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. v1.2 (22 September 2026): addendum (points 1–10) — second-shell exchange J2/J1 and the J2-meter r(g); coordination correction z = 5; the falsifier 3/2 withdrawn; whirling = 3′ (tangential, 116.57°), hedgehog = 3 (Galois pair); rational centroid of a glass; RKKY x-diagram; resolvability and hyperfine limits; Debye–Waller ratio of the Tsai shells (ordering confirmed, magnitude open); network check on the published 1/1 refinement (J′ ≈ J1) and the exact N-point pair (5 + 6J′/J1) ∓ √5; corrections to the source record (PRL 130, 176701 is the Au–Ga–Dy quasicrystal; the Tb4 site of the 2/1 refinement); k_F(e/a) calibrated on the universal Θp/dG curve (λ = 0.55); the three e/a regions of Tamura's phase diagram from a Luttinger–Tisza analysis of the network, and Sato's ferromagnetic structure with net moment (√5+1)/6 from the tangential Ising network. The v1.1 PDF is kept unchanged.