Openai 2026 irrationality exponent pi 2

../

corollary_1_2: The manuscript's stated consequence: the Flint–Hills series, the sum over n >= 1 of 1/(n^3 sin^2 n) with n in radians, converges, deduced from Theorem 1.1 by a dyadic spacing argument. Claims checked only.

corollary_5_2: The manuscript's generalized Flint–Hills criterion: for fixed a, b > 0 the sum over n >= 1 of 1/(n^a |sin n|^b), n in radians, converges if and only if a > max(1, b); the convergence half rests on Theorem 1.1. Claims checked only.

theorem_1_1: The manuscript's main claim: for every nu > 2, all large enough denominators q satisfy |pi - p/q| >= q^(-nu) for every integer p, so the irrationality exponent of pi is 2, with an ineffective threshold. Claims checked only.

OpenAI, The irrationality exponent of $\pi$ is $2$, OpenAI Math Release preprint, September 24, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision fd4aeeb2e), folder preprints/The-irrationality-exponent-of-pi-is-2-September-24-2026; the held PDF, paper.pdf in the release, is retained as openai_2026_irrationality_exponent_pi_2.pdf, and the release's TeX bundle in that folder is the TeX source cited on this card. The license term was read in the release's root LICENSE file at that revision on 2026-10-12.

@misc{OAI:The-irrationality-exponent-of-pi-is-2-September-24-2026,
  author = {{OpenAI}},
  title = {{The irrationality exponent of $\pi$ is $2$}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-irrationality-exponent-of-pi-is-2-September-24-2026/paper.pdf}{OAI:The-irrationality-exponent-of-pi-is-2-September-24-2026}},
  year = {2026}
}

The held PDF has 23 pages with a text layer; its title page is dated September 24, 2026 and its metadata records a creation date of 21 September 2026, three days earlier. The TeX source is build/main.tex (264 lines), which inputs four section files under build/sections/: interpolation.tex (584 lines), determinant.tex (432), conclusion.tex (156) and series.tex (155), 1,591 lines in all, with the bibliography build/references.bib (269 lines, 25 entries).

Attestation. The release README states that the collection holds manuscripts "produced by an internal OpenAI model", that it "includes results at different stages of verification", that "Not all have accompanying Lean formalizations" and that "Some of the unformalized results could have issues". Its account of the method says that "The vast majority of results were obtained with the same procedure using an unreleased internal OpenAI model", with on average three hours of "ChatGPT Pro thinking compute" per result; the exceptions it names (a zero-free region for the Riemann zeta function and the Hodge conjecture for CM abelian varieties) do not include this manuscript's subject. The README also lists this manuscript's family among the families for which the release gives "abridged summaries of the model's reasoning" (reasoning_traces/irrationality-exponent-of-pi.pdf in the release); that file is not held here and was not read. The manuscript's own README carries only the title, the author line "OpenAI", the date and the citation block, and adds no sentence about authorship or assistance. The title page names OpenAI as the sole author, and the manuscript has no acknowledgment or funding note. These are the source's own statements, recorded as attestations 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.

Formalization. The release's Lean page for its family (lean/docs/017.md) says that the formalization proves the irrationality exponent of $\pi$ is exactly two, in the form that for every $\nu>2$ all sufficiently large positive denominators $q$ satisfy $|\pi-p/q|\ge q^{-\nu}$ for every integer numerator $p$, and that it also states the supremum characterization through infinitely many rational approximations; it adds that "The paper's convergence consequence for the Flint–Hills series is outside this selected statement", so Corollaries 1.2 and 5.2 are outside it. The comparator statement lean/ComparatorChallenges/PiExponent.lean (theorem OAI.PiExponent.main, its proof left as sorry as in a challenge file) reads, in words: for every real $\nu>2$ there is an integer $Q\ge2$ with $q^{-\nu}\le|\pi-p/q|$ for all integers $p$ and all integers $q\ge Q$; and the supremum of the set of $\nu>0$ for which infinitely many rationals $r$ with reduced denominator at least $2$ satisfy $0<|\pi-r|<\operatorname{den}(r)^{-\nu}$ equals $2$. Its configuration PiExponent.json names the solution module OAI.NumberTheory.PiExponent.Main and permits only the axioms propext, Quot.sound and Classical.choice; the release tree holds that module as lean/OAI/NumberTheory/PiExponent/Main.lean, whose contents were not read. At revision fd4aeeb2e the release's catalog lean/formalization.yaml lists no entry for this comparator under status.main_results and does not name this manuscript among its sources; the catalog's own review field reads status: unchecked. All of this is read statically from the release's files; nothing was built, replayed or audited for fidelity in this repository. No Lean file here is a proof of any Erdős problem.

