Fragment XIII — The Field‑Theoretic Spectral Count

This paper completes the field‑theoretic stability analysis of neutrino coherence knots within the General Connectivity framework by performing the full spectral count of the linearised operators that govern small perturbations of the confined standing wave. Building on the Vakhitov–Kolokolov slope evaluation of Fragment XII, it derives the exact radial operators from the GC energy density and proves that both amplitude and phase operators have zero negative eigenvalues. The coherence knot is therefore a true local minimum of the augmented energy rather than a saddle, overturning the standard focusing‑NLS expectation of a single negative direction and strengthening the stability result. The proof combines an operator‑monotonicity argument, a hand‑computable Rayleigh‑Ritz bracket, and a fully specified finite‑difference protocol whose predicted outcome is fixed in advance. The result shows that orbital stability follows from the minimiser property alone, with the VK slope reduced to a consistency check. The paper parallels the scaling‑sign correction of Fragment XII, strengthens the GC ontology by confirming lattice confinement as essential, and identifies the spectral gap above the gauge mode as the single remaining numerical quantity to be measured.