A Bethe-Ansatz Source of Early Token Correlations: A marginal-preserving construction for parallel proposals

Parallel token proposals need a joint law, not merely accurate token marginals. We construct coordinate-Bethe-ansatz laws from the periodic XXZ chain and compare them with their maximum-entropy pairwise projections at identical one- and two-site moments. In an eight-site, three-excitation sector, symmetry makes the full probability-law difference one-dimensional, determined by a contiguous three-site moment δ. Along the Bethe branch, this gap changes sign and vanishes exactly at Delta=(3+√3)/2, where the eigenstate law becomes pairwise. A nonlinear two-token readout preserves separate token marginals yet exposes a distance 2|δ|; a direct three-token readout preserves all one- and two-token marginals yet exposes 4|δ|. At Delta=1.5 these deterministic binary distances are 0.02654573 and 0.05309145. We give an exact target-alignment formula and prove a strict speculative accepted-prefix advantage throughout an unrestricted open neighborhood of a constructed target. These are finite-law existence results, not measured language-model or wall-clock gains.