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The irrationality exponent of pi is 2

Theorem 1.1: $\mu(\pi)=2$ ; for every $\nu>2$ there is $Q(\nu)$ with $|\pi-p/q|\ge q^{-\nu}$ for all $p\in\mathbb Z$ and $q\ge Q(\nu)$ . Corollary: the Flint Hills series converges, and $\sum n^{-a}|\sin n|^{-b}$ converges iff $a>\max\{1,b\}$ . The bound is NOT effective (no explicit $Q(\nu)$ ), and it is NOT a bound of the form $c/q^2$ : bounded partial quotients for $\pi$ are not claimed. The method is an interpolation-determinant argument, not an improvement of the integral constructions behind earlier numerical bounds.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The irrationality exponent of pi is 2 · Irrationality exponent of pi

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Registry verification: lean checked · announcement · candidate

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VibeMathed record: The irrationality exponent of pi is 2

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the TeX source read against the conjecture as posed; it gives, for every real $\nu>2$ , a threshold $Q(\nu)$ with $|\pi-p/q|\ge q^{-\nu}$ for all integers $p$ and $q\ge Q(\nu)$ , which is $\mu(\pi)=2$ . The threshold is ineffective. The proof was not refereed. Lean: the release's Comparator challenge PiExponent (OAI.PiExponent.main) with solution module OAI.NumberTheory.PiExponent.Main, both present at the pinned commit; the challenge is not in the formalization catalogue. Its statement was read here and covers the headline: the eventual lower bound for every $\nu>2$ and the supremum characterization over rationals equal to 2. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. The manuscript notes Carella (2022) also claimed exponent two and points out a sign error in that paper's proof of a stronger claim.