Companions. The release files this manuscript as the only member of its family, and at revision fd4aeeb2e it holds no earlier dated version of it, so there is no companion to record.

Read status: claims checked for Theorem 1.1, Corollary 1.2 and Corollary 5.2, together with the definition of the irrationality exponent and the statements of Theorem 2.1 (separated-weight interpolation), Proposition 3.4 (determinant comparison), Lemma 4.1 (choice of constants) and Lemma 5.1 (dyadic spacing), read clause by clause in the release's TeX source: in main.tex, the definition at lines 50--56, thm:main at lines 64--71 with the remark at lines 73--78 and thm:flinthills at lines 90--92; geom:interpolation at lines 24--57 of sections/interpolation.tex; det:comparison at lines 353--370 of sections/determinant.tex; par:constants at lines 9--21 of sections/conclusion.tex; and series:dyadic at lines 11--21 and series:generalized at lines 97--104 of sections/series.tex. The proofs were read for their structure only and no step was checked; nothing here is independently reviewed.

Contents

The numbering below is the PDF's (results are numbered within sections).

• Title page and Section 1, Introduction (pp. 1--4). Defines the irrationality exponent $\mu(x)$ of an irrational $x$ as the supremum of the $\nu>0$ for which $0<|x-p/q|<q^{-\nu}$ has infinitely many solutions in coprime $p,q$ with $q\ge2$, notes $\mu(x)\ge2$ by pigeonhole, and cites Waldschmidt's Open Diophantine problems (p. 265) for the conjecture $\mu(\pi)=2$. States Theorem 1.1 (p. 1), with the remarks that it covers unreduced fractions, that its threshold is ineffective, and that exponent $2$ is weaker than bounded partial quotients. Introduces the Flint–Hills question (popularized by Pickover, discussed by Alekseyev) and states Corollary 1.2 (p. 1), placing it between Alekseyev's necessary condition $\mu(\pi)\le5/2$ (his Corollary 4) and Meiburg's sufficient condition $\mu(\pi)<5/2$ (his Theorem 2.5), and previews Corollary 5.2. Section 1.1 (p. 2) lists earlier upper bounds for $\mu(\pi)$ as context, not inputs: Mahler 1953 ($42$, eventually $30$), Mignotte 1974 ($21$, eventually $20$), Chudnovsky's Hermite–Padé work, Hata 1993 ($8.01604539\ldots$), Salikhov ($7.606308\ldots$), Zeilberger–Zudilin ($7.103205334137\ldots$) and a September 2026 preprint of Bai claiming $7.101862832357$. On the series it cites two separate 2026 preprints, Dan's on generalized series and López Zapata's continued-fraction criterion, as leaving convergence undecided, records the withdrawal in September 2026 of a convergence claim of Mantzakouras and López Zapata, and says Carella's 2022 claim of exponent two and bounded partial quotients is not used, with a footnote objecting to a sign in Carella's displayed proof of the stronger assertion. The outline in Section 1.2 (pp. 2--4) relates the proof to Laurent's interpolation determinants, Roth's separated denominators, Philippon's zero estimates, Farhi's Roth lemma and Demailly's Seshadri-constant viewpoint. The closing conventions list the algebraic-geometry inputs (Bézout with isolated local multiplicities, ordinary blowups, nef plus ample, Proj recovery, Serre vanishing, smooth projective models of curves) and state that no earlier irrationality bound for $\pi$ is assumed.
