{
  "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 $G$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22339",
  "author_total": 1,
  "authors": [
    {
      "name": "Dane Wachs"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aey1fq9mkhpke.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22339",
  "fulltext_url": null,
  "id": "aey1fq9mkhpke",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22339",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GM"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7214038476",
  "pdf_url": "https://arxiv.org/pdf/2609.22339",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aey1fq9mkhpke.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22339",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22339"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Advanced Mathematical Identities",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A note on a recent claimed proof of the irrationality of Catalan's constant",
  "title_text": "A note on a recent claimed proof of the irrationality of Catalan's constant",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/aey1fq9mkhpke",
  "venue": "arXiv",
  "versions": []
}