Fragment XIV — Imaginary‑Time Relaxation of the Coherence Knot

This paper completes the stability analysis of neutrino coherence knots within the General Connectivity framework by relaxing the Gaussian variational ansatz to the exact standing wave through imaginary‑time gradient flow on the augmented energy. It proves that the flow is governed by a strict Lyapunov function, converges exponentially at a rate set by the spectral gap previously bounded in Fragment XIII, and preserves the Morse index along the entire path. A perturbation bound shows that the gauge eigenvalue of the phase operator vanishes at the same rate as the residual, closing the caveat left by the spectral count and ensuring that the linearised operators are evaluated on the true profile. The Vakhitov–Kolokolov sign on the exact standing wave is obtained by three independent routes: the operator count carried along the flow, the gauge‑mode perturbation bound, and the concavity of the minimum‑energy curve via the envelope theorem. The paper provides a complete finite‑difference protocol whose outputs are constrained a priori by the analytic theorems, leaving only the converged slope and spectral gap to be measured. The results strengthen the GC ontology by confirming that the coherence knot is a true local minimum of the augmented energy, with no negative directions, and that the defocusing sign of the quartic term is the structural reason for its stability.