The finite degree formula for normalized volumes

Let $f\colon \big(x\in (X, \Delta_X)\big)\to \big(y\in (Y, \Delta_Y)\big)$ be a finite surjective morphism between klt singularities such that $K_X+\Delta_X=f^*(K_Y+\Delta_Y)$.

We show that the normalized volumes satisfy \[\widehat{\mathrm{vol}}(x, X, \Delta_X)=\mathrm{deg}(f)\cdot \widehat{\mathrm{vol}}(y, Y, \Delta_Y).\] This proves a conjecture in [LLX20, Zhu25, XZ26a].