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Statement

Catalan's constant is $G=\beta(2)=\sum_{j\ge0}(-1)^j/(2j+1)^2$, where $\beta(s)=\sum_{j\ge0}(-1)^j(2j+1)^{-s}$ for $s>0$ is the Dirichlet beta function; equivalently $G=L(2,\chi_{-4})$ for the odd quadratic character of conductor $4$ (p. 1).

Theorem 1.1. $G$ is irrational.

The statement is qualitative. The manuscript says its argument gives no irrationality measure, which "would require additional control of rational approximations" (p. 3). Its closing paragraph (p. 42) draws two consequences, not separately labeled: by Agol's theorem, $4G$ is the least volume of an orientable complete finite-volume hyperbolic 3-manifold with exactly two cusps (curvature $-1$), attained by the Whitehead-link and $(-2,3,8)$-pretzel-link complements, so that minimum and both link volumes are irrational; and, by Voight's volume formula with $\zeta_{\mathbb Q(i)}(2)=\zeta(2)G$, every orientable arithmetic hyperbolic 3-orbifold defined over $\mathbb Q(i)$ has volume a positive rational multiple of $G$, hence irrational, for example $\operatorname{vol}(\mathrm{PSL}_2(\mathbb Z[i])\backslash\mathbb H^3)=G/3$.

Formalization. The release's Lean page for the family describes a formalization of this statement, with the comparator theorem OAI.InternalCatalan.catalan_irrational. In words, it says that the real number given by Mathlib's unconditional sum over all natural numbers $j$ of $(-1)^j/(2j+1)^2$ is irrational; the terms are absolutely summable, so that sum is $G$. The hyperbolic-volume consequences are not part of it. It was read statically and not built or audited here, and the catalog lists no entry for it at this revision (see the card).

Source. OpenAI, Catalan's constant is irrational, release folder preprints/Catalans-constant-is-irrational-September-24-2026; TeX file build/sections/introduction.tex, label thm:main (lines 16--18), PDF p. 1; the proof closes in Section 8 (build/sections/conclusion.tex lines 3--16, PDF p. 42). Provenance and attestations are on the card.

Read depth. Claims checked: the statement and the definition of $G$ (introduction.tex lines 3--8 and 16--18), the definition of $\Delta_N$, the statements of Propositions 2.1, 3.4, 4.1 and 7.1, which the closing proof combines, and of Proposition 2.2, which feeds the constant of Proposition 3.4, and the closing proof itself were read clause by clause in the TeX source. The proof (pp. 3--42) was read for its structure only, as summarized below; no step was checked, and none of the finite certificates (the pivots modulo $101$, the root brackets, the value bounds) was recomputed. The hyperbolic-volume consequences were read as stated and not checked against Agol or Voight. Nothing here is independently reviewed.

Proof pointer

The proof is by contradiction. Suppose $G\in\mathbb Q$. For each $N$ the manuscript builds a determinant $\Delta_N$ of size $n=48N$ (Section 2). Its rows come from integer polynomials $P_r$, $D_r$ (Chebyshev polynomials at $1/t$ times $(1-t)^{2N}t^{63N-1}$), built from one rational expression and its image under the involution $w\mapsto1/w$ of the parametrization $t=2w/(1+w^2)$. Its columns are filtered by $s^{7N+k}(1-s)^{4N}$. Each entry combines two double-integral moments with kernel $1/(1-ts)$, one weighted by $|t|/\sqrt{1-t^2}$ on $(-1,1)$, the other on $(0,1)$. Each moment is an explicit element of $\mathbb Q+\mathbb QG+\mathbb Q\zeta(2)$. With $f=\sqrt{1-t^2}$, the rows satisfy the Taylor contact $tP_r/f-D_r=O(t^{59N})$, so the $\zeta(2)$ part cancels in every entry used (Proposition 2.1). Hence every $\Delta_N$ is rational.

The lower bound is arithmetic. For a nonzero rational $\Delta_N$, the product formula gives $\log|\Delta_N|=\sum_pv_p(\Delta_N)\log p$, so bounds on denominators give a lower bound. Proposition 2.2 handles $p=2$. Section 3 handles odd primes up to $65N$: digit reductions modulo $p$ of the rational moment arrays (Lemma 3.1), pairing of columns, a central-layer reduction (Lemma 3.2) and integral column elimination (Lemma 3.3) track denominators $p^2$ and $p$ separately. Summing these losses with the prime number theorem gives Proposition 3.4: the lower limit of $\log|\Delta_N|/n^2-\tfrac12\log2$ along nonzero determinants exceeds $-2.29084$. Nonzero determinants exist along the primes: for $N=p$ a large prime, Proposition 4.1 uses Frobenius and two palindromic bases over $\mathbb F_p$ to reduce the scaled matrix to a block form. Its determinant is a power of $2$ times powers of the determinants of three fixed rational $48\times48$ matrices. Gaussian elimination modulo $101$ shows these three determinants are nonzero, so $v_p(\Delta_p)=-96p$ exactly.

The upper bound is analytic and needs no hypothesis on $G$. Andréief's identity turns $\Delta_N$ into a multiple integral. Cauchy's double alternant and the change of variable $x=t/(1+\sqrt{1-t^2})$ leave a row determinant that mixes evaluations at $x_i$ and $1/x_i$. When $\sum_i(1-x_i^2)/(1+x_i^2)\le40N-1$, a bounded interpolating function (Lemma 5.1) and a Hilbert-space compression bound that determinant (Proposition 5.2); otherwise Hadamard's inequality does (Proposition 5.3). Proposition 6.1 expands the logarithmic interactions in Chebyshev polynomials and bounds them by quadratic terms in two trial sequences plus two one-variable suprema. Section 7 chooses explicit rational trial sequences and certifies those suprema exactly: Descartes root counts, root brackets of width $2\cdot10^{-10}$, rational interval values and a derivative bound. This gives Proposition 7.1: the upper limit is at most $-2.290939875$.

Along large primes the two limits contradict each other, since $-2.29084>-2.2909$. The manuscript compares only these rounded values and states no margin. Computed for this page, not taken from the manuscript: the exact lower constant of Proposition 3.4, $-\tfrac{8609}{4608}-\tfrac{2809}{4608}\log2\approx-2.29081$, exceeds the upper constant $-2.290939875$ of Proposition 7.1 by about $1.3\cdot10^{-4}$. A margin this small leaves little room for error in the constants of Propositions 2.2, 3.4 and 7.1.

Dependencies

External inputs, taken at statement level; none was checked here. The prime number theorem $\theta(y)\sim y$ (cited to Zagier's account of Newman's proof); Andréief's integration identity (cited to Forrester); Cauchy's double alternant (cited to Krattenthaler); the Chebyshev expansion of the logarithmic kernel, Haagerup's identity as in Garoufalidis and Popescu (Lemma 3.1); the Laurent formulas for Chebyshev polynomials (DLMF, equations 18.5.1--18.5.2). Descartes' rule of signs is cited and its needed form proved in the text, and Sarason's interpolation viewpoint is cited as context with the finite-dimensional facts proved directly. The hyperbolic consequences rest on Agol (2010, Theorem 3.6) and Voight (2021, Definition 38.3.4, Theorem 39.1.13 and Example 39.1.16). Sun's preprint announcing the same theorem is cited and explicitly not used.

Bears on

The manuscript names no Erdős problem, and no problem page is identified as using this result.