Local Coherence Hessians: A Structural Classification of Spectral Gaps
Internal ID: CGI-RSR-000003
Author(s): Barry L. Petersen
Document Type: Research Paper
Publication Date: May 2026
Original Creation Date: February 17, 2026
Status: Public
Domains: Mathematics, Physics
Research Topics: Mathematical physics, Clay Millennium Problems, Yang-Mills
Abstract
We provide a structural classification of spectral gaps for local coherence functionals on periodic lattices. For nearest-neighbor phase interactions, the quadratic Hessian at aligned configurations is a weighted graph Laplacian. In infinite volume this operator is gapless, with the lowest nonzero eigenvalue scaling like L^−2. A uniform spectral gap requires strictly positive onsite quadratic curvature. We show that multi-channel phase locking provides a canonical intrinsic source of such curvature on relative channel modes: when the channel-coupling graph is connected, all relative-channel fluctuations are uniformly gapped, independent of lattice size. After locking, the remaining degree of freedom is the synchronized phase. We prove a sharp dichotomy: if the restricted functional on the synchronized manifold retains global shift symmetry, the model is structurally gapless; if it exhibits positive onsite curvature, the full multi-channel Hessian admits a uniform spectral gap. This isolates the precise structural origin of mass scales in local coherence models: locality alone is insufficient; quadratic curvature in the final soft sector is necessary and sufficient.
Available Document
Citation:
Petersen, B. L. (2026). Local Coherence Hessians: A Structural Classification of Spectral Gaps. Zenodo. https://doi.org/10.5281/zenodo.19969141
Source Code and Supporting Materials
None.
Summary and Notes
This document is written in standard mathematical-physics language. It studies local coherence functionals, Hessians, graph Laplacians, spectral gaps, periodic lattices, and multi-channel phase locking without requiring the reader to adopt the full Coherence Geometry framework.
The CG role is organizational rather than linguistic: it motivates the focus on coherence channels, phase locking, compatibility constraints, and curvature as structural sources of effective excitation scales.
Rather than beginning with fields as primitive objects, the approach studies how collective phase-locking, local coupling, and onsite curvature generate or fail to generate uniform spectral gaps.
From this viewpoint, a mass gap is interpreted structurally as robust quadratic curvature that remains after internal modes organize into stable locked sectors.
Scope Notes
This document is written in standard mathematical-physics language. It studies local coherence functionals, Hessians, graph Laplacians, spectral gaps, periodic lattices, and multi-channel phase locking without requiring the reader to adopt the full Coherence Geometry framework.
The CG role is organizational rather than linguistic: it motivates the focus on coherence channels, phase locking, compatibility constraints, and curvature as structural sources of effective excitation scales.
Rather than beginning with fields as primitive objects, the approach studies how collective phase-locking, local coupling, and onsite curvature generate or fail to generate uniform spectral gaps.
From this viewpoint, a mass gap is interpreted structurally as robust quadratic curvature that remains after internal modes organize into stable locked sectors.
Related Work
This paper is part of the Coherence Geometry research sequence on spectral gaps, local stability, and projection of CG mechanisms into mathematical physics. Additional Yang–Mills-specific translation work has not yet been released.