Openai 2026 catalans constant irrational

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theorem_1_1: The manuscript's only theorem: Catalan's constant G, the sum of (-1)^j/(2j+1)^2 over j >= 0, is irrational; a qualitative statement with no irrationality measure. Claims checked only.

OpenAI, Catalan's constant is irrational, OpenAI Math Release preprint, September 24, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision fd4aeeb2e), folder preprints/Catalans-constant-is-irrational-September-24-2026; the held PDF, paper.pdf in the release, is retained as openai_2026_catalans_constant_irrational.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:Catalans-constant-is-irrational-September-24-2026,
  author = {{OpenAI}},
  title = {{Catalan's constant is irrational}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Catalans-constant-is-irrational-September-24-2026/paper.pdf}{OAI:Catalans-constant-is-irrational-September-24-2026}},
  year = {2026}
}

The held PDF has 44 pages with a text layer; its title page is dated 24 September 2026 and its metadata records a creation date of 24 September 2026. The TeX source is build/main.tex (44 lines), which inputs eight section files under build/sections/ (2,957 lines in all: introduction.tex 208, foundations.tex 434, odd-primes.tex 672, nonvanishing.tex 413, real-place.tex 420, energy.tex 288, certificate.tex 471 and conclusion.tex 51) and cites build/references.bib (28 entries); it is the text read here.

Attestation. The release README states that the repository holds manuscripts "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". Its account of the method says that "The vast majority of results were obtained with the same procedure using an unreleased internal OpenAI model"; 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 family. 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, assistance or verification. The title page names OpenAI as the sole author, and the manuscript carries no acknowledgment. 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/005.md, says the formalization proves that $G=\sum_{j\ge0}(-1)^j/(2j+1)^2$ is irrational, "the mathematical assertion in the paper's title", and links one comparator statement, lean/ComparatorChallenges/Catalan.lean. That file states, with its proof left as sorry, the theorem OAI.InternalCatalan.catalan_irrational: in words, the real number obtained as Mathlib's unconditional sum over all natural numbers $j$ of $(-1)^j/(2j+1)^2$ (an absolutely summable family, so this sum is $G$) is irrational. Its configuration Catalan.json names the solution module OAI.NumberTheory.Catalan.Main, no definitions, and the permitted axioms propext, Quot.sound and Classical.choice. The comparator statement is Theorem 1.1 alone; the manuscript's intermediate propositions and its hyperbolic-volume consequences are not part of it. The release's catalog lean/formalization.yaml at this revision lists no entry for this comparator under its status.main_results, and 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 lists this manuscript as the only member of its family, in both its manuscript map and its overview, and holds no earlier dated version of it; the release history records no revision of it. There are no companions.

Read status: claims checked for Theorem 1.1, read clause by clause in the release's TeX source (build/sections/introduction.tex, the definition of $G$ at lines 3--8 and label thm:main at lines 16--18), together with the definition of the determinant $\Delta_N$ (foundations.tex lines 9--89), the statements of Proposition 2.1 (prop:rationality, foundations.tex lines 243--247), Proposition 2.2 (prop:two-adic, lines 271--280), Proposition 3.4 (prop:finite-lower, odd-primes.tex lines 516--526), Proposition 4.1 (prop:nonvanishing, nonvanishing.tex lines 7--14) and Proposition 7.1 (prop:real-upper, certificate.tex lines 7--16), the closing proof (conclusion.tex lines 3--16) and the paragraph on hyperbolic volumes (conclusion.tex lines 18--51); the proofs were read for their structure only and no step was checked; no certificate was recomputed; nothing here is independently reviewed.

Contents

Result numbers and page numbers below are the PDF's; results are numbered within sections.

