Irrationality measure of π lowered to 7.101862832357, the first improvement on Zeilberger–Zudilin (2019), with GPT-5.6 Sol and GPT-6 Astra; four weeks later OpenAI claimed the exact value 2
Event confirmedAwaiting review
The takeaway
The best known bound on how well π can be approximated by fractions moved for the first time since 2019, in a preprint whose author says OpenAI models helped with parameter search, proofs and computation. Claude-assisted papers then moved the ζ(3) and ζ(2) records the same way. OpenAI’s 6 Oct release goes much further, claiming the conjectured exact value μ(π) = 2 with a Lean proof, which would make all these upper bounds obsolete.
Status
- Claim
Event confirmedAwaiting review
- Our reporting
- High confidence
- Verification
- Preprint only
- Importance
- 3 of 5
- Last verified
- 12 October 2026
Your AI and this story
- GPT-6 Astra133 days after its cutoff
- Claude Opus 5.572 days after its cutoff
- Gemini 3.8 Flash163 days after its cutoff
- Grok 4.7102 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 72 days before it.
Key facts
- Result: μ(π) < 7.101862832357 (Bai, arXiv 2609.11276, v1 10 Sep, v2 11 Sep 2026); previous record μ(π) ≤ 7.103205334137… (Zeilberger & Zudilin, arXiv 1912.06345), which refined Salikhov’s 7.606308…
- Improvement: ‘more than 0.001342501780’, about 0.01890%; the point A1 = A2 = 1857/2785 is ‘a strict two-dimensional local minimizer’ of the auxiliary bound function in its chamber, ‘not a claim that the point is a global optimizer among all constructions’
- Disclosure (abstract): ‘OpenAI’s GPT-5.6 Sol and GPT-6 Astra assisted throughout the research and preparation of this article, including parameter exploration, proof development and checking, computation, and manuscript drafting and revision. Responsibility for all mathematical claims and for the final text rests with the author.’ Acknowledgments thank Bin Dong and Haocheng Ju of Peking University ‘for their guidance’
- Follow-up 1: Niedbala Giraudin, arXiv 2609.37459 (28 Sep, v3 5 Oct): one closed-form law μ(a0,a1,b) for the Salikhov–Zeilberger–Zudilin–Bai family that reproduces all three records, with computational evidence that Bai’s point is the global optimum of the family (statements labelled PROVED / VERIFIED / CONJECTURE); ‘Technical assistance: Claude (Anthropic)’
- Follow-up 2 (ζ(3)): Niedbala Giraudin, arXiv 2609.26980 (22 Sep): μ(ζ(3)) < 5.5138800, the first improvement on Rhin–Viola’s 5.513891 (2001), by moving their parameter point ‘the mechanism … recently used by Bai for π’; the author calls the gain of 1.07·10⁻⁵ ‘by itself … of little interest’. Acknowledgements: ‘This work was carried out with the assistance of an AI system (Claude, Anthropic); every claim in it has been checked by independent computation’; ancillary C verifier (verify_zeta3.c)
- Follow-up 3 (ζ(2)): Niedbala Giraudin, arXiv 2610.02912 (2 Oct): μ(ζ(2)) < 5.0193784, below the Lean-checked 5.0495243 of Sept 2026; ‘Technical assistance: Claude (Anthropic)’ (see New record bound on the irrationality measure of ζ(2), μ ≤ 5.0495243, released as a 92k-line Lean proof written by Claude; beaten a week later by a Claude-assisted paper)
- Overtaken by a stronger claim: OpenAI’s 6 Oct 2026 catalogue, family 017 ‘The irrationality exponent of π is 2’ (preprint dated 24 Sep 2026, Lean formalization of the exponent), proves ‘the conjecture that the irrationality exponent of π is 2’ and hence convergence of the Flint–Hills series Σ 1/(n³ sin² n); not yet independently reviewed (see OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems)
- Spin-off: a pseudonymous ‘Dan Brown’ preprint (dated 7 Oct, footnote ‘Author of The Da Vinci Code’, posted on GitHub Pages) adapts the interpolation-determinant argument of the μ(π) = 2 paper to claim μ(log α) = 2 for every positive rational α ≠ 1, so μ(log 2) = 2 (previous record 3.5746, Marcovecchio), formalized in Lean 4 over Mathlib; HN thread 21 points; author identity unknown, unreviewed
Show 1 more
- Verification: unrefereed preprints; the Niedbala Giraudin papers ship standalone verification programs; no expert reaction found as of 11 Oct 2026
What happened
The irrationality measure of a number says how well it can be approximated by fractions. For π only upper bounds are known, and they come from carefully tuned integrals: Hata, then Salikhov (7.606…), then Zeilberger and Zudilin (7.103205…, posted in December 2019). Yufei Bai added two free exponents to the Zeilberger–Zudilin integral and found a better parameter point, giving μ(π) < 7.101862832357. The abstract itself carries a generative-AI disclosure naming GPT-5.6 Sol and GPT-6 Astra.
