On cellular rational approximations to $\zeta (5)$

We analyse a certain family of cellular integrals, which are period integrals on the moduli space ${\mathcal{M}}_{0,8}$ of curves of genus zero with eight marked points, and give rise to simultaneous rational approximations to $\zeta (3)$ and $\zeta (5)$. By exploiting the action of a large symmetry group on these integrals, we construct an infinite $effective$ sequence of rational approximations $p\mathrm{/}q$ to $\zeta (5)$ satisfying $$0<\mid \zeta (5)-\frac{p}{q}\mid <\frac{1}{q^{0.86}}\mathrm{.}$$