Openai 2026 positive lower density large prime gaps

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corollary_1_2: The manuscript's claimed answer to the Erdős--Prachar question behind Problem 968: the indices at which p_n/n increases have positive lower asymptotic density, from Theorem 1.1 at C=2; claimed, not verified here.

theorem_1_1: The manuscript's main claim: for every fixed C>0 at least c(C)N indices n at most N have p_{n+1}-p_n > C log p_n, by adjacent-interval sieve weights resting on Bombieri--Vinogradov; claimed, not verified here.

OpenAI, Positive lower density of large prime gaps, OpenAI Math Release preprint, September 25, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision adc7f1241), folder preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026; the held PDF, main.pdf in the release, is retained as openai_2026_positive_lower_density_large_prime_gaps.pdf, and the release's TeX bundle in that folder is the TeX source cited on this card.

@misc{OAI:Positive-lower-density-of-large-prime-gaps-September-25-2026,
  author = {{OpenAI}},
  title = {{Positive lower density of large prime gaps}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026/main.pdf}{OAI:Positive-lower-density-of-large-prime-gaps-September-25-2026}},
  year = {2026}
}

Attestation, recorded as the source's own statements and not as this corpus's review: the release's root README says its manuscripts were "produced by an internal OpenAI model", that the collection "includes results at different stages of verification", that not all of them have Lean formalizations, and that "Some of the unformalized results could have issues". The manuscript's own README carries only the title, the author line "OpenAI", the date and the citation block, and adds no statement about how the text was produced or checked. The title page names no individual author and the text names no referee, reader or prior circulation. No refereed publication, arXiv version or independent review of the manuscript is recorded here and nothing on this card is independently reviewed.

Formalization, read statically from the release's catalog; not built, replayed or audited for fidelity in this repository. The release's lean/formalization.yaml, its catalog of papers with a formalized main result, does not list this manuscript; the entry for this manuscript in the release's manuscript map does link a Lean page, which says that the formalized supplement proves that the indices $n$ with $p_n/n<p_{n+1}/(n+1)$ have positive lower asymptotic density, "the prime-ratio corollary associated with the paper's theorem", and names the comparator statement file lean/ComparatorChallenges/PrimeGaps.lean. That file's statement, OAI.Problem344.large_gaps_and_ratio_density (where Problem344 is the release's own numbering, not an Erdős problem number), conjoins the counting bound of Theorem 1.1 (for every real $C>0$ some $c>0$ and $N_0$ have $cN\le\#\{1\le n\le N:C\log p_n<p_{n+1}-p_n\}$ for all $N\ge N_0$) with the positivity of the lower asymptotic density of the ratio-increase set of Corollary 1.2, with $p_n$ defined as the $n$th prime through Nat.nth Nat.Prime (n - 1). The challenge file PrimeGaps.json names a solution module, OAI.NumberTheory.PrimeGaps.RatioCorollary; it was not compiled, searched or audited here. Whether a release declaration settles the problem is recorded on the problem's claim pages, not on this card.

The release files this manuscript alone and lists no companion, alternate proof or consequence paper for it.

Read status: claims checked for Theorem 1.1 and Corollary 1.2, and for the statements of Proposition 2.1, Lemma 2.2, Proposition 3.1, Lemma 4.1, Proposition 4.2, Proposition 5.1 and Lemmas 5.2--5.4, read clause by clause in the TeX source (main.tex; sections/01-introduction.tex, labels thm:main and cor:ratios; sections/02-counting.tex; sections/04-weights.tex; sections/05-cancellation.tex; sections/03-moments.tex, which is the printed Section 5) on 2026-10-07; the proofs were read for their structure only and no step was checked; nothing here is independently reviewed. Page numbers below are those of the held PDF (19 pages); the PDF numbers its results by section, as Theorem 1.1 and Corollary 1.2, and the converter's rendering beside the PDF prints the same labels.

