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An Irrationality Measure for Catalan's Constant

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by Sean Litsey

Version 4 / Oct 11, 2026 / CC BY 4.0

Abstract

Building on OpenAI's proof that Catalan's constant G=∑j≥0(−1)j/(2j+1)2G=\sum_{j\ge0}(-1)^j/(2j+1)^2 is irrational, we prove ∣G−a/b∣>b−878|G-a/b|>b^{-878} for every integer aa and every sufficiently large integer bb. Thus μ(G)≤878\mu(G)\le878. Using Chebyshev's estimates for final prime selection gives 2 6322\,632; denominator and content asymptotics still use PNT. We change the determinant parameters, obtaining a 20N20N matrix and a polynomial of degree at most 18N18N. Uniform physical prime-column pools and sparse residue relations sharpen its common denominator. Exact energy certificates and separate treatment of free, fixed and marked populations give a uniform mixed-integrand bound. Stronger supplied energy trials, central-gap quadrature and jointly paid atom weights control the same determinant polynomial on ∣x−G∣≤e−(519037/5000)N|x-G|\le e^{-(519037/5000)N}. An explicit uniform residue-kernel construction supplies additional common coefficient content of quadratic logarithmic rate 16/6316/63. Coefficient divisibility recycles denominator prime squares; selecting the largest prime with p2∤bp^2\nmid b completes the improved measure criterion. We retain the earlier 48N48N construction and its 1 9811\,981 bound to explain the methods. Every ingredient of the new global theorem is formalized in Lean~4 with only the standard axioms and no unproved steps.

Provenance statement

This work was produced entirely by Claude Opus 5.5 and GPT-6.1 Sol. This followed a conversation between the human author about why OpenAI's proof of the irrationality of Catalan's constant did not give an irrationality measure.

Tools used

Anthropic
Claude OpusVersion 5.5
Lean
LeanVersion 4
OpenAI
CodexVersion GPT-6.1 Sol

References

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Version history

  1. v1Submitted by Sean LitseyInitial depositSupersededOct 10, 2026
  2. v2Submitted by Sean LitseySubmitted as a new versionSupersededOct 12, 2026
  3. v3Submitted by Sean LitseySubmitted as a new versionSupersededOct 14, 2026
  4. v4Submitted by Sean LitseySubmitted as a new versionCurrentOct 16, 2026