Local Coherence Hessians: A Structural Classification of Spectral Gaps
Internal ID: CGI-RSR-000003
Document Type: Research Paper
Publication Date: May 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
In the Coherence Geometry (CG) framework, gauge-type systems are modeled as interacting coherence channels governed by a variational stability structure.
Rather than beginning with fields as primitive objects, the approach studies how collective phase-locking, compatibility constraints, and local curvature can generate effective excitation structure.
From this viewpoint, a mass gap is interpreted as the emergence of robust quadratic curvature after internal modes organize into stable locked sectors.
Related Work
Not yet available.