Associativity certificates for Kontsevich's star-product $\star$ mod $\bar{o}({\mathrm{\hslash }}^k)$: $k⩽6$ unlike $k⩾7$

The formula $\star$ mod $\bar{o}({\mathrm{\hslash }}^k)$ of Kontsevich’s star-product with harmonic propagators was known in full at ${\mathrm{\hslash }}^{k⩽6}$ since 2018 for generic Poisson brackets, and since 2022 also at $k=7$ for affine brackets. We discover that the mechanism of associativity for the star-product up to $\bar{o}({\mathrm{\hslash }}^6)$ is different from the mechanism at order $7$ for both the full star-product and the affine star-product. Namely, at lower orders the needed consequences of the Jacobi identity are immediately obtained from the associator mod $\bar{o}({\mathrm{\hslash }}^6)$, whereas at order ${\mathrm{\hslash }}^7$ and higher, some of the necessary differential consequences are reached from the Kontsevich graphs in the associator in strictly more than one step.