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 $Vol(-K_X)\le C_n{\epsilon }^{-(2^n-n-1)}$ whenever $X$ admits an $\epsilon$-lc log Fano boundary. The exponent is optimal in every dimension at least two, even for toric Fano varieties of Picard number one. In particular, the optimal exponent for fourfolds is eleven. We also prove an anticanonical interpolation theorem with optimal exponent $2^n-1$. The proof combines signed discrepancy estimates under projection, finite morphisms to projective space, and the canonical bundle formula. A reduction to a base bounded independently of $\epsilon$, followed by a volume estimate along a flag, yields the sharper volume exponent.