This compendium timestamps nine results (A19–A27) on the icosahedral two-shell Heisenberg magnet, extending three deposited notes: FM φ² (doi: 10. 5281/zenodo. 21177470), AFM √ (5/2) (doi: 10. 5281/zenodo. 21178025), and the two-shell theory with golden instability point (doi: 10. 5281/zenodo. 21204778). Main results: (A19) The exact antiferromagnetic ground state is the icosahedron seen in perpendicular space: each spin points along the Galois-conjugated position of its own site, Sᵢ ∝ σ (rᵢ) ; the conjugation swaps first and second neighbours. (A20) Exact operator identity A₂ = P·A₁ (P = inversion) and self-duality g₂M (1/g₂) = P·M (g₂) ; a dual golden point at φ⁺²; phase diagram 3′ | 5 | 3 with the quintet phase occupying a golden window of logarithmic width 4·ln φ. (A21–A22) Both massive magnon branches collapse at the golden points with exponent 1/2 while their ratio stays pinned at √ (5/2), which is exchange-, collapse- and duality-invariant in both odd phases; the quintet window hosts a continuously degenerate classical ground manifold. (A23) Collinear members exist exactly; quantum order-by-disorder selects them everywhere except at the self-dual point, where both Galois triplets become simultaneous zero modes. (A24) The S=1/2 model at the self-dual point g₂=1 is exactly solvable: H = ½S (S+1) − Σ sₐ (sₐ+1) ; ground space of 15 singlets at E₀ = −6 J₁, gap exactly 1 J₁; the duality holds fully quantum mechanically; second-order splitting has exact rational coefficients −21/8, −11/8, −3/8. (A25–A26) The inelastic spectrum from the symmetric singlet consists of exactly two delta lines (ω = 1 and 2 J₁) with even/odd parity channels; the 15-fold degeneracy is spectroscopically silent; the visible branches are exactly one 3ᵤ and one 3′ᵤ copy with slopes 1 ∓ √5/2; a uniform field leaves the singlets degenerate up to Bc = 1 J₁, followed by an exact integer magnetization staircase. (A27) Inter-cluster coupling: first-order perturbation vanishes identically; a single bond shifts all product states uniformly by −43/576 J₃²/J₁; the Galois channel transmits only through antipodal contacts; all magnetization plateaus satisfy the Oshikawa–Yamanaka–Affleck criterion; classical Monte Carlo shows quantum and thermal order-by-disorder select differently. All numerical claims verified by exact diagonalization or machine-precision linear algebra. A journal version is in preparation.
Arber Gishto (Sun,) studied this question.