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on Irrationality of the Riemann zeta value at 7

The statement

The Riemann zeta function $\zeta(s)$ is defined for $\operatorname{Re}(s) > 1$ by $\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$ and admits an analytic continuation to $\mathbb{C} \setminus \{1\}$ . A real number is irrational if it is not equal to any ratio of two integers. Prove that $\zeta(7)$ is irrational.

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.