Re-Phase Vol.5.2: State-Equation Compatibility — Transition Relations, Common Parameters, and State Augmentation

This volume develops the state-equation compatibility layer of finite-horizon path-space compatibility in the Re-Phase framework.It treats state equations as generators of transition relations and studies when a chosen state representation is rich enough to make path gluing sound. The volume distinguishes edgewise variables from pathwise common variables, explains false gluing caused by forgotten parameters, memory, or latent modes, and shows how state augmentation restores representation-sound first-order compatibility.The note also clarifies the projection order induced by augmentation: one should augment, intersect, and then forget, rather than forget, relax, and then intersect. This prepares later temporal recoverability, target-tube severity, moving-frame, timing, communication, and computational compatibility layers.