$1 - (1-x^M) ^ {2^M} > (1 - (1-x)^M) ^{2^M}$ 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.

翻译：对于所有$x \in [0,1]$及$M > 1$，本文证明了$1 - (1-x^M) ^ {2^M} > (1 - (1-x)^M) ^{2^M}$。这一结果证实了Wu和Siegel于2019年提出的关于极化码的猜想：当$W$为任意二进制擦除信道且$M = 2^m$时，$W^{0^m 1^M}$比$W^{1^m 0^M}$更可靠。证明过程依赖于三个关键要素：$m$无需为整数的巧妙松弛条件、精心构造的六变量常微分方程，以及涉及函数复合的格林定理的天才推广。该不等式具有最优性——当$M = 2^m - 1$时结论不成立，这揭示了极化码与Reed-Muller码之间的偏差界限。