Re-Phase Vol.1.4: Observation, Compatibility, and Recoverability

This volume formulates the static Compatibility Problem of Re-Phase. Re-Phase stands for Relative Phase Reconstruction. The central idea of this volume is that an observation does not generally reveal a configuration directly. Instead, it restricts a configuration space by eliminating configurations incompatible with the observed information. Given compatibility relations Rᵢ ⊂ X × Oᵢ and observations oᵢ ∈ Oᵢ, each observation determines a compatible set Kᵢ (oᵢ). The accumulated admissible set is the intersection A (o₁,. . . , oₙ) =∩ᵢ Kᵢ (oᵢ). The Compatibility Problem asks what can be reconstructed from this admissible set. The volume separates the question into three layers: existence, identifiability, and structural recoverability. A central case of Re-Phase is that the admissible set A may not be a singleton, while a structural feature remains recoverable. In feature-map form, this is expressed by |A|>1 but |ρₛ (A) |=1. This volume is intended as the static foundation for later Re-Phase volumes on local-to-global compatibility, observation design, and temporal compatibility.