Re-Phase Vol.5.1: Forward–Backward Path Compatibility — Reachability, Survivability, and Admissible Path Projections

This volume develops the forward–backward projection layer of finite-horizon path-space compatibility in the Re-Phase framework. It introduces backward survivable-compatible sets, combines them with forward reachable-compatible sets, and characterizes admissible path projections through their joint compatibility. The note also isolates the gluing mechanism behind this characterization and explains how false gluing can occur when higher-order or memory-dependent constraints are represented with insufficient state information.