{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-06-openai-catalan-constant-irrational/","md":"https://postcutoff.com/e/2026-10-06-openai-catalan-constant-irrational/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-06-openai-catalan-constant-irrational","date":"2026-10-06","date_precision":"day","short_title":"OpenAI: Catalan's constant is irrational (Lean)","deck":null,"takeaway":"Family 005 of OpenAI's 6 Oct 2026 release is a 44-page manuscript, dated 24 Sep, proving that Catalan's constant G = 1 − 1/9 + 1/25 − 1/49 + … is irrational, one of the classical open questions about special values. Its Lean statement is one line on top of Mathlib and passed the community comparator runs. Frank Calegari sees 'a bunch of new ideas, or at least new optimizations', and outside authors have already turned the method into explicit irrationality measures.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"OpenAI: Catalan's constant is irrational (24 Sep)","url":"https://github.com/openai/math/blob/main/preprints/Catalans-constant-is-irrational-September-24-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":2,"title":"Lean scope note for family 005","url":"https://github.com/openai/math/blob/main/lean/docs/005.md","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"Lean comparator challenge: Catalan.lean","url":"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/Catalan.lean","type":"code","group":"primary","domain":"github.com"},{"n":4,"title":"arXiv 2609.04176: Zhi-Wei Sun, Catalan's constant is irrational (3 Sep 2026)","url":"https://arxiv.org/abs/2609.04176","type":"paper","group":"primary","domain":"arxiv.org"},{"n":5,"title":"Hexagon 2610.00187: Sean Litsey, An Irrationality Measure for Catalan's Constant","url":"https://hexagonmath.org/abs/2610.00187v5","type":"paper","group":"primary","domain":"hexagonmath.org"},{"n":6,"title":"oai-math-revised: comparator run on all challenges (LEAN_CHECKS.md)","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/docs/LEAN_CHECKS.md","type":"code","group":"primary","domain":"github.com"},{"n":7,"title":"Persiflage (Frank Calegari): A Brief Digest of Some New OpenAI Results (8 Oct)","url":"https://galoisrepresentations.org/2026/10/08/a-brief-digest-of-some-new-openai-results/","type":"discussion","group":"reactions","domain":"galoisrepresentations.org"},{"n":8,"title":"oai-math-revised: community notes on the Catalan manuscript (follow-ups, Lean)","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Catalans-constant-is-irrational-September-24-2026/AGENT_NOTES.md","type":"discussion","group":"reactions","domain":"github.com"}],"official":6,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"],"title":"OpenAI's release proves Catalan's constant is irrational, with a Lean proof of the exact statement","summary":"\"Catalan's constant is irrational\" (OpenAI, 24 Sep 2026, released 6 Oct, family 005) proves Theorem 1.1, \"Catalan's constant is irrational\", for G = Σ (−1)^j/(2j+1)² = β(2) = L(2, χ₋₄). Earlier results only showed that some value among β(2), β(4), …, β(10) is irrational. The proof builds large determinants from moments that are rational combinations of 1, G and ζ(2), cancels the ζ(2) part, and plays denominator bounds against analytic bounds, with nonvanishing certified by a finite computation. The main theorem is formalized in Lean and the comparator check passed; no expert has yet published a full human reading. Follow-ups derive irrationality measures such as μ(G) ≤ 740.","key_facts":["Paper: 'Catalan's constant is irrational' (OpenAI, 44 pp., dated 24 Sep 2026). Abstract (verbatim): 'We prove that Catalan’s constant G = Σ_{j≥0} (−1)^j/(2j + 1)² is irrational.'","Prior state (paper, §1.1): Rivoal and Zudilin proved that infinitely many even beta values are irrational and that one of β(2), …, β(14) is; Zudilin reduced this to β(12) and Lai–Zhou to β(10). 'Such family results guarantee irrational members without identifying the particular value β(2).' Zudilin's determinantal criterion explained why known approximations fail for G","Method (paper, §1.2): determinants of size 48N whose entries are moments of two kernels, 'Each moment is a rational linear combination of 1, G and ζ(2)'; high Taylor contact 'cancels the ζ(2) term in every entry', so G ∈ ℚ would make all determinants rational; denominator bounds then contradict an integral bound. Nonvanishing comes from a Frobenius reduction to three fixed rational matrices and 'An exact finite certificate'","Earlier claim: Zhi-Wei Sun posted 'Catalan's constant is irrational' on arXiv on 3 Sep 2026 (2609.04176). The OpenAI paper (verbatim): 'Sun [22] has also announced a proof of the same qualitative irrationality statement. No result from that preprint is used here.'","Lean: comparator challenge Catalan.lean states `theorem catalan_irrational : Irrational (∑' j : ℕ, (-1 : ℝ) ^ j / ((2 * j + 1 : ℕ) : ℝ) ^ 2)` using only Mathlib, standard axioms only; solution module OAI.NumberTheory.Catalan.Main. The series converges absolutely, so Mathlib's tsum is the usual sum G (our reading of the statement). The oai-math-revised comparator run (25–26 Oct) accepted it","Frank Calegari (Persiflage, 8 Oct, verbatim): 'there are clearly a bunch of new ideas, or at least new optimizations, that we were not previously aware of'; 'certain optimizations are taking place without being motivated at all'; 'it’s amusing to see this paper cite two crank papers, both of which give nonsense proofs that Catalan’s constant is rational'. He is co-organising an irrationality conference at SLMath in January where these methods will be studied, and notes no analogous OpenAI proof for L(2, χ₋₃) yet","Follow-ups: Sean Litsey (Hexagon 2610.00187, v5): 'Building on OpenAI’s proof that Catalan’s constant … is irrational, we prove |G − a/b| > b^−740', i.e. μ(G) ≤ 