Endpoint permanent gates and cycle-forest families for the Lih--Wang chord inequality

We prove that the Lih--Wang half-interval chord inequality holds for every nonnegative row-stochastic matrix A of order n >= 2 whose permanent is at least 2^(2-n). The endpoint criterion yields a direct-sum theorem from permanent lower bounds on the blocks. For weighted directed cycles whose cycle-element incidence graph is a forest, the inequality holds in every order when there are at least two components counting fixed points, in all orders n >= 64, and for single-cycle and shared-center families in the remaining connected small-order range. The proof combines analytic rook and entropy bounds with exact rational Bernstein certificates. The full Lih--Wang conjecture and connected multi-center forests below order 64 remain outside the result. No optimality or first-priority claim is made. Prior order-three, order-four, conditional order-six and specified two-by-two star-block overlaps are identified in the manuscript. Status: internally reviewed preprint; external mathematical review and formal peer review pending. The archive includes frozen semantic sources, TeX/PDF artifacts, independent verification scripts, exact certificates, hashes and reproduction instructions. The original internal-draft labels are retained in the frozen semantic sources for hash identity.