Mathematical Foundations of Coherence Formation and Stability

This paper develops a mathematical framework for coherence formation and stability in energy-minimizing phase systems. It proves the Unconstrained Coherence Convergence Principle (UCCP), which describes collapse toward a single dominant coherence state in the absence of additional constraints, and then studies how constraints alter that behavior. The paper derives the Structured Coherence Refinement Corollary (SCRC), showing how phase constraints, coherence thresholds, locality, and energy limits can stabilize multiple distinct coherence basins rather than allowing collapse into one global attractor. It also proves the Constrained Stability of Structured Coherence Theorem (CS-SCT), establishing persistence of structured coherence states under local constraints. Together, these results extend the Global Coherence Theorem by formalizing how coherence refinement can remain structured under constraint. 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.