Description This short note introduces the path-space reading of Re-Phase compatibility. Earlier volumes developed admissible fibers over static candidate spaces, linear spaces, syndrome fibers, and operator domains. Vol. 3. 4 showed that operator equations can be read as compatibility conditions on function candidates, and also noted that transformations can be read as compatibility conditions once the candidate space is lifted to path space. Vol. 5 makes this bridge explicit, but only at the exact-compatibility layer. A path is treated as a candidate object: = (x₀, , xT). Stage conditions and transition conditions are not two different primitives. Both are compatibility conditions imposed on the path candidate. The distinction is not a difference of primitive status, but only of arity: a stage condition constrains one component of a path, while a transition condition constrains an adjacent pair. The admissible path set consists of all paths whose components satisfy the stage conditions and whose adjacent pairs satisfy the transition compatibility relations: A₀: ₓ (x₀, , xT) ₖ Fₖ\ (0 k T), (xₖ, x₊+₁) ₓ䂵\ (₀ ₊<ₓ). The note also distinguishes the global path view from the forward reachable-set view. The forward reachable-compatible set (Rₖ) gives candidates reachable from the past, but in generalₖ (A₀: ₓ) Rₖand equality need not hold. A candidate may be reachable from the past without being extendable to a full admissible path up to the final stage. A full forward-backward characterization is deferred to later Vol. 5. x. Persistent recoverability is defined as recoverability over admissible paths: A₀: ₓ, h (A₀: ₓ) B. This volume does not develop differential equations, control theory, stochastic processes, probabilistic transitions, optimization, numerical integration, filtering, smoothing, dynamic programming, or computational search. These are deferred to later Vol. 5. x and Vol. 6 layers. The purpose of this note is modest: to lift Re-Phase compatibility from static candidates to path candidates.
Takashi Ito (Wed,) studied this question.