Catalan's constant G = 0.9159655941…, defined by the series 1/1² − 1/3² + 1/5² − 1/7² + …, appears in the volume formulas of hyperbolic octahedra, domino tiling counts, and models of galactic mass distributions. Unlike π and e, whose irrationality has long been settled, G's status has been open since Catalan's 1865 memoir — 161 years.
On September 3, Sun Zhiwei (Zhi-Wei Sun) of Nanjing University posted a 20-page paper on arXiv whose title is simply the conclusion: "Catalan's constant is irrational". Sun is a prolific combinatorial number theorist with 235 papers on arXiv. The abstract's entire method description is half a sentence: "via using suitable weights".
What the Proof Looks Like
Reading the paper's HTML in full, its skeleton is a five-stage pipeline:
1. Define "weighted tail terms" for G, dividing T_m by (2m+1); 2. Use weighted finite differences to build a full-rank residual matrix; 3. Perform a "Pascal-Cauchy factorization" via the Cauchy–Binet formula; 4. Do p-adic prime-power bookkeeping, splitting the estimate into three segments with S taken at ⌊B/20⌋; 5. Assuming G = a/q, construct a nonzero integer N_B and prove 0 < |N_B| < 1 — an integer cannot fit between 0 and 1, a contradiction. The closing margin δ₀ exceeds 0.00966.
This shape is classical: construct rational approximations to a constant that are "too good to be true," push until the denominators cannot bear it, and extract a contradiction. Apéry's 1978 proof that ζ(3) is irrational used the same family of ideas.
The Protagonist Hiding in the Acknowledgments
The paper's most striking part is the acknowledgments. Quote: "The author's many rounds of conversations with AI provide the basis of this paper." The required numerical data was AI-generated, and the closing line reads: "The whole proof has passed the verification of Chatgpt 5.6 Solar."
There is no record of human review anywhere. The arXiv category is also telling: math.GM, the catch-all "General Mathematics" — major number theory results typically land in math.NT. Third-party sites have mislabeled it as math.NT.
The community did not wait for referees. Within 24 hours, the single substantial comment on a 3-point Hacker News thread identified a suspected fatal flaw: recurrence (1.4) yields 1/(2i+1)², but later step (2.12) uses (2X+3)² — off by one slot. Under the former reading, K(i) does not vanish to zero, the full-rank argument collapses, and everything downstream fails. r/math saw "major error" discussion the same week. On X, someone (relayed via forums, not directly verifiable) claimed the method could equally "prove" that 1 is irrational — a fatal defect if true.
These are accusations. Accusations need adjudication, and adjudication needs people.
The 1978 Control Group
Invoking Apéry as a precedent of a mocked proof later vindicated is a misapplication. At Apéry's lecture the audience did jeer, but within two months Cohen, Lenstra, and van der Poorten had checked the proof line by line, and on August 18 Cohen lectured on the full details. Human verification: two months.
This time? The verifier was a chatbot, the fault-finders are anonymous netizens. Both sides' reliability is discounted. Mathematics has no jury protocol ready for AI.
The genuine prior progress should also be stated: Rivoal and Zudilin proved in 2003 that at least one of β(2) through β(12) is irrational — a step toward G, but not G itself. In 2024, Calegari–Dimitrov–Tang proved via 218 pages that the "cousin" constant L(2, χ₋₃) is irrational — the Catalan constant of the Q(√−3) world, still one number field away from G itself.
Wikipedia still says today that G's irrationality is unknown. Until human experts check the proof line by line, the problem remains open. Who will take on those two months?
This story's real protagonist is not the constant — it is "verification" itself, whose institution is changing. This site previously covered Anthropic pushing forward the Riemann zero lower bound (178634291). News of AI colliding with classical number theory will multiply; this one puts the question of "how to verify after the collision" on the table for the first time.