• Title page and Section 1, Introduction (pp. 1--3). Defines $G=\beta(2)=\sum_{j\ge0}(-1)^j/(2j+1)^2$, the value at $2$ of the Dirichlet beta function, equivalently $L(2,\chi_{-4})$, and states Theorem 1.1 (p. 1). Section 1.1 reviews the family results that guarantee irrational even beta values without naming one (Rivoal--Zudilin, Theorems 1 and 2; Zudilin 2019, Theorem 1; Lai--Zhou, Theorem 6.1; Fischler 2020), the criterion that nonzero integer forms $A_mG-B_m$ must tend to zero, Apéry's and Beukers's proofs for $\zeta(2)$ and $\zeta(3)$, Zudilin's Apéry-like recurrences for $G$ whose integer forms do not tend to zero, the denominator bounds of Rivoal and of Krattenthaler--Rivoal, the identity of Krattenthaler--Zudilin, Nesterenko's approximations, the Viola--Marcovecchio exponent $0.6293\ldots$ and an Eskandari preprint (September 2026) with $0<|G-p_m/q_m|\le q_m^{-0.62}$, which the text calls bounds for particular constructions, not irrationality measures. It compares Calegari, Dimitrov and Tang's linear independence of $1,\pi^2,L(2,\chi_{-3})$ and Calegari's $2$-adic analogue of $G$, and records that Sun (arXiv, September 2026) has announced a proof of the same statement, from which nothing is used. Section 1.2 outlines the method: determinants $\Delta_N$ of size $n=48N$ whose entries are combinations of moments in $\mathbb Q+\mathbb QG+\mathbb Q\zeta(2)$, rows with Taylor contact that cancels $\zeta(2)$, a product-formula lower bound against an integral upper bound, with Zudilin's determinantal criterion (2017) as precedent; it defines the normalized logarithm $\mathcal L_N=\log|\Delta_N|/(48N)^2-\tfrac12\log2$ and quotes the two constants $-2.29084$ and $-2.290939875$. It says the argument is qualitative and gives no irrationality measure. Section 1.3 states that the only prime-distribution input is the prime number theorem.
• Section 2, Polynomial rows and mixed moments (pp. 3--10). Fixes $n=48N$, $a=11N$, $b=7N$, $q=g=4N$, $h=2N$, $L=59N$, $C=63N$, $H=65N$, $A=19N$, the parametrization $t=2w/(1+w^2)$, $f=\sqrt{1-t^2}$, and rows $R_r=(1-t)^ht^{C-1}w^{r-g}$ for $0\le r<n$. Averaging $R_r$ with its image under $w\mapsto1/w$ gives $P_r$, and the image minus $R_r$, times $t/(2f)$, gives $D_r$; both are integer polynomials (Chebyshev polynomials $T_d$, $U_{d-1}$ at $1/t$, $d=|r-g|$, times $(1-t)^ht^{C-1}$, with the sign of $r-g$ on $D_r$), and $tP_r/f-D_r=O(t^L)$. Defines the two moment functionals $M(i,j)$ (a double integral over $t\in(-1,1)$, $s\in(0,1)$ with weight $|t|/\sqrt{1-t^2}$ and kernel $t^is^j/(1-ts)$) and $Z(i,j)$ (over $(0,1)^2$), the raw columns $F_j(r)=M(P_r,j)-\tfrac32Z(D_r,j)$, and $\Delta_N$ as the $n\times n$ determinant against the filtered columns $s^{b+k}(1-s)^q$ (eq. (7), p. 5), an integer binomial filter of the raw columns. Section 2.1 evaluates $M$ as a rational array plus $4G$ and $\tfrac32\zeta(2)$ times central-binomial kernels, from the starting values $4G$ and $\pi^2/4$, and $Z$ as a rational array plus $\zeta(2)$ on the diagonal. Proposition 2.1 (p. 7): every raw entry lies in $\mathbb Q+\mathbb QG$, so $G\in\mathbb Q$ makes every $\Delta_N$ rational. Section 2.2, Proposition 2.2 (p. 8): under $G\in\mathbb Q$, $v_2(\Delta_N)\ge-\tfrac{505}{4608}n^2-O_G(n\log(n+2))$, and the same for every full raw minor.
• Section 3, Odd primes and the finite-place lower bound (pp. 10--19). Assuming $G\in\mathbb Q$: Lemma 3.1 (digit reductions, p. 11) gives the residues of $p^2$ times the rational moment arrays modulo $p$ for $2\sqrt H<p\le H$; Section 3.2 pairs raw columns whose sum has at most one power of $p$ in its denominator, giving a bound for $2\sqrt H<p\le H/2$; Lemma 3.2 (central layer, p. 14) and Lemma 3.3 (elimination over the local integers, p. 15) give $v_p(\Delta_N)\ge-\min(2n,n+R,2R+S)$ for $H/2<p\le H$ with explicit counts $R,S$. Proposition 3.4 (finite-place bound, p. 17): along any sequence $N\to\infty$ with $\Delta_N\ne0$, the lower limit of $\log|\Delta_N|/n^2-\tfrac12\log2$ is at least $-\tfrac{8609}{4608}-\tfrac{2809}{4608}\log2$, which exceeds $-2.29084$. The proof tabulates a piecewise-linear loss $d(p/N)$ on $[0,65]$ with integral $8609/2$, shows the primes below $2\sqrt H$ cost $o(n^2)$, sums with the prime number theorem (cited to Zagier's account of Newman's proof), adds the prime $2$ and applies the product formula.
