Openai 2026 joint dickman law consecutive integers

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corollary_1_2: The Erdős--Turán comparison statement for the largest prime factors of consecutive integers, deduced in the OpenAI release from the joint Dickman law by continuity of the limiting law; the claimed resolution of Problem 371.

theorem_1_1: The joint Dickman law for consecutive integers in ordinary natural density, claimed by the OpenAI release through a mixed decorrelation of bin characters amplified into a divisor graph; the claimed resolution of Problem 928.

OpenAI, The joint Dickman law for consecutive integers, OpenAI Math Release preprint, September 24, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision adc7f1241), folder preprints/The-joint-Dickman-law-for-consecutive-integers-September-24-2026; the held PDF, paper.pdf in the release, is retained as openai_2026_joint_dickman_law_consecutive_integers.pdf, and the release's TeX bundle in the same folder is the TeX source cited below.

@misc{OAI:The-joint-Dickman-law-for-consecutive-integers-September-24-2026,
  author = {{OpenAI}},
  title = {{The joint Dickman law for consecutive integers}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-joint-Dickman-law-for-consecutive-integers-September-24-2026/paper.pdf}{OAI:The-joint-Dickman-law-for-consecutive-integers-September-24-2026}},
  year = {2026}
}

Attestation, as the source states it. The release's own README says the collection's manuscripts were "produced by an internal OpenAI model", that the collection "includes results at different stages of verification", that "Not all have accompanying Lean formalizations" and that "Some of the unformalized results could have issues"; it also says that most results used "the same procedure using an unreleased internal OpenAI model" with, on average, "three hours of ChatGPT Pro thinking compute" per result. The manuscript's own README in the release adds nothing beyond the title, the author line "OpenAI", the date and the citation block; the title page names no person. These sentences are recorded here as the source's historical attestations of its own provenance, not as this corpus's review. No refereed publication, no arXiv version and no independent review of the manuscript is recorded here and nothing on this card is independently reviewed.

Formalization, as the release lists it. The release's Lean documentation page for its family "Independent largest prime factors of consecutive integers" names this manuscript as its only paper and says that the formalization proves the joint Dickman law in ordinary natural density, for every $0<a,b<1$ the density of $n$ with $P^+(n)\le n^a$ and $P^+(n+1)\le n^b$ tends to $\rho(1/a)\rho(1/b)$, and that each ordering $P^+(n)<P^+(n+1)$ and $P^+(n+1)<P^+(n)$ has natural density $1/2$. It names the comparator statement file lean/ComparatorChallenges/JointDickman.lean, which states three theorems (joint_law, with both thresholds based at $n$; increasing_order; decreasing_order) over a Dickman function defined there by an iterated delay-equation construction, and whose companion JointDickman.json names the solution module OAI.NumberTheory.JointDickman.PaperMain and permits the axioms propext, Quot.sound and Classical.choice. That module restates the three theorems and refers each to an unconditional theorem of the release's own OAI/NumberTheory/JointDickman/ tree (1,502 Lean files). The tree's Amplification/PublishedInputs.lean declares the two short-interval theorems (Theorems 2.4 and 2.5 here) as "propositions to be supplied as hypotheses" that "are not axioms or proofs of the cited analytic theorems", and the other cited analytic inputs are declared the same way in the tree's Analysis/, Arithmetic/ and Amplification/ subfolders; its ConditionalMain.lean states the three laws as theorems taking those hypotheses as arguments, and its UnconditionalMain.lean states them without hypotheses, with proof terms that invoke the modules the release labels as discharged inputs and proved short-average estimates. The release's catalog file lean/formalization.yaml carries no entry for this manuscript. All of this is read statically from the release's Lean documentation page, comparator files and source tree; not built, replayed or audited for fidelity in this repository. Whether a release declaration settles the problem is recorded on the problem's claim pages, not on this card.

The release groups this manuscript alone in its family; it has no companion manuscript in the release.

