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 $Δ=(3+\sqrt3)/2$, 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 $Δ=1.5$ 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.