A note on a recent claimed proof of the irrationality of Catalan's constant

Abstract

The preprint arXiv:2609.04176 (Z.-W. Sun, 3 September 2026) claims a proof that Catalan's constant $G$ is irrational: it constructs for each $B$ a number $\hat q_B$ from weighted tails of the series and asserts that, if $G=a/q$, the integer $N_B=q^S H_B^{\min}\hat q_B$ satisfies $0<|N_B|<1$ for large $B$ (its Theorem 9.1). We report four exact computations. First, the printed proof of the rank theorem (Theorem 2.1) carries $T_i$ where its own recurrence requires $T_{i+1}$; the slip is repairable, and the statement is verified exactly at 44 parameter pairs with $S\le 4$, $B\le 30$. Second, with the paper's definitions and the worst-case denominator, attained at an explicit rational $a/q$, a lower bound for the quantity the derivation of Theorem 9.1 controls equals $+1.49$ to $+1.79\,B^2$ at eighteen indices $20\le B\le 119$, where the derivation claims at most $-0.00966\,B^2+o(B^2)$. Third, the paper's own denominator bound (Corollary 5.2) holds in every tested instance and is nearly sharp, so the discrepancy lies in the asymptotic ledger of Sections 6-9. Fourth, an exact identity locates it: the prime 2 contributes $v_2(F_D)\log 2=(2\log 2+o(1))B^2$ to the quantity, because the positive part at 2 vanishes while $|\hat q_B|$ carries the full power of 2 in $F_D$; no display in the ledger carries a term of this order, and Remark 9.3 assigns it to the derivation of the odd-prime constant $c_{\rm odd}=0.006$ without showing how. At the tested parameters the odd-prime layers agree with the paper's constants to within about $0.1\,B^2$, and the remainder of the discrepancy is the size of the residual minor at those indices. These computations test the paper's bounding method, which is uniform in $(a,q)$; they show that the published derivation does not justify the claimed estimate, so the proof of the main theorem is incomplete. Nothing is concluded about the irrationality of $G$.