Read status: claims checked for Theorem 1.1 and Corollary 1.2, together with the supporting statements Lemma 2.1, Proposition 2.2, Lemma 2.3, Theorems 2.4--2.5 (the two imported short-interval theorems) and Lemma 11.1, read clause by clause in the TeX source (sections/introduction.tex, labels thm:main and cor:comparison; sections/labels.tex, labels lem:labels-marginal, prop:mixed, lem:labels-short, thm:labels-real-MR, thm:labels-complex-MRT; sections/distribution.tex, label lem:distribution-dickman and the displays eq:distribution-fixed and eq:distribution-upper-tail) on 2026-10-07; the proofs in Sections 2--11 were read for their structure only and no step was checked; nothing here is independently reviewed.

Contents

The manuscript is 84 pages (title and contents p. 1, references pp. 81--84). Throughout, $P^+(n)$ is the largest prime factor of $n\ge2$, $P^+(1)=1$, and $\rho$ is the Dickman--de Bruijn function, continuous on $[0,\infty)$ with $\rho(u)=1$ on $[0,1]$ and $u\rho'(u)=-\rho(u-1)$ for $u>1$.

• Section 1, Introduction (pp. 2--6). States Theorem 1.1 (the joint Dickman law with moving thresholds $n^a$, $n^b$, limit through all real $X$) and Corollary 1.2 (natural density $1/2$ for $P^+(n)<P^+(n+1)$ and for the reverse ordering), and says the corollary follows from the continuity of the limiting marginal with no quantitative separation estimate. Section 1.1 surveys earlier work: Erdős and Pomerance 1978 (formulation of the joint independence problem, positive lower density $0.0099$ for each ordering, and their Theorem 1 that $P^+(n)/P^+(n+1)$ is rarely within $X^{\pm\delta}$ of $1$); the lower-density bounds $0.05544$ (de la Bretèche, Pomerance and Tenenbaum 2005, Section 3) and $0.05866$ (an observation of Fouvry recorded there), $0.1063$ and $0.1356$ (Wang 2017, 2018), $0.2017$ (Lü and Wang 2025) and $0.280$ (Yang 2026, Theorem 1.4); Teräväinen 2018 (Theorem 1.14, the product law in logarithmic density; Theorem 1.16, logarithmic density $1/2$ for the ordering; Theorem 1.19, each nondegenerate rectangle of normalized values in $(0,1)^2$ has positive lower natural density); Tao and Teräväinen 2019 (Remark 3.3, equation (50): the product law for ordinary averages outside a set of scales of logarithmic density zero; Corollary 1.16 for the ordering); Wang 2021 (the natural-density joint law assuming Elliott--Halberstam for friable integers); Jiang, Lü and Wang 2022 (averaged-over-shift forms); and Tao and Teräväinen 2026 (Theorem 1.8, a joint law with an explicit error term, valid outside a thin set of scales). The manuscript locates its own contribution as "the unconditional ordinary limit at every sufficiently large scale" (p. 3) for fixed parameters, with no quantitative error term. Section 1.2 outlines the method and names its antecedents: Tao's logarithmic two-point Chowla argument, Helfgott and Radziwiłł's prime-divisibility graphs, Pilatte's amplification, the mixed decoupling of Tao and Teräväinen 2026, and the cut-norm sampling of Frieze--Kannan, of Alon, Fernandez de la Vega, Kannan and Karpinski, and of Borgs, Chayes, Lovász, Sós and Vesztergombi.
• Section 2, Large-prime labels and their short averages (pp. 6--14). For fixed $J\ge2$ the primes in $(x^{1/J},x]$ are cut into bins $\mathcal B_{k,x}=(x^{k/J},x^{(k+1)/J}]$, and two completely multiplicative labels $f_x$, $g_x$ of modulus one are built from two independent phase vectors; $F_x=f_x-\mu$ is the centered label. Lemma 2.1 (p. 6): the joint distribution of the bin counts on $\alpha x<n\le\beta x$, $n\equiv a\pmod q$, converges to a limit depending only on $J$, computed through mixed factorial moments as explicit simplex integrals of $\prod