Contents

• Section 1, Introduction (pp. 1--3). Defines $p_n$, $d_n=p_{n+1}-p_n$ and the lower asymptotic density $\liminf_N|A\cap[1,N]|/N$; recalls the unbounded-gap results of Westzynthius [18], Erdős [4], Rankin [15], Ford, Green, Konyagin and Tao [7], Maynard [14] and Ford, Green, Konyagin, Maynard and Tao [6], and notes that a lower bound for the largest gap gives no positive proportion of large gaps. States Theorem 1.1 (for every fixed $C>0$, $\#\{n\le N:d_n>C\log p_n\}\ge c(C)N$ for all $N\ge N_0(C)$) and Corollary 1.2 (the set of $n$ with $p_n/n<p_{n+1}/(n+1)$ has positive lower asymptotic density), with the corollary's five-line proof from the theorem at $C=2$ and the prime number theorem; attributes the question to Erdős and Prachar [5, p. 256]. A subsection distinguishes counting prime-free intervals by integer starts from counting gaps by their starting prime, reads Theorem 5 of Bazzanella, Languasco and Zaccagnini [1] as giving positive prime-start proportions for thresholds below $2/3.454=0.579\ldots$ and their Theorem 3 as reaching longer intervals only by integer starts, cites Tao's MathOverflow answer [16] for the passage from interval measure to gap counts, and places Gallagher's conditional Poisson statistics [8] and Jha's conditional Poisson-tail results [12] as comparisons. A second subsection outlines the method: average over $X<m\le2X$ with $h=\lfloor\lambda\log X\rfloor$; build a nonnegative weight giving a prime in $(m,m+h]$ substantial weighted mass and the interval $(m+h,m+2h]$ arbitrarily small weighted prime mass; the last prime in $(m,m+h]$ then starts a gap longer than $h$ and is selected by at most $h$ values of $m$. Each weight is the square of a signed combination of smooth divisor sums over subsets of the second interval; a prime in a shift set forces that divisor coordinate to one, which couples terms of neighboring dimensions, and an alternating family makes those terms nearly cancel. Named antecedents are the correlation method of Goldston and Yıldırım [10, equation (2.14)] and Maynard's multidimensional sieve [13].
• Section 2, From two adjacent intervals to gap counts (pp. 3--5). Fixes $L=\log X$, $h=\lfloor\lambda L\rfloor$, $J=\{1,\ldots,h\}$, $I=\{h+1,\ldots,2h\}$, $\vartheta(n)=(\log n)\mathbf 1_{n\text{ prime}}$ and $V_B(m)=L^{-1}\sum_{b\in B}\vartheta(m+b)$. Proposition 2.1 (p. 3): for each fixed $\lambda>0$ there is $K_\lambda>0$ such that for every $\varepsilon>0$ there are weights $W_X\ge0$ on $(X,2X]$ with $\mathbb E_XW_X\to1$, $\mathbb E_X(W_XV_J)\to\lambda$, $\limsup\mathbb E_X(W_XV_I)\le\varepsilon$, $\limsup\mathbb E_X(W_XV_J^2)\le K_\lambda$ and $\mathbb E_XW_X^2=O_{\lambda,\varepsilon}(1)$, with $K_\lambda$ independent of $\varepsilon$. Lemma 2.2 (p. 3): if at least $\delta X$ starts $m\in(X,2X]$ have a prime in $(m,m+h]$ and none in $(m+h,m+2h]$, then at least $\delta X/h$ primes in $(X,2X+h]$ have next-prime gap exceeding $h$. The proof of Theorem 1.1 (p. 4) chooses $\lambda>\max\{C,1\}$ and $\varepsilon=\lambda^2/(2K_\lambda)$, uses two weighted Cauchy--Schwarz steps to get a proportion $\delta>0$ of starts with a prime in $J$ and none in $I$, applies Lemma 2.2, and converts the count on $(X,2X+h]$ with $X=\lfloor p_N/3\rfloor$ into the index count through the prime number theorem, giving $c(C)=\delta/(4\lambda)$.
• Section 3, A square with a small marked moment (pp. 5--6). Defines the smooth divisor sums $D_F(m;\mathbf b)=\sum_{d_i\mid m+b_i}\mu(d_1)\cdots \mu(d_j)F(\log d_1/L,\ldots,\log d_j/L)$, fixes $\tau=1/8$, takes symmetric $f_j\in C_c^\infty$ supported in $\{\sum t_i<\tau\}$ for $1\le j\le k$ with a scalar $f_0$, their cumulative integrals $F_j$ and the one-coordinate integral $Tf_{j+1}$, and the signed sum $Z(m)=\sum_{j\le k}\sum_{S\subset I,|S|=j}D_{F_j}(m;S)$ with $\alpha_j=\lambda^j/j!$. Proposition 3.1 (p. 6): as $X\to\infty$, $\mathbb E_XZ^2\to w=\sum_j\alpha_j\|f_j\|_2^2$, $\mathbb E_XZ^4=O(1)$, $\mathbb E_X(Z^2V_J)\to\lambda w$, $\mathbb E_X(Z^2V_I)\to v=\lambda\sum_j\alpha_j\|f_j+Tf_{j+1}\|_2^2$, and $\limsup\mathbb E_X(Z^2V_J^2)\le C_*w$ with $C_*=\lambda+\lambda^2c_G^2$ independent of $k$ and of the family. The exact deletion identity (display (8) of the PDF) rewrites $Z$ on the event that $m+a$ is prime, $a\in I$, as a sum over subsets of $I\setminus\{a\}$ with $\widetilde F_j=F_j+F_{j+1}(\cdot,0)$; the normalization $W_X=Z^2/w$ then satisfies Proposition 2.1 once $v/w\le\varepsilon$.
