Optimal Self-Dual Inequalities to Order Polarized BECs

$1-(1-x^M{)}^{2^M}>(1-(1-x{)}^M{)}^{2^M}$ is proved for all $x\in [0,1]$ and all $M>1$. This confirms a conjecture about polar code, made by Wu and Siegel in 2019, that $W^{0^m{1}^M}$ is more reliable than $W^{1^m{0}^M}$, where $W$ is any binary erasure channel and $M=2^m$. The proof relies on a remarkable relaxation that $m$ needs not be an integer, a cleverly crafted hexavariate ordinary differential equation, and a genius generalization of Green’s theorem that concerns function composition. The resulting inequality is optimal, $M$ cannot be $2^m-1$, witnessing how far polar code deviates from Reed–Muller code.