• Section 4, Nonvanishing along the prime sequence (pp. 19--25). Proposition 4.1 (p. 19): under $G\in\mathbb Q$, for every sufficiently large prime $p$, $v_p(\Delta_p)=-96p$, so $\Delta_p\ne0$. The first three subsections take $N=p$, use two bases of a space of palindromic polynomials over $\mathbb F_p$ and Frobenius to express each scaled row through two rows of a fixed rational $49\times48$ matrix $\mathcal B$ built at $N=1$, and factor the reduced determinant as a power of $2$ times $\det\mathcal B_0$, $\det(\mathcal B_0+\mathcal B_1)^{(p-1)/2}$ and $\det(\mathcal B_0-\mathcal B_1)^{(p-1)/2}$. Section 4.4 certifies the three fixed determinants nonzero by Gaussian elimination modulo $101$, with rational recipes for the entries and the full pivot list (Table 1, p. 24).
• Section 5, The real-place determinant estimates (pp. 25--30). Without any rationality hypothesis: Andréief's identity, applied twice (cited to Forrester), and Cauchy's double alternant (cited to Krattenthaler) write $\Delta_N$ as a multiple integral whose row determinant mixes evaluations at $x_i$ and $1/x_i$, $x=t/(1+\sqrt{1-t^2})$. Lemma 5.1 (p. 26) builds, when $\sum_i(1-x_i^2)/(1+x_i^2)\le40N-1$, an interpolating function bounded by $10e^{12}n$ on the closed unit disk; Proposition 5.2 (interpolation estimate, p. 28) bounds the row determinant in that case by a compression of the multiplication operator to an $n$-dimensional subspace of $H^2$, avoiding any inverse Vandermonde bound; Proposition 5.3 (Hadamard estimate, p. 29) covers every configuration. Section 5.4 reduces $\log|\Delta_N|/n^2-\tfrac12\log2$ to the larger of two suprema of explicit integrand majorants, up to $O(\log n/n)$.
• Section 6, A uniform energy bound (pp. 30--34). Proposition 6.1 (p. 31) bounds the limit superior of each supremum by $-\tfrac{11}{16}\log2$ plus quadratic norms of two freely chosen trial sequences and two one-variable suprema, through the Chebyshev expansion of the logarithmic kernel (cited to Garoufalidis--Popescu, Lemma 3.1), damping, completed squares, and the inequality $-a^2\le b^2-2ab$.
• Section 7, An exact certificate for the two barriers (pp. 34--42). Proposition 7.1 (p. 34): without hypothesis, the upper limit of $\log|\Delta_N|/n^2-\tfrac12\log2$ is at most $-2.290939875<-2.2909$. The section gives explicit rational trial sequences, counts the stationary points of the two one-variable functions by Descartes' rule of signs (proved in the text), isolates them in brackets of width $2\cdot10^{-10}$, evaluates logarithms and arctangents by rational series with explicit remainders, lists fifty certified value bounds (Table 2, pp. 39--40), and passes to global suprema by a derivative bound.
• Section 8, Conclusion (p. 42). The proof of Theorem 1.1 combines Propositions 2.1, 4.1, 3.4 and 7.1 along large primes and compares the rounded constants $-2.29084>-2.2909$. The manuscript states no margin; computed for this card, the exact lower constant of Proposition 3.4, about $-2.29081$, exceeds the upper constant $-2.290939875$ of Proposition 7.1 by about $1.3\cdot10^{-4}$. A closing paragraph deduces, from Agol's identification of $4G$ as the least volume of an orientable complete finite-volume hyperbolic 3-manifold with exactly two cusps, that this minimum is irrational, and, from Voight's volume formula, that every orientable arithmetic hyperbolic 3-orbifold defined over $\mathbb Q(i)$ has volume a positive rational multiple of $G$, hence irrational.
• References (pp. 42--44), 28 items, including the Sun and Eskandari preprints named above.

The manuscript presents its finite checks (the pivots modulo $101$ in Section 4.4; the trial sequences, root brackets and value bounds in Section 7) as exact rational computations specified in the text, and says that "supplementary programs reproduce them" (p. 3). The release folder at this revision holds only the README, the PDF and the TeX build directory: no programs and no verification/ folder. Nothing was recomputed here. The manuscript flags no input as conditional.

Bears on

The manuscript names no Erdős problem, and no problem page is identified as using it. The problem pages were searched for Catalan's constant, the Dirichlet beta function, $L(2,\chi_{-4})$ and hyperbolic volumes, and the statements of the irrationality problems were read; none asks about $G$ or these values, and none records an argument that takes the irrationality of $G$ or this determinant method as an input.