Uniqueness of positive harmonic profiles outside convex cones for general stable operators

We prove uniqueness, up to a positive factor, of positive harmonic functions outside a proper closed convex cone with nonempty interior for every nondegenerate symmetric stable operator. The spectral measure may be singular or supported on finitely many directions. Each profile is assumed continuous, zero on the cone, and of polynomial growth strictly below the operator's order. The proof controls the entire growth norm by one interior value. This follows by normalizing at a maximizing scale and using a lower barrier whose exponent is smaller than the chosen growth weight. A comparison of the resulting normalized profiles then yields proportionality.