A Multi-Phase Extension of Complex Numbers and the Global Coherence Theorem
Internal ID: CGI-RSR-000009
Document Type: Research Paper
Publication Date: May 2026
Status: Public
Domains: Foundations, Mathematics
Research Topics: Coherence Geometry Canon
Abstract
We introduce µ-numbers (short for Multi-Phase Numbers), a generalization of complex numbers that extends traditional amplitude-phase representations to multi-phase systems under a single amplitude constraint. This framework unifies real, complex, and multi-phase constructs while preserving local coherence without requiring global orthonormality constraints. A central result of this work is the Global Coherence Theorem (GCT), which guarantees that systems of µ-numbers remain structurally stable and interference-free under local bounded interactions, even as they grow arbitrarily large. We rigorously prove that multi-phase coherence remains intact while ensuring bounded energy evolution, making µ-numbers naturally suited for infinite-dimensional settings. Key contributions include the formal definition of µ-numbers, the introduction of a local synergy function governing amplitude-phase interactions, rigorous bounds on amplitude evolution, an illustrative example of amplitude stabilization, and a proof of coherence for infinite sets. This work establishes a mathematical foundation for structured stability in high-dimensional multi-phase systems.
Available Document
Citation:
Petersen, B. L. (2026). A Multi-Phase Extension of Complex Numbers and the Global Coherence Theorem. Zenodo. https://doi.org/10.5281/zenodo.20116654
Source Code and Supporting Materials
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Summary and Notes
This paper is listed in the Foundation Papers section because it serves as a source document for CDR-00.
Related Work
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