Re-Phase Vol.5.3: Pathwise Compatibility Lifting — Path Fibers, Bridge Worlds, and Nonempty-Guarded Recoverability

This volume develops Pathwise Compatibility Lifting as the finite-horizon path-space extension of the product, fiber, and projection viewpoint introduced earlier in the Re-Phase series. It separates pointwise compatible choices from pathwise compatible lifts, showing that local compatibility at each time does not necessarily form a temporally compatible path. The volume introduces path fibers, lift status, forward and backward compatibility refinements, bridge worlds, online prefix failure, live-cone obstruction, and complete but non-local infeasibility certificates. It also defines nonempty-guarded path-feature recoverability and illustrates the framework through finite witness examples, including dead-end collapse, multivalued-to-unique lifting, ambiguous but recoverable lifts, missing-cell feature narrowing, edge-correlation counterexamples, and a constant-velocity observation-thinning example. This work does not claim a new reachability algorithm, a new relational calculus, or a replacement for existing automata, CSP, filtering, or model-checking methods. Its role is to provide a Re-Phase diagnostic interface for collapse, ambiguity, feature-effective narrowing, target recoverability, and the later selector and forgetting layers.