Parallel token proposals in language-model decoding need a joint law, not merely accurate token marginals. We construct a family of coordinate-Bethe-ansatz laws from the periodic XXZ chain and compare each with its maximum-entropy pairwise projection at identical one- and two-site moments. In the eight-site, three-excitation sector, symmetry makes their entire probability-law difference one-dimensional, determined by a contiguous three-site moment. Along the Bethe branch, this gap varies with anisotropy, changes sign, and vanishes exactly at $Δ=(3+\sqrt{3})/2$, where the eigenstate law becomes exactly pairwise. A nonlinear readout transfers a nonzero gap into the first two token positions while preserving their separate marginals. At $Δ=1.5$ the deterministic binary token laws differ by 0.02654573 in total variation. Exact speculative verification then gives a strict accepted-prefix advantage for a constructed target and a neighborhood within targets sharing its first-token marginal. This is a mathematical existence result. It does not establish a language-model gain or wall-clock speedup.

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