On the Allen--Cahn equation

We prove that finite Morse index implies $O(R^{n-1})$ energy growth for bounded entire solutions of the quartic Allen–Cahn equation in dimensions $4\le n\le 9$, uniformly in the center of the ball. Combined with the cited finite-index classification results, this gives planar rigidity for $4\le n\le 7$ and, for nonconstant solutions, multiplicity-one stable minimal cone blowdowns through dimension nine. The proof uses an additive interaction estimate for adjacent transition layers, a weighted curvature and defect estimate with exponent $p=9\mathrm{/}4$, and a common test function for penalized reach and ambient stability. The local interaction coefficient $48$ is optimal in the class specified by the low-defect hypotheses.