dv_i/v_i$; hence $f_x$ has a mean $\mu$, and $F_x(un)=F_x(n)$, $g_x(un)=g_x(n)$ for every fixed multiplier $u$ once $x$ is large. Proposition 2.2 (p. 9), mixed decorrelation: $x^{-1}\sum_{n<x}\overline{g_x(n)}F_x(n+1)\to0$ through integer scales, the statement from which Section 11 derives the joint law. Lemma 2.3 (p. 9): weighted short averages of $F_x\cdot G_{B,v}$ over $L_B\to\infty$ consecutive shifts in a residue class, with origins of size $Tx$, have mean square tending to zero in the iterated limit $x\to\infty$ then $B\to\infty$. Its proof interpolates the centered label by finitely many real nonnegative multiplicative functions (a Vandermonde interpolation the manuscript attributes in idea to Teräväinen 2018, Section 4), resolves the residue condition by Dirichlet characters, and imports Theorem 2.4 (the real short-interval theorem of Matomäki and Radziwiłł 2016, Theorem 1, in mean-square form) for the principal character and Theorem 2.5 (the complex short-average theorem of Matomäki, Radziwiłł and Tao 2015, Theorem A.1, in the corrected version) with Lemma 2.6 (divergence of the pretentious distance from $n^{it}$ for a function agreeing with a fixed nonprincipal character on $p\le x^{1/J}$, using the Vinogradov--Korobov bound cited from Ford 2002, (1.2)) for the nonprincipal ones. Section 2.6 supposes Proposition 2.2 fails, passes to a subsequence and extracts a bounded profile $W(t,w)$ on $(0,\infty)\times\widehat{\mathbb Z}$ with $\beta_*=|\int\phi W|>0$ for a fixed bump $\phi$.
• Section 3, Arithmetic preliminaries at an auxiliary log scale (pp. 14--26). Fixes $T=\lfloor B^{0.32}\rfloor$, $P_0=B^{1000}$, the prime range $\mathcal P=\{P_0<p\le e^{4B}\}$, "rough" integers and the fair-split weights $A_0$, $K_0$. Lemma 3.1 and Lemma 3.2 (Selberg--Delange expansions for $\mu^2(n)z^{\omega(n)}$, $z\in\{1/4,1/2\}$, with and without a moving roughness cutoff, cited to Granville and Koukoulopoulos 2019, Theorem 1, and Koukoulopoulos 2019, Theorem 13.2, with the Dirichlet zero-free region and Siegel's bound from Koukoulopoulos 2019, Theorems 12.3 and 12.10; the manuscript notes that the character-twisted constants "need not be effective", p. 15). Proposition 3.3: local laws for the coefficient weight and for random products of primes selected with probability $z/p$. Lemma 3.4: upper sieves in intervals and rectangles with reducing local weights, from the bounded-dimension fundamental lemma (Ford's sieve lecture notes 2023, Theorems 2.4 and 3.6). Lemmas 3.5--3.6: first and second coefficient moments and two- and three-form bounds with the loss factor $\Sigma(j)=\prod_{p\mid j}(1+C_2/p)$. Definition 3.7 and Lemma 3.8: regular prime sets (prefix counts within $\tau\ell$ of $g\ell/2$ on a grid, tail counts at least $0.4\log(B/Y)-C_*$) and the loss $O(\epsilon_B+e^{-c_3C_*})$ from imposing regularity.
• Section 4, Amplifying a mixed correlation (pp. 26--36). Defines the nonnegative divisor weight $D_B$ on $\widehat{\mathbb Z}$ from the two factorizations $n=am$, $n+1=cl$ with $e^B<c<e^{2B}$ and $Tc<a<2Tc$. Lemma 4.1 (mean at least $d_0>0$, bounded $L^2$ norm), proved through a reduction to independent fair splits (Lemma 4.2), an addition-product concentration bound (Lemma 4.3) and a two-split second-moment calculation. Lemma 4.4: the profile survives the weight. Proposition 4.5: after Cauchy--Schwarz removes $g_x$, the energy $I_{2,x}$ satisfies $\liminf_B\liminf_x I_{2,x}/(BT)\ge c_5>0$ while its diagonal is negligible.
