Divisor representations of practical numbers

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doorn_corollary_3_4_reconstruction: Reconstructs the claimed step that the modulus of Lemma 3.3 extends a practical n = 2^E V to a practical An with h(An) at most h(n) + 4.

doorn_lemma_3_1_reconstruction: Reconstructs the claimed extension step: if every residue modulo A is a short sum of divisors of a practical n avoiding multiples of A, then An is practical and h(An) grows by at most the length of those sums.

doorn_lemma_3_2_reconstruction: Reconstructs the claimed elementary criterion: a weighted sum of residue collision probabilities of the divisor sets below one forces every residue modulo A to be z0 + 2z1 + 4z2 + 8z3 with the zi divisors of V.

doorn_lemma_3_3_reconstruction: Reconstructs the claimed averaging argument: for large k, a random product of t(k) primes in (Q(k), 2Q(k)] makes the criterion sum of Lemma 3.2 smaller than one, so some modulus admits four-divisor representations.

doorn_proposition_4_1_reconstruction: Reconstructs the claimed iteration of the extension step and the recurrence for log omega that gives, for every large x and odd prime p*, a practical n in [x, x^2) prime to p* with h(n) at most c0 (log log x)^2 - 1.

doorn_theorem_1_1_reconstruction: Reconstructs the claimed deduction of the (log log n)^2 bound for infinitely many practical numbers from the uniform Proposition 4.1, and records what it would settle for Problem 18.

evidence/: Independent focused reviews of the Problem 18 reconstruction pages and their distinct grade, with no executable evidence and no tier.

hughes_corollary_3_reconstruction: Reconstructs the consecutive-divisor gap bound for n! from the imported Berend–Harmse estimate, stating the exact imported form and the two facts about its error term that the step count uses.

hughes_lemma_4_reconstruction: Reconstructs the greedy step for an integer whose consecutive divisors have ratio at most two, and supplies a proof of that ratio property for n!.

hughes_remark_6_reconstruction: Reconstructs the subset-count lower bound h(n!) >> (log n)^2 from the Chebyshev estimate log tau(n!) << n/log n.

hughes_theorem_1_reconstruction: Reconstructs the count of greedy steps below and above the square root of n! that bounds h(n!) by (2 log 2 + o(1)) n/log n from the factorial divisor gap.

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What the folder holds

Problem 18 asks, for practical $m$ (every smaller positive integer is a sum of distinct divisors of $m$) and $h(m)$ the number of distinct divisors that always suffice, (a) whether infinitely many practical $m$ have $h(m)<(\log\log m)^{O(1)}$, (b) whether $h(n!)<n^{o(1)}$, and (c) whether $h(n!)<(\log n)^{O(1)}$. This folder holds author-recorded source-proof reconstructions, one page per result, each with its Source, Standing, Definitions, Statement and Proof, written in the corpus's own words with imported theorems stated in the form used.

The pages prefixed hughes_ reconstruct, from Hughes (2026), the greedy upper bound Theorem 1, $h(n!)\le(2\log2+o(1))\,n/\log n$, with its two ingredients Lemma 4 (the greedy step, with a compilation-supplied proof that consecutive divisors of $n!$ have ratio at most $2$) and Corollary 3 (the divisor gap of $n!$ from the imported Berend–Harmse estimate), and the lower bound Remark 6, $h(n!)\gg(\log n)^2$.

The pages prefixed doorn_ reconstruct, from the van Doorn note (2026), the chain Lemma 3.1 (extending a practical number by a modulus), Lemma 3.2 (a Cauchy–Schwarz and Plancherel criterion for four-divisor representations of every residue), Lemma 3.3 (a random squarefree modulus meets the criterion), Corollary 3.4 (one extension costs four divisors), Proposition 4.1 (the iteration and its recurrence) and Theorem 1.1, infinitely many practical $n$ with $h(n)\le c_0(\log\log n)^2$, $c_0=14/\log2$. Every doorn_ page is labeled claimed: the note is a site proof claim without documented independent acceptance.

Where things stand

Status unchanged. Question (b) is proved by the site-accepted Lean proof filed as the accepted record, which has no written proof, so it is not reconstructed here. Question (a) carries the claimed van Doorn chain reconstructed here and the earlier Price claim, whose write-up is not held and could not be reconstructed. Question (c) is open between Remark 6's $(\log n)^2$ and the $n^{o(1)}$ of (b); Theorem 1 is superseded as a bound and kept for its explicit constant. The problem page keeps status: open.

Reviewed. Each reconstruction page was independently reviewed, as it stood at 2026-09-28T05:03:27Z, by a focused review filed under evidence/verify/, and the ten reviews were checked by a distinct grade. The graded verdicts, as the grade records them: Corollary 3.4, fidelity faithful and argument sound conditional on Lemma 3.1 and Lemma 3.3 as claimed inputs; Lemma 3.1, faithful and sound; Lemma 3.2, faithful with corrections (C1) and sound; Lemma 3.3, faithful with corrections (C2, C5) and sound on the imported lower bound $\pi(2Q)-\pi(Q)\gg Q/\log Q$, Hölder's inequality and Lemma 3.2 as a claimed input; Proposition 4.1, faithful with corrections (C5) and sound conditional on Lemma 3.1, Lemma 3.3 and Corollary 3.4 as claimed inputs and on the prime count in $(Q,2Q]$ and Stirling's weak form; Theorem 1.1, faithful with corrections (C5) and sound as a deduction from the statement of Proposition 4.1, a claimed input; Corollary 3, faithful and sound on the second-hand Berend–Harmse import, consumed exactly as the preprint prints it; Lemma 4, faithful with corrections (C3, C4) and sound, including the compilation-supplied proof that consecutive divisors of $n!$ have ratio at most $2$; Remark 6, faithful and sound on Chebyshev's bound; Theorem 1, faithful and sound on Lemma 4, Corollary 3 and the two elementary asymptotics the page proves. No report was graded void. The corrections C1 to C5 were applied, so the current text of the Lemma 3.2, Lemma 3.3, Proposition 4.1, Theorem 1.1 and Lemma 4 pages differs from the reviewed text at the places the grade names; the other five pages are the reviewed text. No tier is assigned, the van Doorn results remain claims, and the problem's status is unchanged. After the review, line wrapping was normalized on the reconstruction pages; no formula or sentence changed.

Mechanism. Both arguments turn a divisor-gap or residue-covering property of a highly composite $N$ into a short representation by subtracting or adjoining. Hughes's route is greedy: subtract the largest divisor below the remainder; when consecutive divisors of $N$ have ratio $1+\eta$ the remainder shrinks by a factor $2\eta$, so the number of steps is a sum of $1/\log(1/\eta)$ over the logarithmic scale, and the Berend–Harmse gap $\log(1/\varepsilon_j)\asymp(\log j)^2$ on the window $[\sqrt{(j-1)!},\sqrt{j!}]$ gives $n/\log n$. The van Doorn route is multiplicative: multiply a practical $n=2^EV$ by a squarefree modulus $A$ whose every residue is $z_0+2z_1+4z_2+8z_3$ with $z_i\mid V$; each such step adds $4$ to $h$ and multiplies $\omega(V)$ by $1+\Theta(1/\log\omega(V))$, so $j$ steps reach $\log\omega\asymp\sqrt j$ and $\log\log n\asymp\sqrt{h}$. The residue-covering step is proved by a second-moment (Cauchy–Schwarz and Plancherel) bound on the divisor sets modulo $d\mid A$, averaged over random choices of $A$, in place of the sum-product exponential-sum input the Price claim uses.