Noncoplanar Order in the Galois-Conjugate Triplet and a √(5/2) Magnon Signature for Antiferromagnetic Icosahedral Quasicrystals

The first antiferromagnetic icosahedral quasicrystal, Au56In28.5Eu15.5 (T_N = 6.5 K), was reported in 2025 with no published spin-wave spectroscopy. We solve the classical nearest-neighbour Heisenberg antiferromagnet on the icosahedral cluster exactly: the Luttinger–Tisza bound is attained, the ground state is noncoplanar with all bond angles equal to arccos(−1/√5), and — the central structural result — the three-dimensional order parameter spans precisely the Galois-conjugate irreducible representation 3′ of the icosahedral group, i.e. the phason triplet of the cut-and-project formalism. Linear spin-wave theory then yields three Goldstone modes and two massive multiplets carrying the rational representations 5 and 4, with energies E5 = 2√((5−√5)/5)·JS and E4 = 4·sin(π/5)·JS, whose ratio is exactly E4/E5 = √(5/2) = 1.5811, parameter-free. Together with the ferromagnetic companion prediction φ² = 2.618 (DOI 10.5281/zenodo.21165182), both magnetic sectors of icosahedral quasicrystals are governed by the single angle π/5: the cosine side (φ = 2cos(π/5)) in the ferromagnet, the sine side (4sin(π/5)) in the antiferromagnet.