Random cubic graphs admit nonsynchronized stable Kuramoto equilibria

For a uniformly random labelled simple cubic graph on an even number of vertices, we give a proof claim that the probability that every local minimum of the homogeneous attractive Kuramoto energy is synchronized tends to zero. This is the opposite of the limit proposed in problem MD-06 of \emph{OpenProblemsInNLA} and Conjecture~3 of \emph{Randomstrasse101: Open Problems of 2024}. A two-by-two sign pattern on a cycle of length divisible by four, together with a decaying nonlinear profile on its outward binary trees, has uniformly positive edge cosines and an exponentially small boundary gradient. A Laplacian spectral gap converts this truncated profile into an exact local minimum. Fixed-length cycle counts and the spectral gap of random regular graphs then establish the claimed probability limit. The analytic argument does not depend on numerical optimization. Supplementary rational existence certificates and a symbolic finite construction are checked with exact arithmetic.