Openai 2026 poisson dirichlet law prime predecessors

../

theorem_1_1: The claimed main theorem: over primes p up to x, the normalized logarithms of the prime factors of p-1, listed with multiplicity in decreasing order, converge in every finite joint distribution to PD(1) (Ford-Konyagin-Luca).

theorem_7_1: The claimed arithmetic core of the manuscript: over primes p weighted by a fixed nonnegative smooth compactly supported cutoff of (p-1)/x, the count of ordered tuples of distinct primes from prime slot intervals (depending on x) with lower endpoints at least x^eps whose upper endpoints multiply to at most x^(1-eps), dividing p-1, matches its reciprocal-sum mean up to o(x/log x); Section 8 turns it into Theorem 1.1.

OpenAI, The Poisson-Dirichlet law for prime predecessors, 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-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026; the held PDF, paper.pdf in the release, is retained as openai_2026_poisson_dirichlet_law_prime_predecessors.pdf, and the release's TeX bundle in the same folder is the TeX source cited below.

@misc{OAI:The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026,
  author = {{OpenAI}},
  title = {{The Poisson--Dirichlet law for prime predecessors}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/paper.pdf}{OAI:The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026}},
  year = {2026}
}

Attestation, as the release states it. The release's root README says the 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 adds nothing beyond the author line "OpenAI", the date and the citation block; the manuscript carries no statement on how it was produced and no author names. These are the source's own attestations, recorded here as history and not as this corpus's review. No refereed publication, arXiv version or independent review of the manuscript is recorded here and nothing on this card is independently reviewed.

The release's Lean catalog (lean/formalization.yaml) lists no formalization for this manuscript, and the release has no Lean page for its family (011).

Companions. The release groups this manuscript with Weighted dilation graphs, smooth shifted primes and totient fibers (its card), which the text calls S and uses as the source of its block sieve and of the dilation-graph operator theorems imported in Section 6; S is the family member that addresses totient fibers, and this manuscript says S's Theorem 1.2 (a count of $x^{1-o(1)}$ primes with smooth predecessors) does not imply the positive limiting proportion claimed here. The third family member, Prime predecessors with an even number of prime factors, is not cited by this manuscript and has no card in this library as of the read date.

Read status: claims checked for Theorem 1.1, for the fixed-$u$ consequence (1.3) the introduction draws from it, and for the statements of Theorems 3.1, 6.1 and 7.1, read clause by clause in the TeX source (sections/01-introduction.tex lines 8--48 and 78--97, sections/03-determinant.tex lines 37--63, sections/06-correlations.tex lines 55--82, sections/07-extraction.tex lines 3--31 and sections/08-poisson-dirichlet.tex in full) on 2026-10-07; the proofs were read for their structure only and no step was checked; nothing here is independently reviewed.

Contents

The PDF has 74 pages: a table of contents on pp. 1--2, eight sections and an appendix, and 28 references on pp. 73--74. Results are numbered by section.

