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Claim. Write $F(t)=\sum_{n\ge1}(t^n-1)^{-1}$. For coprime integers $a>b\ge1$ with $\log b/\log a<\theta^*$, $F(a/b)$ is irrational, where $\theta^*=0.4056830213\ldots$ is an explicit constant defined from thirteen intervals of a periodic indicator function; in particular $F((31/4)^r)$ is irrational for every integer $r\ge1$. This is Theorem 2.1 of W. Cook, Irrationality of $F(31/4)$ and the Exact Normalized Hankel Order (manuscript, September 2026, 14 pp., the preprint link). The argument takes the linear forms of W. Zudilin, Heine's basic transform and a permutation group for $q$-harmonic series, Acta Arith. 111 (2004), 153--164, uses the polynomial conclusion of its Lemma 7 before the integer specialization, and cancels the cyclotomic factors of the coefficients before clearing the denominator at $a/b$; the manuscript attaches no priority claim to the extension, noting that Zudilin's 2016 paper on generalized $q$-logarithms already remarks that rational bases can be treated under an unspecified logarithmic restriction. The same forms bound the irrationality exponent of $F((31/4)^r)$ uniformly in $r$ (Corollary 2.3), and Section 3 determines the exact order and leading coefficient of Zudilin's normalized Hankel determinants, which does not enlarge the region. The manuscript states that AI agents did most of the research and drafting, and that Lean checks finite subclaims, not the irrationality theorem; the forum comment announcing it (11 September 2026, the discussion link) says that the AI system Astra wrote it up.

Covers. Every $t=a/b>1$ in lowest terms with $\log b/\log a<\theta^*$ of Problem 1049. This adds the band $\frac12-\frac1{\pi^2}\le\log b/\log a<\theta^*$ to the region of Bundschuh and Väänänen; $31/4$ lies in it, since $\log4/\log31=0.40369\ldots$. It does not cover $3/2$, which the manuscript says lies outside its region.

Acceptance. None. The thread has no reply to the comment, and no review, publication or formal proof of the irrationality theorem is known.

Depends on. Nothing in this wiki.