragraph and its proof were seen; the proof was not needed. (3) The Problem 18 page has no Statement heading; its statement sits in a lead section that also holds a Status paragraph, which was seen. (4) The canonical failure-modes section preceding the audit checklist in docs/verification.md was seen. Besides these, a listing of the evidence/verify/ folder taken after this report was written showed untracked review files of other pages; their names were seen and their contents were not read. Not read: the folder's _index.md, anything under any evidence/ folder, the Lemma 3.1, Corollary 3.4, Proposition 4.1 and Theorem 1.1 reconstructions, the card's result pages, other reviews, anything outside the repository.

Restatement

Conventions. All logarithms are natural; $c_0=14/\log2$; $D(n)$ is the set of positive divisors of $n$ and $\omega(n)$ the number of its distinct prime factors; for $k\ge3$,

$$Q(k)=k^6\log k,\qquad t(k)=\Bigl\lfloor\frac{k\log2}{7\log k+3\log\log k}\Bigr\rfloor ,$$

the latter equal to the source's $\lfloor14k/(c_0(7\log k+3\log\log k))\rfloor$ since $14/c_0=\log2$. A residue $c$ modulo $A$ "has a representation (3.2)" when $c\equiv z_0+2z_1+4z_2+8z_3\pmod A$ for some $z_0,z_1,z_2,z_3\in D(V)$; the note's display writes $z_i\mid V$, and in the note's convention (p. 2) and in the proof of Lemma 3.2, which sums over $D(V)$, this means positive divisors, so the page's reading is the printed one.

Claim. There is an absolute integer $k_0$ such that for every integer $k\ge k_0$, every odd prime $p_*$ and every positive odd squarefree integer $V$ with exactly $k$ prime factors, each at most $2Q(k)$, there exists an odd squarefree integer $A>1$ with $\gcd(A,p_*V)=1$, exactly $t(k)$ prime factors, all in $(Q(k),2Q(k)]$, such that every residue class modulo $A$ has a representation (3.2) with divisors of this $V$. The order of quantifiers is: $k_0$ first and absolute; then $k$, $p_*$, $V$; then $A$, which may depend on all three; then the residue $c$. The page states exactly this, and it agrees clause by clause with the lemma as printed on p. 4.

Checklist

• Quantifiers and scope. Pass. The statement on the page carries the source's quantifiers unchanged, including $A>1$, the exact count $\omega(A)=t(k)$, the half-open interval $(Q(k),2Q(k)]$ and the threshold independent of $p_*$ and $V$; the proof delivers exactly these, since $t\ge1$ for large $k$ and every estimate depends on $k$ alone.
• Circularity. Pass. The proof consumes Lemma 3.2, the prime number theorem, Hölder's inequality and elementary inequalities; none restates the claim.
• Model and convention changes. Pass. The random modulus is an existence device: $\mathbb E_IS<1$ for a nonnegative $S$ forces some $I$ with $S<1$, and that $I$ is the actual object handed to Lemma 3.2. No relaxed system stands in for the real one.
• Finite and statistical overreach. Pass. Every expectation is bounded by an explicit inequality (the collision bound, (3.3), Hölder, the binomial expansion); nothing is a heuristic average, and no independence is assumed beyond what a uniform subset supplies.
• Uniformity. Pass for the proof body: $R\le2k$ and the exclusion of at most $k+1$ primes make $|\mathcal P|$ and $\rho$ depend on $k$ alone, and the error in the prime number theorem is a function of $Q(k)$. The Qualifications bullet misstates the strength of the lower bound consumed (F1); the proof body uses the correct strength.
• Extremal conclusions. Inapplicable: the page claims no infimum, supremum, attained value or sharpness.
• Consequences and composition. Pass. The single "hence" (from $\mathbb E_IS<1$ to Lemma 3.2's conclusion) was checked separately; Lemma 3.2's interface is met at full strength ($V_1,V_2$ coprime positive odd, $A>1$ odd, $S<1$) and its conclusion is exactly the clause claimed.
• Computation. Inapplicable: the page runs no code and cites no evidence program.
• Reproduction. Inapplicable: there are no rerun commands or coverage claims.
• Source and verdict fidelity. Pass with corrections. Statement, definitions and every proof step match p. 4; the standing sentence claims only an author-recorded reconstruction of a claimed result. The locator for $c_0$ is missing (F2), one supplied convention is unmarked (F3), and one Qualifications sentence is false as written (F1).