• Section 1, Introduction (pp. 2--5). For a prime $p\ge3$ the prime factors of $p-1$ are listed with multiplicity in decreasing order $q_1(p)\ge q_2(p)\ge\cdots$, padded by $1$, and $V_j(p)=\log q_j(p)/\log(p-1)$; the stick-breaking fragments $B_1=1-U_1$, $B_j=(\prod_{i<j}U_i)(1-U_j)$ of independent uniforms have decreasing rearrangement $(L_1,L_2,\ldots)$ with the law $\mathrm{PD}(1)$ (display (1.1)). Theorem 1.1 (p. 3): for every fixed $k$ and bounded continuous $F$ on $[0,1]^k$, the average of $F(V_1(p),\ldots,V_k(p))$ over primes $3\le p\le x$ tends to $\mathbb E F(L_1,\ldots,L_k)$, through all real $x$. The text says this resolves Conjecture 5 of Section 6 of Ford, Konyagin and Luca (2010; filed as ford_2010_prime_chains_pratt_trees) with the same conventions. The history paragraph cites Dickman, de Bruijn, Billingsley (1972), Donnelly--Grimmett (1993) and Arratia--Kochman--Miller (2014) for ordinary integers; Erdős (1935, 1956), Pomerance (1980), Baker--Harman (1998, threshold $x^{0.2961}$) and Lichtman (2022, threshold $x^{0.2844}$ on $(x,2x]$) for smooth shifted primes; and Granville's conjectural asymptotic (1.3), $\#\{p\le x:P^+(p-1)\le x^{1/u}\}\sim\pi(x)\rho(u)$ for fixed $u\ge1$, which the text says Theorem 1.1 resolves (the change from $x$ to $p-1$ in the threshold is handled by restricting to $\varepsilon x<p\le x$). It places the result against Bharadwaj--Rodgers (2026, Theorem 7: the full law for shifted primes under Elliott--Halberstam; unconditionally level one half and correlations on the half simplex), against Ford (2025) and Gorodetsky (2026) on small prime divisors of shifted primes, and notes that Theorem 1.1 supplies only the root law of the Ford--Konyagin--Luca Pratt-tree model and that the Carmichael construction of Alford, Granville and Pomerance (1994) needs a separate progression-distribution input. Two paragraphs describe the method (below) and Figure 1 maps the dependence of the steps.
• Section 2, Conventions and analytic inputs (pp. 5--9). $L=\log x$, $W=\exp(L^{0.24})$, $V(W)=\prod_{p\le W}(1-1/p)$; an integer is rough when it has no prime factor at most $W$. Lemma 2.1 (prime estimates: the prime number theorem with arbitrary logarithmic savings, Mertens, and Siegel--Walfisz for moduli up to $(\log y)^C$, constants not effective) is cited to Tao's lecture notes; the manuscript proves Lemma 2.2 (divisor moments and coefficient-independent Dirichlet-polynomial mean squares) on the page and Lemma 2.3 (long prime polynomial: $\sum_{p\in I}\chi(p)p^{-1+it}\ll L^{-A}$ for $x^\tau/2\le N\le x^\eta$, $q\le L^C$, $L^{B_0}\le|t|\le x^2$) and Lemma 2.4 (logarithmic phases on progressions) in Appendix A, and Lemma 2.5 (rough integers in long intervals and progressions, with characters) by Bonferroni truncation; Lemma 2.6 (block sieve) is imported from S, Lemma 2.9.
• Section 3, A determinant estimate with marks on one side (pp. 9--18). Theorem 3.1 (p. 10): with $K$ fixed disjoint prime bands $[\exp(L^{a_i}),\exp(2L^{a_i})]$, $0.1<a_1<\cdots<a_K<0.2$, the marked weight $\mathcal W(h)=q^{\omega(h)-K}\prod_i\omega_i(h)/V_i$, a centered coefficient $\alpha_m=m^{iv}(\mathbf 1_{m\text{ prime}}-\mathbf 1_{m\text{ rough}}/(V(W)\log m))$, and an arbitrary rough coefficient $\beta_n$ of size $L^C$ on scales $H_m,H_n\ge x^\delta$ with $H_mH_n\asymp x$, the bilinear sum $\sum\alpha_m\beta_nF(mn-1)\mathcal W(mn-1)$ is $O(XL^{-D_*})$ once $K$ is large in terms of $\delta,C,D_*,q$, with the needed $K$ independent of the $a_i$. The proof divides out one tuple of marks, applies Cauchy's inequality, and arrives at a condition that two primitive lattice vectors have small determinant; Lemma 3.3 (root residues) and Lemma 3.4 (replacement of the root average by independent projective lines) prepare the moment.
