The Roof Inside the House: Shell Algebras of the Golden Polytopes, Integral Disagreement, and Guests as Multiplicities (INDYNA Research Note 28)

For the two-shell operator M(g)=A1+gA2 on the golden polytopes we determine when the shells commute. The icosahedron, dodecahedron and 600-cell commute in every pair of angle shells (association schemes: "united"); the icosidodecahedron is mildly non-commuting (commutator eigenvalues λ2=48 on all twelve non-commuting pairs, carried by its rational ±2 families, whence its exact mirror line); the 120-cell is deeply non-commuting. Time laws. On commuting shell pairs the two shells are Galois partners family by family: present =a+σ a (rational), time =a-σ a (√5-part); the true clock √(|N|) is √5 in 3D (icosahedron, dodecahedron), 4 and 6 in the 600-cell, and √5, 1 and √(11) (=√(m2(H4))) on the commuting sector of the 120-cell; rational shells give an antipodal "inversion time". Guests. The nearest-shell spectrum of the 120-cell lives in six number fields: Q(√5), Q, Q(√2) (silver, -1±√2, ×48), Q(√(13)) (bronze, ×16), Q(√(21)) (×16) and two cubic fields (x3-x2-7x+8, ×36; x3-x2-7x+4, ×25; discriminants 733, 1509, non-cyclic). The commuting sector is exactly gold∪rational; every foreign field sits in the commutator. Integrality. The orbital (coherent) algebra of H4 on the 600 cells has rank 45 with integer structure constants; the 120-cell distinguishes 45 relations over 31 angles, nine of them directed. For Galois-partner shells the commutator is σ-odd ("time") and its square σ-even ("present"), whence λ2 rational: 12 and 48 are theorems. In general every commutator vanishes on the golden blocks: the permutation representation contains twenty irreducibles, all ten golden ones with multiplicity one; multiplicity >1 occurs only on rational blocks (irreps 48 (m=2), 16 (m=2, ×2), 25 (m=3), 36 (m=3)). Guests are repetitions: a rational block of multiplicity m produces eigenvalues of degree m — the quadratic metals for m=2, the cubics for m=3. Gold is united because it never repeats. Part of the INDYNA Notes series (Notes 16–34). All numbers reproduce from the INDYNA law register (master data record, doi:10.5281/zenodo.21891088). Licence CC-BY-4.0.