Re-Phase Vol.1.9: Compatibility Lattice and Admissible Hierarchy

This volume formulates the context-poset structure underlying admissible refinement in the Re-Phase framework. For each context S ⊆ I, the admissible set A_S is defined by intersecting the compatible sets associated with the observations or constraints in S. This creates an order-reversing admissible hierarchy: when S ⊆ T, the refined context T corresponds to a smaller admissible set A_T ⊆ A_S. The volume defines recoverability regions in the context poset using a pre-specified feature map and reconstruction-acceptable target region. The nonemptiness condition is included so that collapsed contexts are not counted as recoverable. It also introduces minimal sufficient contexts, maximal compatible contexts, minimal incompatible contexts, obstruction boundaries, and design paths as chains through the context poset. Vol.1.9 should be read as the Level 0 order-structure layer of Re-Phase. It defines the design arena on which later observation and constraint design problems can be formulated. It does not assume topology, metric geometry, smoothness, probability, spectral structure, temporal transport, optimization, or numerical algorithms.