Galois Conjugation in Q(sqrt(5)) and Acoustic Phonon Scaling in Magnetically Ordered Icosahedral Quasicrystals

The two three-dimensional irreducible representations of the icosahedral group are Galois conjugates over Q(√5); in the cut-and-project description of icosahedral quasicrystals they are carried by physical space E∥ (phonons) and perpendicular space E⊥ (phasons). We point out a sharp spectroscopic consequence for the recently discovered magnetically ordered icosahedral quasicrystals: linear spin-wave theory on the icosahedral cluster fixes the magnon spectrum, up to one overall scale, to {0, 5−√5, 6, 5+√5}·JS, so the two Galois-conjugate branches must appear at the energy ratio (5+√5)/(5−√5) = φ² = 2.6180…, independent of exchange constant, spin, temperature, and sample. The cubic reference value is 3/2. For ferromagnetic i-Au65Ga20Gd15 (T_C = 23 K) the branches are estimated at ~0.6/1.3/1.6 meV, resolvable on cold-neutron time-of-flight spectrometers. A secondary geometric prediction — the acoustic φ-scaling ladder, exact in the long-wavelength limit — is stated with its precise domain of validity.