A Multi-Phase Extension of Complex Numbers and the Global Coherence Theorem

This paper introduces μ-numbers, or multi-phase numbers, as an extension of real and complex amplitude-phase representations to systems with multiple phase components under a shared amplitude structure. It defines the μ-number space, develops local synergy rules for amplitude-phase interaction, and studies bounded coherence behavior in multi-phase systems. The central result is the Global Coherence Theorem (GCT), which establishes structural stability for systems of μ-numbers under local bounded interactions, including arbitrarily large and countably infinite systems. The paper also develops bounds on amplitude evolution and provides an illustrative stabilization example. It serves as one of the foundational mathematical documents for Coherence Geometry. This record contains the original hash-committed foundation paper associated with Coherence Geometry Canon CDR-00. The PDF is released in the same form referenced by the CDR-00 provenance record so that the public file remains consistent with the recorded SHA-256 hash. Some terminology, notation, or exposition may differ from later canonical usage; where terminology differs, CDR-00 and later canonical CG documents should be treated as the current terminology reference.