An Irrationality Measure for Catalan's Constant
Building on OpenAI's proof that Catalan's constant $G=\sum_{j\ge0}(-1)^j/(2j+1)^2$ is irrational, we prove $|G-a/b|>b^{-740}$ for every integer $a$ and every sufficiently large integer $b$. Thus $\mu(G)\le740$. Using Chebyshev's estimates for final prime selection gives $2\,220$; denominator and content asymptotics still use PNT. The earlier bounds $759$, $2\,276$, $878$ and $2\,632$ remain proved. We change the determinant parameters, obtaining a $20N$ matrix and a polynomial of degree at most $18N$. Uniform physical prime-column pools and sparse residue relations sharpen its common…
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