Local Coherence Hessians: A Structural Classification of Spectral Gaps

This record contains a Coherence Geometry manuscript studying the structural origin of spectral gaps in local coherence models on periodic lattices. For nearest-neighbor phase interactions, the quadratic Hessian at an aligned configuration is a weighted graph Laplacian. In infinite volume this operator is structurally gapless, with the lowest nonzero eigenvalue scaling like \(L⁻²\). The manuscript shows that a uniform spectral gap requires strictly positive onsite quadratic curvature. Multi-channel phase locking provides an intrinsic source of such curvature for relative-channel modes: when the channel-coupling graph is connected, all relative-channel fluctuations are uniformly gapped independently of lattice size. After locking, the remaining synchronized phase obeys a sharp dichotomy. If the restricted functional retains global shift symmetry, the model remains structurally gapless; if the synchronized sector has positive onsite curvature, the full multi-channel Hessian admits a uniform spectral gap. This manuscript is part of the Coherence Geometry Clay-problem research series and is categorized under the Yang–Mills existence and mass gap problem. It is presented as a structural coherence-geometric classification of how mass-gap-like spectral gaps arise in local lattice models, not as an accepted resolution of the Yang–Mills problem.