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on Irrationality measure of $\Gamma(1/4)$ ($C_{7b}$)

The statement

For a real number $\gamma$ , its irrationality exponent $\mu(\gamma)$ is defined by

\mu(\gamma) := \inf\Bigl\{c\in\mathbb{R}:\ \Bigl\lvert\gamma-\frac{a}{b}\Bigr\rvert\le \lvert b\rvert^{-c}\ \text{has only finitely many solutions }(a,b)\in\mathbb{Z}^2\Bigr\}.

[Zud2004-def-mu]

We define

C_{7b}\ :=\ \mu\bigl(\Gamma(1/4)\bigr).

For $p/q\in\mathbb{Q}$ in lowest terms with $q>0$ , write

h(p/q):=\log\max(\lvert p\rvert,q).

This agrees with the absolute logarithmic height used by Bruiltet. [Suk2013-def-hpq] [Bru2002-def-h] [Bru2002-def-places]

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.