Weakest steps

W1: averaging the divisor sum over a random modulus. With $F(J)$ the factor $(M_{a_J}(X_1)M_{a_J}(X_2)M_{a_J}(X))^{1/3}$ and $I$ a uniformly random $t$-subset of $\mathcal P$, the divisor sum satisfies, term by term,

$$S=\sum_{\varnothing\ne J\subseteq I}a_J^{2/3}F(J) \le\sum_{s=1}^{t}w^s\sum_{J\subseteq I,\ |J|=s}F(J),$$

because $a_J\le(2Q)^{|J|}$. For a fixed $s$-subset $J$ of $\mathcal P$, $\Pr(J\subseteq I)=\binom{|\mathcal P|-s}{t-s}/\binom{|\mathcal P|}t$, and counting the pairs $(I,J)$ with $J\subseteq I$ in two ways gives $\binom Nt\binom ts=\binom Ns\binom{N-s}{t-s}$ with $N=|\mathcal P|$, so

$$\mathbb E_I\sum_{J\subseteq I,\ |J|=s}F(J) =\sum_{|J|=s}F(J)\,\frac{\binom ts}{\binom{|\mathcal P|}s} =\binom ts\,\mathbb E_{|J|=s}F(J),$$

which is the page's sentence about a random $s$-subset of a random $t$-subset. It presupposes $t\le|\mathcal P|$, true for large $k$ because $t\le k$ and $|\mathcal P|\sim k^6/6$ (F4). Composition: this turns the divisor sum into the binomial sums that the rest of the proof estimates.

W2: the second and fourth of the four terms. These are the steps whose margin is a power of $u=\log k$ rather than of $k$. From $\rho\le Ck^{-5}$ and $w=2^{2/3}k^4u^{2/3}$, $1+w\rho^{1/3}\le C'k^{7/3}u^{2/3}$ for $k\ge3$, so $\log(1+w\rho^{1/3})\le\tfrac73u+\tfrac23\log u+C''$ with absolute constants. By the definition $\tau=k\log2/(7u+3\log u)$ one has the exact identity $\tfrac13k\log2=\tau(\tfrac73u+\log u)$, hence, with $t\le\tau$,

$$\log\bigl(2^{-k/3}(1+w\rho^{1/3})^t\bigr) \le-\tau\bigl(\tfrac73u+\log u\bigr) +\tau\bigl(\tfrac73u+\tfrac23\log u+C''\bigr) =\tau\bigl(-\tfrac13\log u+C''\bigr),$$

and $\tau\asymp k/u\to\infty$ while $-\tfrac13\log u+C''\to-\infty$, so the second term tends to zero. For the fourth, $tw\rho\le\tau w\rho\ll(k/u)\,k^4u^{2/3}\,k^{-5}=u^{-1/3}$, so $(1+w\rho)^t-1\le e^{tw\rho}-1\ll u^{-1/3}\to0$. The $u$-power of $\rho$ decides both: a bound $\rho\ll k^{-5}u^{\beta}$ would leave $\tau(\tfrac{\beta-1}3\log u+O(1))$ for the second term, which tends to $-\infty$ only for $\beta<1$, and $tw\rho\ll u^{\beta-1/3}$ for the fourth, which tends to zero only for $\beta<\tfrac13$. So the argument as written consumes $|\mathcal P|\gg k^6u^{-\beta}$ for some $\beta<\tfrac13$; the loss-free bound $|\mathcal P|\gg k^6$ suffices and is what $\rho\le2k/|\mathcal P|\ll k^{-5}$ uses, while the page's $k^6/u$ (F1) is $\beta=1$ and closes neither term. For comparison the first and third terms have linear negative exponents: their logarithms are at most $k\log2(-\tfrac23+\tfrac47+o(1))=-\tfrac2{21}k\log2+o(k)$ and $k\log2(-\tfrac13+\tfrac2{21}+o(1))=-\tfrac5{21}k\log2+o(k)$, which are the source's $-4k/(3c_0)$ and $-10k/(3c_0)$ since $c_0=14/\log2$; here $\tau(4u+\tfrac23\log u+O(1))=\tfrac47k\log2\,(1+O(\log u/u))$ and $\tau(\tfrac23u+\tfrac23\log u+O(1))=\tfrac2{21}k\log2\,(1+O(\log u/u))$. Composition: the four bounds together give $\mathbb E_IS\to0$ uniformly, hence $\mathbb E_IS<1$ beyond an absolute threshold.