• Section 4, Signed memory and the determinant moment (pp. 18--33). Proposition 4.1 (p. 19): the signed long moment of the determinant operator is at most $L^{-E_0N}$, so the determinant indicator can be replaced by its major-arc kernel with error $O(XYL^{-A})$. The proof builds an exact primewise identity for an operator that remembers a prime between two of its uses (the lifespan picture of Figure 2), a symmetric memory space, truncation and adjoints, absolute bounds, a lattice box, the two Schur sides, and then restores global distinctness of prime labels by grouping equalities by rank; the text compares the construction with the lifespan expansion of S, Section 3, and says every operator estimate is proved here.
• Section 5, The major term of the determinant estimate (pp. 33--39). Proposition 5.1: the unrestricted major sum is $O(XYL^{-D})$. Lemma 5.2 (uniform cancellation of the centered coefficient against characters of modulus $\le L^{A_0}$ and phases up to $2XL^B$) and Lemma 5.3 (a small prime polynomial is large only on $\exp(O(L^{0.91}))$ unit intervals) supply the two cases; with Proposition 4.1 this completes Theorem 3.1.
• Section 6, Two-sided marked correlations (pp. 39--51). Theorem 6.1 (p. 40): for fixed $d$, $\varepsilon$ and prime slots $I_1,\ldots,I_d$ with lower endpoints at least $x^\varepsilon$ and product of upper endpoints at most $x^{1-\varepsilon}$, the centered divisor statistic $F_x(n)=\mathcal W(n)(\sum_{\mathbf l}\mathbf 1_{l\mid n}-C_x)$ correlates with any $G(n+1)$ of size $L^C\tau(n_*)^C$ depending only on the inactive part $n_*$, both sides marked by $W_1$, at $O_A(L^{-A})$ in logarithmic average. Propositions 6.2 and 6.3 are imported from S (Theorem 3.5, Corollary 3.11 and Lemma 3.4; Theorem 4.1 and Lemma 4.3) with their exact hypotheses restated; the new work is the endpoint estimates: removal of shared labels, comparison multipliers and local Mellin energy (Lemma 6.4), a truncated divisor model for low frequencies (Lemma 6.5), and the high-frequency factorization, which cites Matomäki--Radziwiłł (2017) and Soundararajan (2009) for the exceptional-time argument.
• Section 7, Extraction of the prime statistic (pp. 51--60). Theorem 7.1 (p. 51): for the same slots and a smooth cutoff $\Phi$, $\sum_p\Phi((p-1)/x)(f_x(p-1)-C_x)=o(x/\log x)$, where $f_x(u)$ counts the ordered tuples of distinct slot primes whose product divides $u$ and $C_x$ is the corresponding reciprocal sum. Lemma 7.2 (summed progression remainder up to $x^{c_0}$ for the marked weights, by Bonferroni truncation of the band weights and residue counting in one progression) feeds a presieve of $u+1$ by the block sieve; Theorem 3.1 turns each prime slot of a composite term into a rough slot; candidate active groups with independent coin tosses and Theorem 6.1 bound the marked composite contribution; averaging over $b$ disjoint marking arrays removes the marks by a mean-square bound. The order of limits (fix $h$, then $K$, then the arrays, then $x\to\infty$, then $b\to\infty$, then $h\to\infty$) is stated explicitly.
• Section 8, From interior statistics to the full law (pp. 60--63). From Theorem 7.1: the factorial measures of ordered tuples of labeled factor masses converge on the open simplex to $\prod_i dt_i/t_i$; size-biased sampling gives the interior density $h_d(t)=\prod_i(1-t_1-\cdots-t_{i-1})^{-1}$, shown to have total mass one by the triangular change of variables (8.10), which excludes boundary mass; a sorting bound and the expected undrawn mass $2^{-d}$ pass to the decreasing rearrangement over dyads $x<p\le2x$; a finite dyadic decomposition gives all real $x$ and removes $p=2$. The text credits the viewpoint to Donnelly--Grimmett and Arratia--Kochman--Miller and gives the passage in full.
• Appendix A, Prime polynomials and logarithmic phases (pp. 64--72). Lemma A.1 restates Lemma 2.3. Lemma A.4 is a degree-uniform power-sum bound by the Vinogradov mean-value iteration in Linnik's $p$-adic form (exposition cited to Wooley 2012, without efficient congruencing; compared with Stechkin 1975, Ford 2002 and Khale 2024); Lemma A.5 restates Lemma 2.4; Lemmas A.6--A.8 give a bound near $\mathrm{Re}\,s=1$, a local logarithmic derivative and a zero-free strip $\mathrm{Re}\,s\ge1-c(\log L)^2/L$, $1\le|\mathrm{Im}\,s|\le x^3$, for characters of modulus at most $L^C$; Mellin inversion then yields Lemma A.1.

