Exact Two-Shell Theory of the Icosahedral Spin Cluster: a J₂ Meter, an Invariance Theorem, and a Golden Instability Point

We extend the parameter-free magnon predictions for magnetically ordered icosahedral quasicrystals (DOI 10.5281/zenodo.21177470, 10.5281/zenodo.21178025) to the full three-coupling Heisenberg model on the icosahedral cluster. The distance classes realize the Galois conjugation structurally: the second-neighbour adjacency carries the swapped assignment, A₂ = σ(A₁). Consequences, all exact: (i) the ferromagnetic Galois-branch ratio obeys an invertible Möbius law — a measured deviation from φ² determines J₂/J₁; (ii) both massive antiferromagnetic multiplets factorize over coupling-matrix eigenvalue gaps, so E₄/E₅ = √(5/2) holds identically in all exchange ratios (invariance theorem, proven as a polynomial identity); (iii) the massive modes soften at the golden coupling ratio J₂/J₁ = φ⁻² = 0.3820 — a level crossing at which the order parameter migrates from the phason triplet into the quintet sector, beyond which no exact ratios survive; (iv) the quintet magnon energy equals √10 times the uniform scalar spin chirality of the ground state; (v) magnetoelastic selection rules: E₅/E₄ hybridize only with cluster vibrations of irreps 5/4, while phason-type strain broadens E₄ an order of magnitude more strongly than E₅. Robustness against inter-cluster coupling and dipolar interactions is quantified.