W3: the collision bound and Hölder. Elements of $X_1$, $X_2$ and $X$ are divisors of $V$, so a nonzero difference $\delta$ satisfies $0<|\delta|<V$, and since $V$ is odd every such $\delta$ is even, so $|\delta|\ge2$. If $r$ distinct primes of $\mathcal P$ divide $\delta$, their product exceeds $Q^r$ and divides $\delta$, so $Q^r<|\delta|<V$ (for $r=0$ this reads $1<|\delta|$, also true), whence $r<\log V/\log Q$ and $r\le R=\lfloor\log V/\log Q\rfloor\le k\log(2Q)/\log Q\le2k$, using $V\le(2Q)^k$. For a uniform $s$-subset $J$, $a_J\mid\delta$ exactly when every prime of $J$ divides $\delta$, so

$$\Pr(a_J\mid\delta)\le\binom Rs\Big/\binom{|\mathcal P|}s =\prod_{i=0}^{s-1}\frac{R-i}{|\mathcal P|-i}\le\rho^s ,$$

where for $s\le R$ each factor is at most $\rho=R/|\mathcal P|$ because $R\le|\mathcal P|$, and for $s>R$ the left side is $0$. Counting the $|Y|$ diagonal pairs exactly and the rest in expectation gives $\mathbb E\,M_{a_J}(Y)\le|Y|^{-2}(|Y|+|Y|(|Y|-1)\rho^s)\le|Y|^{-1}+\rho^s$, which is (3.3). Hölder with exponents $3,3,3$ applied to $M_{a_J}(X_1)^{1/3}$, $M_{a_J}(X_2)^{1/3}$, $M_{a_J}(X)^{1/3}$ gives $\mathbb E\,F(J)\le\prod_Y(\mathbb E\,M_{a_J}(Y))^{1/3}$, and with $|X_1|^{-1},|X_2|^{-1}\le\sqrt2\cdot2^{-k/2}$ and $|X|^{-1}=2^{-k}$ this is at most $2^{1/3}(2^{-k/2}+\rho^s)^{2/3}(2^{-k}+\rho^s)^{1/3}$. Composition: this is the factor inserted into W1's binomial sum; the expansion by $(a+b)^\alpha\le a^\alpha+b^\alpha$ into $2^{-2k/3}$, $2^{-k/3}\rho^{s/3}$, $2^{-k/3}\rho^{2s/3}$, $\rho^s$ and the binomial theorem then give the four terms of W2.

Strongest attack

The attack aimed at the two terms of W2, the only places where the exponent of $k$ cancels exactly and the conclusion rests on a power of $\log k$. Two versions were tried. First, against the page's own weakest description of the input: the Qualifications bullet says the prime number theorem is used "only through the lower bound $|\mathcal P|\gg k^6/u$". With that input alone, $\rho\ll k^{-5}u$; then $tw\rho\ll u^{2/3}$, so the fourth term's bound $(1+w\rho)^t-1=o(1)$ is lost, and $\log(1+w\rho^{1/3})\le\tfrac73u+\log u+O(1)$, so the second term's logarithm is bounded only by $\tau\cdot O(1)$, which need not tend to $-\infty$. The argument as written does not close under that input. This refutes the bullet as a description of the argument (F1), but not the argument, whose body uses $|\mathcal P|\sim k^6/6$ and $\rho\ll k^{-5}$. Second, against the body: with $\rho\ll k^{-5}$ the margins are $-\tfrac13\tau\log u\to-\infty$ and $tw\rho\ll u^{-1/3}\to0$, and any prime-count bound weaker by a factor $u^{\beta}$ with $\beta<\tfrac13$ would still close both, so the steps have slack and the attack fails. Further attacks that failed: a difference of two divisors in $X=D(V)$ (rather than in $X_i$) exceeding $V$ in absolute value (impossible, all lie in $[1,V]$); the nested-subset identity in W1 (verified by the double count); Hölder applied with exponents summing to more than one (they are $\tfrac13+\tfrac13+\tfrac13$); the threshold depending on $V$ through $|\mathcal P|$ (at most $k+1$ primes are excluded, and the prime number theorem's error is a function of $Q(k)$); and $A=1$ (excluded since $t\ge1$ for large $k$, so $A>Q>1$).

Premises

• Lemma 3.2 (local claim), consumed through its reconstruction as of the same time, Statement read; also read as printed on physical p. 3 of the held PDF. Interface: $V_1,V_2$ coprime positive odd integers, $V=V_1V_2$, $X_i=D(V_i)$, $X=D(V)$, $A>1$ odd, and $S=\sum_{d\mid A,\,d>1}d^{2/3}(M_d(X_1)M_d(X_2)M_d(X))^{1/3}<1$; then every residue modulo $A$ has a representation (3.2) with $z_i\in D(V)$. Its standing is outside this review's subject and is not assessed here; the page names it as a reconstruction of a claimed result.
• Prime number theorem, imported, no source held; stated on the page as $\pi(2Q)-\pi(Q)\sim Q/\log Q$. Consumed only as the lower bound $\pi(2Q)-\pi(Q)\ge cQ/\log Q$ for $Q\ge Q_0$ with absolute $c,Q_0$, which gives $|\mathcal P|\gg k^6$ after excluding at most $k+1$ primes.
• Hölder's inequality for three nonnegative functions on a finite probability space with exponents $3,3,3$; imported, standard.
• $(a+b)^\alpha\le a^\alpha+b^\alpha$ for $a,b\ge0$, $0<\alpha\le1$; imported, standard (concavity of $x^\alpha$ with value $0$ at $0$).
• Elementary facts used without citation: the binomial theorem, $1+x\le e^x$, the nested uniform-subset identity of W1, and $|D(V)|=2^{\omega(V)}$ for squarefree $V$.
• Explicit assumptions: $k$ exceeds an absolute threshold large enough that $t\ge1$, $u\ge1$, $t\le|\mathcal P|$, $|\mathcal P|\ge k^6/12$, and the $o(1)$ terms of W2 are below the fixed margins.

