Sharp Polynomial Upper Bounds for Anticanonical Volumes

For every positive integer $n$, we prove that there is a constant $C_n$, depending only on $n$, such that $Vol(-K_X) \leq C_n ε^{-(2^n-n-1)}$ whenever $X$ admits an $ε$-lc log Fano boundary. The exponent is optimal in every dimension at least two, even for toric Fano varieties of Picard number one.