Within weeks David Niedbala Giraudin, an independent researcher in France, applied the same idea, displacing a classical parameter point instead of changing the construction, to ζ(3). That gave the first improvement on Rhin and Viola’s 2001 record, though by only about 10⁻⁵. He did the same for ζ(2) and also wrote a note unifying the three π records in one closed-form law. All three papers credit Claude (Anthropic).
Why it matters
Records for classical constants such as π and ζ(3) had stood for years. They fell to small, computation-heavy parameter searches of the kind AI assistants are good at, and the authors say so. The gains are tiny, and the authors themselves say they matter less than what they show: the classical parameter choices were not optimal. For π the episode was short: four weeks after Bai’s bound, OpenAI claimed the exact value 2, which, if it holds up, ends the race of upper bounds altogether.
What is disputed or not yet verified
| Verification | Unrefereed preprints; computer-verified numerics (standalone verifiers for the follow-ups) |
|---|
Sources
8 sources from 4 sites. Numbers match the chips in the text.
8 sources: 7 primary, 1 reaction
Primary
- Bai: The irrationality measure of π is at most 7.101862832357 (arXiv 2609.11276)arxiv.org, paper
- Zeilberger & Zudilin: The irrationality measure of π is at most 7.103205334137… (arXiv 1912.06345)arxiv.org, paper
- Niedbala Giraudin: A closed-form law for the Salikhov–Zeilberger–Zudilin–Bai family (arXiv 2609.37459)arxiv.org, paper
- Niedbala Giraudin: A new upper bound for the irrationality exponent of ζ(3) (arXiv 2609.26980)arxiv.org, paper
- Niedbala Giraudin: A new upper bound for the irrationality exponent of ζ(2) (arXiv 2610.02912)arxiv.org, paper
- OpenAI math catalogue (CONTENTS.md): 017 The irrationality exponent of π is 2github.com, paper
- ‘Dan Brown’: The irrationality measure of the logarithm of a rational number is 2 (PDF, 7 Oct)jdb19937.github.io, paper
Reactions
- Hacker News: The logarithms of rational numbers have irrationality exponent 2news.ycombinator.com, discussion
Changes
- Linked the new entry on OpenAI’s μ(π) = 2 claim (OpenAI model claims the irrationality exponent of π is exactly 2 (Lean-formalized), which implies the Flint–Hills series converges)
- Filed. While processing the leads queue; the ζ(3) and ζ(2) papers had been recorded as having “no AI disclosure”, but both PDFs credit Claude (fixed in New record bound on the irrationality measure of ζ(2), μ ≤ 5.0495243, released as a 92k-line Lean proof written by Claude; beaten a week later by a Claude-assisted paper). our automated search’s arXiv query now also looks for irrationality / record-bound papers.