The Platonic Dichotomy: The Galois Signature is a Property of the Symmetry Group (Research Note 5)

This note timestamps one exact result extending four deposited notes on the icosahedral two-shell magnet (doi:10.5281/zenodo.21177470, 10.5281/zenodo.21178025, 10.5281/zenodo.21204778, 10.5281/zenodo.21210671). Result: The scalar graph-Laplacian spectra of all five Platonic solids are computed exactly. Every solid with cubic or tetrahedral symmetry (tetrahedron, cube, octahedron; group S4) has a purely rational spectrum. Both solids with icosahedral symmetry (group A5) carry the Galois pair over Q(sqrt5): the icosahedron with the eigenvalue pair 5±sqrt5 at the exact ratio phi^2 = 2.618, and its face dual, the dodecahedron, with the pair 3±sqrt5 and a doubled golden ladder — two successive steps of exactly phi^2, total span phi^4. The golden Galois signature is therefore a property of the symmetry group A5, not of any individual solid, and face duality doubles the golden ladder. Consequence for experiment: a magnetic or photonic realization on cubic-symmetry sites cannot display phi^2 at all — the cubic reference value of the earlier notes marks a different Galois class, not merely a different number. This sharpens the falsification criterion of the two-compound neutron protocol. The attached verification script (reproduce_all.py) reproduces the headline exact results of the whole series — icosahedral Laplace spectrum and phi^2, the AFM ratio sqrt(5/2), the negative-space theorem, the self-duality and golden points, the exactly solvable self-dual point (E0 = −6 J1, 15-fold singlet ground space, gap 1 J1), and the Platonic dichotomy — each as a PASS/FAIL check, runtime about one minute. A journal version is in preparation.