Phase-Synchronized Latent Trajectory Optimization with Automata-Constrained Sequence Synthesis: Theory, Thermostats, and Benchmarks

Continuous latent space trajectory optimization in generative architectures and autonomous planners frequently suffers from numerical stiffness, gradient singularities, and structural non-compliance during discrete sequence decoding. We propose a framework unifying goal-conditioned Mahalanobis trajectory synthesis with Deterministic Finite Automaton (DFA) transition constraints. State optimization operates on an open affine domain M ⊆ R^D equipped with a constant, symmetric positive-definite metric tensor G ≻ 0. Continuous state trajectories follow a phase-synchronized regularized potential kernel whose exact analytical spatial gradient eliminates step discontinuities. Dynamic updates are integrated via an underdamped BAOAB Langevin thermostat satisfying the Einstein-Smoluchowski relation. We establish: (1) exponential phase locking within a forward-invariant domain, (2) stationarity of the Gibbs-Boltzmann density for the underdamped Langevin diffusion under standard integrability and ergodicity assumptions, and (3) differential entropy invariance under deterministic symplectic flow. Empirical evaluation on SMILES generation and integrator stability is planned as future work; no experimental performance claims are made here.