This volume formulates the static Compatibility Problem of Re-Phase. The central idea is that an observation does not usually reveal a configuration directly. Instead, it restricts the candidate space by eliminating configurations incompatible with the observed information. The volume introduces compatibility relations, compatible sets, admissible sets, and recoverability over the remaining candidates. It separates the problem into existence, identifiability, and structural recoverability. A key point is that the admissible set need not be a single configuration for a feature to be recoverable: state ambiguity may remain while feature ambiguity vanishes. This is a reference-corrected release of Vol.1.4, preserving the mathematical content while correcting bibliographic references and DOI assignments.