Mathematical Foundations of Coherence Formation and Stability

Internal ID: CGI-RSR-000010
Document Type: Research Paper
Status: Public
Domains: Foundations, Mathematics
Research Topics: Structure formation, stability, coherence basins

Abstract

We establish a mathematical framework for coherence formation and stability in energy-minimizing phase systems. First, we prove the Unconstrained Coherence Convergence Principle (UCCP), which states that in the absence of constraints, coherence structures inevitably collapse into a single dominant state due to self-reinforcing phase align-ment. This follows from the Global Coherence Theorem (GCT), which ensures that phase-aligned systems governed by shared amplitude constraints evolve toward stable coherence. However, the GCT does not account for the structured stability of coherence under energy minimization. To address this, we derive the Structured Coherence Refinement Corollary (SCRC), which demonstrates that coherence structures remain distinct when phase constraints, coherence thresholds, and energy limits are introduced. Finally, we prove the Constrained Stability of Structured Coherence Theorem (CS-SCT), establishing that structured coherence states remain stable under local constraints rather than merging into a single attractor. These results extend the GCT by formalizing the conditions under which coherence refinement progresses in a structured manner, providing a rigorous foundation for coherence stability in generalized phase systems.

Available Document

DOI: 10.5281/zenodo.20116648

Citation:
Petersen, B. L. (2026). Mathematical Foundations of Coherence Formation and Stability. Zenodo. https://doi.org/10.5281/zenodo.20116648

Source Code and Supporting Materials

None.

Summary and Notes

This paper is listed in the Foundation Papers section because it serves as a source document for CDR-00.

Related Work

CGI-RSR-000009