A KL-Aware Tied-Block Inequality and the Constructive Closure of Linear-Ordering Polytope Membership at n ≤ 7 for Sigmoid-Tilt Models

For the Bradley-Terry (BT) sigmoid-tilt family on K_n, the tilted edge-marginal vector p~(u,s) = (sigma(u_i - u_j + s_ij ln 2))_ij is the natural carrier of the base-2 moment-generating functional Phi_s(u) = log2 E_{P*(u)}[2^{s.X}]. We give a constructive, exact-arithmetic closure of the small-n theory of this functional. The analytic engine is a Bregman identity for the independent BT exponential family, Phi_s = s.p~ - (1/ln2) KL_indBT(p~ || p*), equivalently the tilt-path-average identity Phi_s(u) = int_0^1 s.p~(u,ts) dt. Our main result is the KL-aware tied-block inequality: for every facet a.x