740, with a Lean formalization; its provenance says the work 'was produced entirely by Claude Opus 5.5 and GPT-6.1 Sol'. Community notes list further Zenodo and Hexagon preprints with weaker measures and extensions to other beta values (Jha, Carreira, Krisman)"],"key_numbers":[],"tags":["math","number-theory","irrationality","catalan-constant","dirichlet-beta","diophantine-approximation","lean","openai","openai-math-release"],"science":{"field":"mathematics","subfield":"number theory (irrationality of special values)","problem":"Is Catalan's constant G = β(2) irrational?","result":"Proved irrational (OpenAI manuscript with a Lean proof of the exact statement); follow-ups give explicit irrationality measures","open_since":"","ai_system":["OpenAI internal model (openai/math release)"],"human_role":"Autonomous AI output released by OpenAI under its fixed procedure (README)","verification":{"key":"lean","label":"Lean-verified","text":"Lean formalization of the main theorem (comparator accepted); no published human verification yet"},"status":"pending","shock":"Catalan's constant was a standard example of a natural constant whose irrationality nobody could prove, alongside Euler's γ and ζ(5)."},"body_md":"## What happened\n\nCatalan's constant, G ≈ 0.9159655, is the alternating sum of the reciprocals of the odd squares. Unlike ζ(2) = π²/6 and ζ(3)\n(Apéry, 1978), it had resisted every attempt: known approximations to G converge fast, but their\ndenominators grow too fast for the classical irrationality criterion. The best results were statements that at least one value in a\nshort list of Dirichlet beta values is irrational.\n\nOpenAI's manuscript proves G irrational directly. It assumes G = a/b, builds determinants of growing size from moments that mix 1, G\nand ζ(2), arranges the construction so that the ζ(2) parts cancel, and shows that the determinants would have to be nonzero rational\nnumbers that are too small for their denominators. The nonvanishing step reduces, along a sequence of primes, to three fixed matrices\nwhose determinant is checked by an exact computation. The main theorem is formalized in Lean against a one-line statement that uses\nonly Mathlib's definitions.\n\n## Why it matters\n\nThe irrationality of Catalan's constant is one of the best-known open questions about specific constants, listed with Euler's γ on\nstandard lists of unsolved problems. A Lean proof of a faithful one-line statement makes this one of the more trustworthy results in\nthe release, although no specialist has yet published a complete human reading of the argument.\n\n## Reactions and follow-ups\n\nFrank Calegari wrote that, as with the AI proof of the irrationality of ζ(5) in September, the paper contains new ideas or at least\nnew optimizations, often unmotivated, and noted that it cites two crank papers claiming G is rational. Zhi-Wei Sun had posted a\ndifferent proof claim on arXiv a month earlier; the OpenAI paper says it uses nothing from it. Outside authors, largely with AI\nassistance, have since pushed the method to explicit irrationality measures (μ(G) ≤ 740 with a Lean development) and probed\nextensions to β(4) and other values.","disputed":[{"label":"Verification","text":"Lean formalization of the main theorem (comparator accepted); no published human verification yet","sources":[]}],"related":[{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":79,"official":13,"filed":"2026-10-07","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-irrationality-exponent-pi-openai","url":"https://postcutoff.com/e/2026-10-06-irrationality-exponent-pi-openai/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI model claims the irrationality exponent of π is exactly 2 (Lean-formalized), which implies the Flint–Hills series converges","deck":null,"takeaway":"Since Mahler's 1953 bound of 42, mathematicians had lowered the irrationality exponent of π only to about 7.1. OpenAI's release claims the conjectured value 2 in a 23-page paper with a Lean proof. Six days on, no expert has publicly assessed it; an outsider has already adapted the Lean proof to log 2.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":9,"official":8,"filed":"2026-10-12","updated":"2026-10-13","orgs":["OpenAI"]},{"id":"2026-09-17-zeta-5-irrational-fauzan-lean-verified","url":"https://postcutoff.com/e/2026-09-17-zeta-5-irrational-fauzan-lean-verified/","date":"2026-09-17","date_precision":"day","short_title":"ζ(5) proved irrational","deck":"Aabir Fauzan's Zenodo preprint, the first such result since Apéry's ζ(3) in 1978, is formally verified in Lean within a week, one formalization written by Claude","takeaway":"It is the most famous number-theory result of the AI-assisted 2026 wave: a problem experts had worked on for about 48 years.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":10,"official":5,"filed":"2026-10-05","updated":"2026-10-05","orgs":["Aalto University","Google DeepMind","Anthropic"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"filed","text":"Created (families 001–125; manuscript introduction, Lean statement, comparator results, Persiflage digest, Sun's arXiv abstract and the Litsey Hexagon page read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-29","run":null,"sources_read":"Families 001–125; manuscript introduction, Lean statement, comparator results, Persiflage digest, Sun's arXiv abstract and the Litsey Hexagon page read","updated":"2026-10-29","human_review":null,"version":{"date":"2026-10-29"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":159,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":98,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":189,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":128,"in_training_data":false}],"short_url":"https://postcutoff.com/s/openai-catalan-constant"}