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Irrationality of Catalan's constant

Catalan's constant is G=β(2)=∑j≥0(−1)j/(2j+1)2=L(2,χ−4)G=\beta(2)=\sum_{j\ge0}(-1)^j/(2j+1)^2=L(2,\chi_{-4}), the simplest even value of the Dirichlet beta function. Odd beta values are rational multiples of powers of π\pi, but nothing was known about the arithmetic of GG itself. Rivoal and Zudilin (2003) proved that infinitely many even beta values are irrational and that at least one of β(2),β(4),…,β(14)\beta(2),\beta(4),\ldots,\beta(14) is, a list later cut to β(2),…,β(10)\beta(2),\ldots,\beta(10); such family results never single out GG. Apery-like recurrences and continued fractions for GG (Zudilin 2003) converge fast, but the integer linear forms they give do not tend to zero once denominators are cleared. Is Catalan's constant irrational?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Diophantine approximation; irrationality of L-values
Posed by
Long-standing open problem with no single poser; the manuscript frames it through Rivoal and Zudilin (Math. Ann. 2003) and Zudilin (2003)
Year posed
—
Years open
—
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
60 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: GG is irrational. Under the hypothesis G∈QG\in\mathbb Q the paper builds mixed determinants ΔN\Delta_N of size 48N48N from moments of two kernels, whose ζ(2)\zeta(2) terms cancel by Taylor contact of Chebyshev rows, so all ΔN\Delta_N are rational; a finite certificate gives Δp≠0\Delta_p\ne0 for large primes pp, prime-by-prime denominator bounds give lim inf⁡Lp>−2.29084\liminf\mathcal L_p>-2.29084, and a real-place integral estimate gives lim sup⁡LN<−2.2909\limsup\mathcal L_N<-2.2909, a contradiction. Corollaries: 4G4G, the minimal volume of a two-cusped orientable hyperbolic 3-manifold, and every arithmetic hyperbolic 3-orbifold volume over Q(i)\mathbb Q(i) are irrational. Not shown: an irrationality measure, transcendence, or anything about β(4),β(6),…\beta(4),\beta(6),\ldots. An earlier preprint by Zhi-Wei Sun (arXiv:2609.04176, 3 September 2026) also claims irrationality; the manuscript says it uses nothing from it, so the first claimed proof may not be this one.

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, using on average about three hours of ChatGPT Pro thinking compute per result, with outputs grouped into families and manuscripts. The README's two exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region write-up, which was human-edited for readability) do not concern this family. The manuscript is credited to OpenAI alone, names no human author and has no acknowledgements. A Lean formalization of the irrationality statement is in the release's lean/ library (lean/docs/005.md). The release does not say how much human review happened before publication.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the TeX source (Catalan's constant is irrational) against the posed question, and the Lean statement. The result is not in the release's formalization catalogue (formalization.yaml), but lean/docs/005.md links ComparatorChallenges/Catalan.json, whose solution_module OAI.NumberTheory.Catalan.Main exists at the pinned commit; theorem OAI.InternalCatalan.catalan_irrational, permitted axioms propext, Quot.sound and Classical.choice. The comparator statement, read here, is Irrational (sum over j in N of (-1)^j / (2j+1)^2) over the reals, which is exactly the headline. Not rebuilt here; the comparator run and axiom check were not reproduced. The paper's argument rests on exact finite certificates (a nonvanishing certificate and rational trial coefficients for the real-place bound); the text says supplementary programs reproduce them, and I did not find such programs in the manuscript's folder. It proves qualitative irrationality only.

Sources

Changelog1 change

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