Equilateral stability and cubic deficit expansions for quantum stars with preferred-orientation coupling

Fix N ≥ 3 and L > 0. Consider the lowest eigenvalue of a compact N-edge star with total length L, Neumann conditions at the leaves, and preferred-orientation coupling of fixed length scale at the centre. We establish a global stability estimate for the equilateral maximizer: the eigenvalue deficit dominates the squared Euclidean deviation of the edge lengths, with an explicit positive coefficient. A scalar phase equation and a quantitative concavity remainder also yield an exponential estimate. For perturbations whose coordinates sum to zero, we derive the cubic expansion of the deficit. Its quadratic coefficient is positive and its cubic coefficient is negative for every N and L. A perturbation of three edges therefore shows that the best global quadratic stability constant is strictly smaller than the local quadratic coefficient. For degenerating stars, the same phase equation gives a finite limit for the negative branch indexed by m exactly when the limiting graph has more than 2m positive-length edges.