• Section 4, Cancellation between adjacent dimensions (pp. 6--9). Lemma 4.1 (p. 7): for every $\lambda,\beta>0$ there is a nonnegative $g\in C_c^\infty((0,\infty))$ with $\int g^2=1$, $\int g=\sqrt\lambda$ and $\int ug(u)^2\,du<\beta$ (a rescaled, normalized $1/u$ profile on $[1,R]$). Proposition 4.2 (p. 7): for every fixed $\lambda>0$ there are families with increasing finite $k$ for which $w\to1$ and $v\to0$; the proof takes alternating product profiles $f_j=(-1)^jQ_j\chi(S_j)/(\sqrt r\sqrt{\alpha_j})$ on the $r=\lfloor\sqrt k\rfloor$ top levels, with $Q_j(\mathbf t)=k^{j/2} \prod g(kt_i)$ and a simplex cutoff $\chi$, so that neighboring levels cancel up to $1-\sqrt{(j+1)/k}$ and the two endpoint levels contribute $O(1/r)$; Chebyshev's inequality under the density $Q_j^2$ controls the cutoff. The proof of Proposition 2.1 (p. 9) sets $K_\lambda=C_*$, freezes one finite family with $v/w\le\varepsilon$, and stresses the parameter order: $k$ is fixed before $X\to\infty$, so no divisor-sum asymptotic uniform in $k$ is needed.
• Section 5, Analytic estimates for the square (pp. 9--18). Defines the singular series $\mathfrak S(\mathcal H)=\prod_p(1-\nu_p(\mathcal H)/p) (1-1/p)^{-|\mathcal H|}$. Proposition 5.1 (p. 9), uniform mixed moments: for fixed smooth compactly supported $F_r$ with support budgets $\rho_r$, shifts of size $O(L^B)$ sharing an equality pattern $\kappa$, and an optional prime mark $\vartheta(m+a_0)$, the average $\mathbb E_X[\chi_\delta\prod_rD_{F_r}]$ equals $L^{-t}\{\mathfrak S(\mathcal H)\mathcal C+O(L^{-1/2}(\log L)^A)\}$ with $t$ the number of distinct shifts, provided $\sum\rho_r<1$ (unmarked) or $<1/2$ (marked); when every fiber of $\kappa$ has at most two elements the constant $\mathcal C$ is an integral of products of signed complete mixed derivatives. Lemma 5.2 (p. 11) takes the Bombieri--Vinogradov theorem in the form of Bombieri, Friedlander and Iwaniec [3, equation (1.4)] and derives a weighted version with uniform shifted endpoints and $K^{\omega(q)}$ weights over $q\le X^\sigma$, $\sigma<1/2$. Lemma 5.3 (p. 12) reduces the average to an exact finite density sum over compatible squarefree divisor tuples with error $O_A(L^{-A})$. The proof of Proposition 5.1 (pp. 13--15) evaluates that sum by Fourier inversion, an Euler product split into zeta factors and a factor whose value at zero is $\mathfrak S(\mathcal H)$, and a comparison that never divides by a possibly vanishing local factor. Lemma 5.4 (p. 15), singular series in boxes: for fixed $s$ and $K$, the sum of $\mathfrak S(\{a_1,\ldots,a_s\})$ over distinct $a_i$ drawn from $s$ intervals of length $h$ inside a window of diameter $\le Kh$ is $h^s(1+o(1))$, a box form of Gallagher's mean [8, equation (3)] proved by truncating the Euler product at $y=(1/4)\log h$. The proof of Proposition 3.1 (pp. 17--18) applies these to $Z^2$, $Z^2V_J$, $Z^2V_I$ (through the deletion identity), $Z^4$ and the detector moment, with the support budgets $2\tau$, $2\tau$, $4\tau$ and $6\tau$ tabulated against the limits $1$ or $1/2$; the detector bound majorizes the two possible primes in $J$ by squares of one-dimensional divisor sums $D_G$ so that no two-mark asymptotic is needed.
• References [1]--[18] (pp. 18--19): Bazzanella, Languasco and Zaccagnini (Trans. Amer. Math. Soc. 362, 2010); Bombieri (1965); Bombieri, Friedlander and Iwaniec (Acta Math. 156, 1986); Erdős (1935); Erdős and Prachar (Abh. Math. Sem. Univ. Hamburg 25, 1962); Ford, Green, Konyagin, Maynard and Tao (2018); Ford, Green, Konyagin and Tao (2016); Gallagher (Mathematika 23, 1976); Goldston and Yıldırım (Integers 3, 2003, and arXiv:math/0504336v1); Granville, Koukoulopoulos and Maynard (2021); Jha (arXiv:2605.23014v2, 2026); Maynard (2015, 2016); Rankin (1938); Tao (MathOverflow answer 332888, 2019); Vinogradov (1965); Westzynthius (1931).

