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

Abstract

For the two-shell operator $M(g)=A_1+gA_2$ 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 $\lambda^2=48$ on all twelve non-commuting pairs, carried by its rational $\pm2$ 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 $=a+\sigma a$ (rational), time $=a-\sigma a$ ($\sqrt5$-part); the true clock $\sqrt{|N|}$ is $\sqrt5$ in 3D (icosahedron, dodecahedron), $4$ and $6$ in the 600-cell, and $\sqrt5$, $1$ and $\sqrt{11}$ ($=\sqrt{m_2(H_4)}$) 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: $\mathbb Q(\sqrt5)$, $\mathbb Q$, $\mathbb Q(\sqrt2)$ (silver, $-1\pm\sqrt2$, $\times48$), $\mathbb Q(\sqrt{13})$ (bronze, $\times16$), $\mathbb Q(\sqrt{21})$ ($\times16$) and two cubic fields ($x^3-x^2-7x+8$, $\times36$; $x^3-x^2-7x+4$, $\times25$; discriminants $733$, $1509$, non-cyclic). The commuting sector is exactly gold$\cup$rational; every foreign field sits in the commutator. Integrality. The orbital (coherent) algebra of $H_4$ 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 $\sigma$-odd ("time") and its square $\sigma$-even ("present"), whence $\lambda^2$ 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 $48\,(m{=}2)$, $16\,(m{=}2,\,\times2)$, $25\,(m{=}3)$, $36\,(m{=}3)$). 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.