One Formula, Three Media: Golden-Ratio Eigenmodes of Icosahedral Networks and the Multiplicity Theorem (Research Note 6)

This note unifies and timestamps the network extension of a deposited series on the magnon spectra of magnetically ordered icosahedral quasicrystals (doi:10.5281/zenodo.21177470, 21178025, 21204778, 21210671, 21223594; journal submission Physical Review B, accession BU15557). One operator identity, sigma(A1) = A2 = P*A1, governs three physical media: magnons in the icosahedral spin cluster, acoustic phonons of the quasicrystal, and designed classical networks (LC circuits, pendulum chains, wave-guide graphs). Main results: (i) a scalar icosahedral network rings at the golden ratio itself, with eigenfrequency ratio omega_3'/omega_3 = phi = 1.618 exactly (cubic control sqrt(3/2) = 1.225); (ii) on metric graphs the full wave equation obeys the exact invariant (1 - cos k1 l)/(1 - cos k2 l) = phi^-2 at all frequencies, with the Galois conjugation acting as a mirror symmetry about the band centre (k1 l + k2 l = pi); (iii) a multiplicity theorem states which realizations are protected — a level is parameter-free if and only if its irreducible representation occurs once, so scalar vertex, edge and face networks qualify while three-dimensional vector displacement networks do not; (iv) a seven-fold ring realizes Galois triples with parameter-free invariants 49 and 14; (v) all signatures survive 5% component disorder at more than 300 sigma separation from the cubic reference. Component-level designs are given for an LC circuit (10 uH, 2.2 nF, 100 pF; exact observable is the inverse-square detuning ratio) and an audible 30-pipe acoustic network. An accompanying verification script (reproduce_networks.py, eight checks, each printing PASS, Python 3 with NumPy) reproduces every headline result in about one minute.