External inputs the proofs rest on: the Bombieri--Vinogradov theorem (through Lemma 5.2), the prime number theorem (in the proof of Theorem 1.1, in the corollary, and for $Q\le h^{1/2}$ in Lemma 5.4), and elementary estimates ($\sum_{p\le y}1/p=O(\log\log y)$, $\zeta(1+u)\le1+1/u$, the Chinese remainder theorem, Chebyshev's inequality). The manuscript says the divisor-sum correlations of Goldston--Yıldırım, Maynard's sieve and the smoothing analysis of Granville, Koukoulopoulos and Maynard are motivation only and that Proposition 5.1 is proved in the paper. The manuscript flags nothing as unproved, numerical, computer-assisted or conditional; its only hypothesis-shaped sentence, "Assume the Bombieri--Vinogradov theorem" in Lemma 5.2, names a cited theorem.

Bears on

• Problem 968: claimed resolution. The problem asks whether $\{n:p_n/n<p_{n+1}/(n+1)\}$ has positive density; Corollary 1.2 claims that this set has positive lower asymptotic density, which is an affirmative answer when "positive density" is read as positive lower density (the reading of the Erdős--Prachar question the manuscript cites at p. 256); the manuscript does not claim that the natural density exists. The claim is unverified here, and the page's status rests on acceptance evidence, not on this card.
• Problem 234: comparison and a partial fact, not stated in the manuscript. With $C=c$, Theorem 1.1 and $\log p_n>\log n$ give that $\{n:(p_{n+1}-p_n)/\log n<c\}$ has upper density at most $1-c(c)<1$ for every $c>0$, so the density $f(c)$ of the problem, where it exists, is below one; the existence and continuity of $f(c)$ that the problem asks for are untouched. The manuscript's own comparison is with the unconditional prime-start proportions of Bazzanella, Languasco and Zaccagnini below the threshold $0.579\ldots$ and with the conditional Poisson statistics of Gallagher. The claim is unverified here, and the page's status rests on acceptance evidence.
• Problem 5: background only. The theorem gives a positive proportion of normalized gaps above every fixed $C$ but no two-sided control of a single gap, so it supplies no limit point of $(p_{n+1}-p_n)/\log n$; the claim is unverified here, and nothing here changes that page, whose status rests on its own acceptance evidence.
• Erdős and Prachar (1961/62): the source of the question. The manuscript cites p. 256 of that paper for the question whether the set of $n$ with $p_n/n<p_{n+1}/(n+1)$ has positive lower density and claims an affirmative answer in Corollary 1.2; the held card records the paper's two theorems and notes that further problems are posed; the specific question is not transcribed there. The claim is unverified here, and nothing on that card is changed by it.