• Section 2, Interpolation with separated weights (pp. 4--11). For positive rational weights $W=(w_0,\dots,w_m)$ on $Y,X_1,\dots,X_m$ and $V=(v_0,\dots,v_m)$ on $t,u_1,\dots,u_m$, defines the polynomials of $W$-degree at most $H$ and the coefficient packets of $V$-weight below $H$. Theorem 2.1 (p. 4), stated for arbitrary complex centers: under $K\theta^m<1$ and $Kw_0\theta^m/v_0<1$, weights $w_1,\dots,w_m$ chosen in turn above thresholds that do not depend on the centers make the jet map along the logarithmic curves $Y=1+t$, $X_i=c_{ji}+u_i+\log(1+t)$ surjective at any $K$ centers with coordinatewise distinct entries, for all large, suitably divisible $H$. Lemma 2.2 (p. 5) compares the weights of coordinate and vector-field normal bases on a component where many derivatives vanish, through a simplex count and a weighted Bézout bound (cited to Mondal's notes, Corollary VIII.3). Proposition 2.3 (p. 7), the curve inequality, bounds the total weighted contact of an irreducible curve with the logarithmic curves at the centers by its weighted degree divided by $1+\sigma$. Section 2.3 (pp. 9--11) passes from curves to jets by a blowup of a weighted projective compactification, the nef-plus-ample criterion (Lazarsfeld, Corollary 1.4.10) and Serre vanishing, with the Stacks Project cited for blowups, Rees algebras and vanishing.
• Section 3, The arithmetic and analytic determinant estimates (pp. 12--17). Assumes $m$ approximations $|\pi-p_i/q_i|\le q_i^{-\nu}$, sets $w_i=\lceil\log q_i\rceil$ and centers $2\mathrm ijp_i/q_i$ in $\mathbb Q(\mathrm i)$, truncates the logarithms, and takes a nonzero square minor $\Delta_H$ of the interpolation matrix using every row, with a simplex count for the number $M$ of rows. Lemma 3.1 (p. 13), the arithmetic lower bound for $\log|\Delta_H|/(MH)$, proves the needed bound $\log\operatorname{lcm}(1,\dots,n)\le4n\log2$ in the text. Lemma 3.2 (p. 14), exact row translation to Taylor tests of entire functions at the periods $2\pi\mathrm ij$. Lemma 3.3 (p. 15), the collision estimate, after Laurent's Taylor-expansion mechanism (his 2000 paper, Theorem 1). Proposition 3.4 (p. 16), the determinant comparison giving three sufficient numerical conditions for a contradiction.
• Section 4, Choice of parameters and proof of Theorem 1.1 (pp. 17--19). Lemma 4.1 (p. 17) supplies the rational constants; the proof (pp. 18--19) fixes the dimension, then the approximations, then lets the degree grow, and closes with the irrationality of $\pi$ and the lower bound $\mu(\pi)\ge2$.
• Section 5, Flint–Hills series and its generalization (pp. 19--21). The dyadic spacing estimate Lemma 5.1 (p. 19), a summability bound for any irrational $\alpha>0$ with $\|q\alpha\|\ge cq^{1-\nu}$ for all $q\ge1$, where $1<\nu<5/2$; the proof of Corollary 1.2 (pp. 19--20); and Corollary 5.2 (p. 20), the criterion for $\sum_n n^{-a}|\sin n|^{-b}$, with its proof (pp. 20--21).
• References (pp. 21--23), 25 entries, including the 2026 preprints of Bai, Dan and López Zapata named above.

The manuscript flags nothing as numerical, computer-assisted or conditional; every constant in the proof is chosen in the text, and the threshold $Q(\nu)$ of Theorem 1.1 is ineffective. The release folder holds only the PDF, the TeX build directory and the README; there is no verification/ folder.

Bears on

The manuscript names no Erdős problem, and no problem page is identified as using it. The pages of the irrationality area and those tagged Diophantine approximation were read for a possible use. The nearest is Problem 495, which asks whether $\liminf_{n\to\infty}n\|n\alpha\|\|n\beta\|=0$ for all real $\alpha,\beta$ (Littlewood's conjecture). That equality would hold for every pair containing $\pi$ if $\pi$ had unbounded partial quotients, but exponent $2$ is consistent with bounded and with unbounded partial quotients, and the manuscript's remark after Theorem 1.1 distinguishes exponent $2$ from bounded partial quotients; the result decides neither case, so it supplies nothing there. The other pages in those areas ask about almost every real number, about every irrational number, or about series and integer sequences unrelated to $\pi$, and a statement about one constant does not enter them.