Bruhat Monotonicity of the Pan–Skandera–Wang Bijection

Pan, Skandera, and Wang defined a bijection from permutations preserving two balanced consecutive blocks to permutations preserving the parity of every position, and conjectured that each input lies below its image in the strong Bruhat order. We prove Conjecture 7.8 of arXiv:2606.13162v1 for every order on which Algorithm 7.5 is defined, namely n ≥ 4. In fact, every order-preserving shuffle of the odd-position and even-position channels of the image is a Bruhat upper bound for the input. The proof derives a two-extreme insertion recursion at even orders, establishes its compatibility with reverse-complement, and proves prefix and suffix counting bounds for arbitrary channel lengths. The natural extension to orders below four and the source algorithm’s length-consistent indices are specified explicitly. The permanent corollary holds for the balanced consecutive cut in all totally nonnegative matrices; the full family of arbitrary consecutive cuts is not established. Version 1 contains the seven-page preprint and its complete editable LaTeX source. OpenAI Codex was used substantially in mathematical derivation, internal proof checking, English drafting, and manuscript preparation, as disclosed in the paper. Internal checks do not constitute external peer review or a machine-checked formal proof.