The Wang Transform Inequality

We prove the Wang transform inequality in every order: if $n$ is a positive integer, $A$ is a nonnegative $n\times n$ matrix whose row and column sums equal one, and $J_n$ has every entry $1/n$, then \[ \operatorname{per}A\ge \operatorname{per}\!\left(\frac{nJ_n+A}{n+1}\right). \] Equality holds exactly when $A=J_n$. We also obtain an explicit positive quadratic gap for every $n\ge2$.