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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 ₀₅₅ mod o (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 kontsevint software by Panzer, and the resulting affine star-product is associative modulo o (⁷). We discover that the Riemann zeta value (3) ²/⁶, which enters the harmonic graph weights (up to rationals), actually disappears from the analytic formula of ₀₅₅ mod o (⁷) because all the Q-linear combinations of Kontsevich graphs near (3) ²/⁶ 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 ₀₅₅^red mod~o (⁷) with only rational coefficients.
Buring et al. (Thu,) studied this question.