Findings

F1. Severity: required. Location: Qualifications, first bullet, "used only through the lower bound $|\mathcal P|\gg k^6/u$". Defect: the bound named is weaker than the one the argument consumes; the proof body uses $|\mathcal P|\sim Q/\log Q\sim k^6/6$ (source p. 4, the display after "The prime number theorem gives") and $\rho\ll k^{-5}$, and only $|\mathcal P|\gg k^6$ yields that. Witness: with $|\mathcal P|\gg k^6/u$ alone, $\rho\ll k^{-5}u$; then $tw\rho\ll(k/u)k^4u^{2/3}k^{-5}u=u^{2/3}$, so the fourth term's bound $(1+w\rho)^t-1=o(1)$ is lost, and $\log(1+w\rho^{1/3})\le\tfrac73u+\log u+O(1)$ makes the second term's logarithm bound $-\tau(\tfrac73u+\log u)+\tau(\tfrac73u+\log u+O(1))=O(\tau)$, which the page's argument cannot drive to $-\infty$ (see W2). The page's Standing paragraph names the correct form, $\pi(2Q)-\pi(Q)\gg Q/\log Q$, and $Q/\log Q=k^6u/(6u+\log u)\asymp k^6$, not $k^6/u$. Replacement: "The prime number theorem is used only through the lower bound $|\mathcal P|\gg Q/\log Q\asymp k^6$, hence $\rho\ll k^{-5}$, which Chebyshev-type estimates also supply; a bound weaker by a factor $u^{1/3}$ or more would not close the fourth term. The note cites the theorem itself."

F2. Severity: suggested. Location: Source paragraph, "physical p. 4". Defect: the Definitions fix $c_0=14/\log2$, which p. 4 does not define; the note defines it in Theorem 1.1 ("where $c_0=14/\log2$", physical p. 1) and in the abstract, and defines $D(n)$ and $\omega(n)$ at the end of Section 1 (physical p. 2). Witness: p. 4 mentions $c_0$ only inside $t(k)$ and the term bounds. Replacement: after "physical p. 4 of the seven-page PDF", add "with $c_0=14/\log2$ from Theorem 1.1 (physical p. 1) and $D(n)$, $\omega(n)$ from the end of Section 1 (physical p. 2)".

F3. Severity: note. Location: Definitions, "For integers $k\ge3$". Defect: the note sets $Q(k)$ and $t(k)$ "for sufficiently large integers $k$" (p. 4); the domain $k\ge3$, the range on which $\log\log k>0$ makes $t(k)$ well defined, is supplied by the page and not marked as such. Replacement: "For integers $k\ge3$ (the note says 'sufficiently large'; $k\ge3$ is the range where $\log\log k>0$)".

F4. Severity: note. Location: The random modulus, "Let $I$ be a uniformly random $t$-element subset of $\mathcal P$". Defect: this needs $t\le|\mathcal P|$, which holds for large $k$ since $t\le k$ and $|\mathcal P|\sim k^6/6$, but the page does not say so; the same fact makes $\binom{|\mathcal P|}s>0$ in the collision bound. Replacement: append "(possible since $t\le k<|\mathcal P|$ for large $k$)".

Verdict

Source fidelity: faithful with corrections. The Statement, the Definitions and every step of the Proof agree with Lemma 3.3 and its proof as printed on physical p. 4, with routine steps filled in and no hypothesis, quantifier, constant or boundary case changed; the required correction (F1) is confined to a Qualifications sentence that understates the consumed prime-count bound, and F2–F4 are locator and labeling matters.

The argument as reconstructed: sound. Each essential deduction was re-derived above (W1–W3), including the exact constants $-2k\log2/21$ and $-5k\log2/21$ of the first and third terms, the $\log u$ margin of the second, and the $u^{-1/3}$ decay of the fourth; all constants are absolute, so the threshold is independent of $p_*$ and $V$, and Lemma 3.2 is applied inside its hypotheses.

Limitations: the review covers Lemma 3.3 and its interface to Lemma 3.2 only; the proof and standing of Lemma 3.2 and the note's remaining results are outside its subject; the prime number theorem and Hölder's inequality are imported and not re-proved; no computation was run; nothing here bears on whether the note's main theorems are correct. This focused review assigns no tier and changes no status.