External inputs the proofs rest on, at statement level: the classical prime estimates of Lemma 2.1 (Siegel--Walfisz, so constants are not effective), the block sieve and the two dilation-graph operator theorems of the companion S (Lemma 2.6, Propositions 6.2 and 6.3), Donnelly--Grimmett and Arratia--Kochman--Miller for the probabilistic framing of Section 8, and the cited exceptional-time technique of Matomäki--Radziwiłł and Soundararajan in Section 6. The manuscript flags nothing as numerical, computer-assisted or conditional; it states that all smoothness parameters in (1.3) stay fixed as $x$ grows and gives no rate of convergence. The release folder holds only the PDF, its build files and the README; there is no verification folder.

Bears on

• Problem 821: a claimed input to the Erdős--Pomerance smooth-shifted-prime route recorded on the Baker--Harman and Lichtman cards (the page's references), not a claimed answer. The problem asks for infinitely many $n$ with more than $n^{1-\epsilon}$ totient preimages; the manuscript states no result about totient fibers and names that question only as the origin (Erdős 1935, Pomerance 1980) of the smooth-shifted-prime question. Its claimed consequence (1.3) gives, for every fixed $u$, a positive proportion $\rho(u)$ of primes $p\le x$ with $P^+(p-1)\le x^{1/u}$, stronger in form than the counts those cards record, which reach one threshold each (Baker--Harman at $x^{0.2961}$, Lichtman at $x^{0.2844}$) and whose corollaries, recorded on those cards, turn each count into a fiber bound at that one threshold. The manuscript draws no fiber conclusion, this corpus has not checked any transfer against (1.3), and the fiber claim of the family is made by the companion S from its own count. Nothing here is verified in this corpus, and the page's status rests on acceptance evidence, not on this card.
• Problem 1057: a claimed input the page lacks. Theorem 1 of Alford, Granville and Pomerance gives $C(x)\ge x^{EB}$ for every $E$ such that a positive proportion of primes $p\le x$ have $P^+(p-1)\le x^{1-E}$ and every admissible progression exponent $B$; the manuscript's consequence (1.3) would make every $E<1$ admissible, which is Erdős's conjecture recorded on the AGP card, so the Carmichael exponent would become the best admissible $B$ alone. The manuscript does not state this consequence; it mentions the AGP construction in one sentence and claims no Carmichael count. The claim is unverified here and the page's status rests on acceptance evidence.
• Alford, Granville and Pomerance (1994): if (1.3) holds as claimed, the set $\mathcal E$ of that card would be all of $(0,1)$, the hypothesis side of its Theorem 1; the manuscript does not state this and nothing is verified here.
• Baker and Harman (1998) and Lichtman (2022): the manuscript cites both as the previous lower bounds $x/(\log x)^C$ at thresholds $x^{0.2961}$ and $x^{0.2844}$ and claims, through (1.3), the asymptotic $\pi(x)\rho(u)$ at every fixed threshold $x^{1/u}$, a stronger claimed form, for the shift $a=1$ only, of their Theorems 1 and 1.1 (which treat every fixed nonzero shift $a$) as counts of smooth shifted primes (not of their totient or Carmichael corollaries, which need further inputs); unverified here.