• Section 5, From mixed amplification to an independent-site kernel (pp. 36--42). Distinct coefficients $a,b$ with $a-b=jc$, $0<|j|\le T$, are reindexed to additive shifts $n'$, $n'+j$ of size $Tx$, giving a weighted graph of shifts with kernel $\mathcal K_j$; Lemma 5.1 bounds its mean edge mass by $\Sigma(j)/T$. Positions are grouped into blocks of $M=\lceil C_6T\rceil$ and compared in cut norm. Proposition 5.3 couples the actual prime-divisibility sets at the block positions to independent site sets $S_i$ (each $p\in\mathcal P$ included with probability $1/p$) and replaces the kernel by the latent kernel $\mathcal L_{ik}$, with Lemma 5.2 giving uniform conditional row means.
• Section 6, A second moment for the latent rows (pp. 42--50). Proposition 6.1: $\sum_{k}\mathbb E\mathcal L_{ik}^2\ll B^{-0.21}$ and the normalized total mass has bounded second moment, via Lemma 6.2 on weighted representation multiplicity (entropy counts of omitted primes, a tilted measure, and a harmonic two-dimensional sieve for the numeric addition).
• Section 7, Smoothing channels on logarithmic and residue space (pp. 50--56). Proposition 7.1: the fair-split channel $U_d$ recording $\log b/B$ and $b\bmod d$ is bounded in operator norm, its nonconstant residue modes are $O(B^{-1/200})$, and its logarithmic output is approximable on a fixed coarse partition; Corollary 7.4 defines the coarse site features $V_l(S)$.
• Section 8, Integral approximation by endpoint features (pp. 56--67). Lemma 8.1 removes the regularity cutoffs at cost $\Sigma(j)(r_B+e^{-c_3C_*})/T$; the relation $a-b=jc$ is detected by additive Fourier analysis, with minor arcs handled by the Montgomery--Vaughan bound (1977, Corollary 1) and major arcs by Proposition 3.3; the Ramanujan-sum identity of Section 8.4 yields the singular series $\mathfrak S(j)=\frac j{\varphi(j)}\prod_{p\nmid j}(1-(p-1)^{-2})$, zero for odd $j$. Proposition 8.2: the integrated comparison of $\mathcal L$ with the finite-feature matrix $\mathcal M_{ik}= (k_B/T)\mathfrak S(|j|)w(s,|j|/T)\sum_lh_lV_l(S_i)V_l(S_k)$, uniform over single-site tests, with error $Ce^{-c_3C_*}+C_\eta\epsilon_1+o(1)$.
• Section 9, Sampling the integral cut comparison (pp. 67--72). Proposition 9.1 upgrades the integrated comparison to expected cut norm, with tests chosen after the matrix is known, by approximating an optimizing sign vector from $m_2=\lfloor MB^{-0.18}\rfloor$ sampled columns (Lemmas 9.2--9.4) and applying McDiarmid's bounded-differences inequality (1989) to a Lipschitz extension (McShane 1934; Caputti 1984).
• Section 10, Vanishing of the main energy (pp. 72--77). Lemma 10.1 approximates each feature $V_l$ in $L^2$ by polynomials in the fair-split transforms $G_{B,v}$; Lemma 10.2 approximates $\mathfrak S$ in mean by a periodic function; Proposition 10.3 subdivides the blocks into intervals of length at most $\delta T$ and reduces the main energy to the short averages of Lemma 2.3, which vanish. The proof of Proposition 2.2 (p. 77) collects the error terms and chooses $C_*$, then $\eta$ and $C_6$, then $\epsilon_1$ and $\epsilon_2$, to contradict the lower bound $c_5$; the order of limits is always $x\to\infty$ before $B\to\infty$.
• Section 11, Marginals, the joint law, and the ordering corollary (pp. 77--81). Finite Fourier inversion over the roots of unity of order $J+1$ turns Proposition 2.2 into factorization of joint bin events; Lemma 11.1 (p. 78) recovers the Dickman marginal $\rho(1/c)$ from the factorial moments by inclusion--exclusion and the delay equation; display (11.7) gives the fixed-threshold law with $X^c$, $X^d$ through real $X$, display (11.8) the fixed-scale upper-tail law $(1-D(c))(1-D(d))$ on $[0,1]^2$ (the law the introduction, p. 2, calls the "upper-tail independence conjecture" of Erdős and Pomerance 1978, p. 311), and the proofs of Theorem 1.1 (p. 80) and Corollary 1.2 (p. 81) follow.

