The Shift Law in Four Dimensions: Triangle Hinges, the 600-Cell Cone, and its Galois Shadow (INDYNA Research Note 20)

Abstract

The Cone Spectral Theorem of Note 17 states that the Regge length-fluctuation operator of the cone over a vertex-transitive polyhedron is the shifted adjacency operator of its edge shell, $H=\beta(A-c\,\mathbb1)$, with $c$ the position-family eigenvalue. Note 17 left four dimensions open, since there the deficit angles live on triangles rather than edges. We show that the theorem ports exactly. For the cones over the $5$-cell, the $16$-cell and the $600$-cell (all cells tetrahedra; hinges are the triangles spanned by the apex and an edge) the Hessian in the radial lengths is $\beta(A_1-c)$ to machine precision, with $c=-1,\ 0,\ 3+3\sqrt5$ respectively — the four-dimensional position eigenvalues — and with exactly four zero modes, the four-dimensional translations. On the $600$-cell all nontrivial families lie below $c$: $115$ negative modes, one positive breathing mode; the family ratios are golden with rational denominators, including a purely rational $\tfrac12$. The conjugate shell $\cos\theta=-1/(2\varphi)$ is the edge shell of the grand $600$-cell $\{3,3,5/2\}$; its cone is branched-flat at $r^\ast=1$ with winding $2$ (dihedral sum $4\pi$), obeys the Shift Law with $c\to\sigma(c)=3-3\sqrt5$, and is strictly stable ($116$ positive modes, none negative): the Kepler–Poinsot pairing of Note 17 acquires its four-dimensional row. The Shift Law is thus not a three-dimensional artefact; the linear foundation for the Regge time slab and for multi-shell (Wong-type) lattices holds in the dimension of general relativity. 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.