Kontsevich's star-product up to order 7 for affine Poisson brackets: where are the Riemann zeta values?

The Kontsevich star-product admits a well-defined restriction to the class of affine – in particular, linear – Poisson brackets; its graph expansion consists only of Kontsevich’s graphs with in-degree $⩽1$ for aerial vertices. We obtain the formula ${\star }_{\text{aff}}\text{ mod }\bar{o}({\mathrm{\hslash }}^7)$ with harmonic propagators for the graph weights (over $n⩽7$ aerial vertices); we verify that all these weights satisfy the cyclic weight relations by Shoikhet–Felder–Willwacher, that they match the computations using the $\mathsf{\text{kontsevint}}$ software by Panzer, and the resulting affine star-product is associative modulo $\bar{o}({\mathrm{\hslash }}^7)$. We discover that the Riemann zeta value $\zeta (3{)}^2\mathrm{/}{\pi }^6$, which enters the harmonic graph weights (up to rationals), actually disappears from the analytic formula of ${\star }_{\text{aff}}\text{ mod }\bar{o}({\mathrm{\hslash }}^7)$ $\mathit{\text{because}}$ all the $\mathbb{Q}$-linear combinations of Kontsevich graphs near $\zeta (3{)}^2\mathrm{/}{\pi }^6$ represent differential consequences of the Jacobi identity for the affine Poisson bracket, hence their contribution vanishes. We thus derive a ready-to-use shorter formula ${\star }_{\text{aff}}^{\text{red}}$ mod~$\bar{o}({\mathrm{\hslash }}^7)$ with only rational coefficients.