The manuscript flags nothing as numerical, computer-assisted or conditional; its only self-declared ineffectivity is the Siegel-type constant in Lemma 3.1. The release folder holds paper.pdf, README.md and a build directory and no verification/ folder.

Bears on

• Problem 928: claimed resolution. Theorem 1.1 asserts that the density of $n$ with $P^+(n)\le n^a$ and $P^+(n+1)\le n^b$ exists for every $a,b\in(0,1)$ and equals $\rho(1/a)\rho(1/b)$. The problem's event uses $(n+1)^\beta$ for the second threshold and strict inequalities; the manuscript's displayed statement uses $n^b$ and weak inequalities, and its Section 11 handles the strict-versus-weak change only for fixed thresholds $X^c$. The passage to the problem's exact form is not written in the manuscript. The claim is unverified here; the page's status rests on acceptance evidence.
• Problem 371: claimed resolution. Corollary 1.2 asserts natural density $1/2$ for $P^+(n)<P^+(n+1)$ and for the reverse ordering, deduced from Theorem 1.1 by continuity of the limiting law. The manuscript records the previous lower density bounds ($0.2017$ by Lü and Wang, $0.280$ by Yang) and Teräväinen's logarithmic-density $1/2$ as the prior state. Unverified here; the page's status rests on acceptance evidence.
• Problem 370: claimed stronger form of a problem the page records as proved. The problem asks for infinitely many $n$ with $P^+(n)<n^{1/2}$ and $P^+(n+1)<(n+1)^{1/2}$. Theorem 1.1 at $a=b=1/2$ gives the set of $n$ with $P^+(n)\le n^{1/2}$ and $P^+(n+1)\le n^{1/2}$ natural density $\rho(2)^2=(1-\log2)^2$; the $n$ with $P^+(n)=n^{1/2}$ are the squares of primes, a set of density zero, and $P^+(n+1)\le n^{1/2}<(n+1)^{1/2}$, so the problem's set would have lower density at least $\rho(2)^2>0$. This deduction is made here, not in the manuscript, which does not name the problem. Unverified here; nothing here changes the page's status, which rests on its acceptance evidence.
• Problem 1201: does not apply. The manuscript treats the single shift $n\mapsto n+1$ and the pair $(P^+(n),P^+(n+1))$; it says nothing about $P^+(n(n+1)\cdots(n+k))$, about longer runs of shifts, or about the problem, and it is not an input the page lacks. The row is recorded because the problem shares the subject. Nothing here bears on the page's status, which rests on acceptance evidence.
• Lü and Wang 2025: comparison. That card records Theorem 1 of the 2018 text, lower density $0.2017$ for each ordering and no density $1/2$; the manuscript cites the same bound as the previous record before Yang 2026 and claims the exact density $1/2$ (Corollary 1.2). Unverified here.
• Teräväinen 2018: claimed upgrade. That card records the logarithmic-density independence of the large prime factors of $n$ and $n+1$; the manuscript cites the same paper's Theorems 1.14 and 1.16 for the product law and the ordering density $1/2$ in logarithmic density, claims both in ordinary natural density (Theorem 1.1, Corollary 1.2) and takes from that paper's Section 4 the idea of interpolating large-prime count functions by real multiplicative functions (proof of Lemma 2.3). Unverified here.
• Erdős and Pomerance 1978: claimed answer to that paper's question. That card records that its authors could not prove the expected density $1/2$ for $P^+(n)>P^+(n+1)$; the manuscript claims it (Corollary 1.2), cites the paper for the joint independence conjecture, its Theorem 1 and the bound $0.0099$, and derives the fixed-scale upper-tail independence law of its p. 311 as display (11.8). Unverified here.