The Shift Law: A Cone Spectral Theorem for Golden Regge Geometries and its Galois Shadows (INDYNA Research Note 17)

Abstract

We prove a structure theorem for the length fluctuations of three-dimensional Regge cones over vertex-transitive polyhedra: the fluctuation operator equals the shifted adjacency operator of the edge shell, $H=\beta\,(A-c\,\mathbb 1)\oplus 0$, where $c$ is the adjacency eigenvalue of the position family and the facet-centre sector decouples exactly (umbrella lemma). All spectral ratios are therefore coupling-independent field elements; for icosahedral shells they are golden. The theorem extends verbatim to branched covers, which realizes the Kepler–Poinsot star polyhedra as flat Regge geometries: the four cones organize into two Galois doublets with $\sigma$-conjugate spectra and conjugate flat radii $(5\pm\sqrt5)/8$, the visible members being conformal-mode saddles and the star members strictly stable. A matter corollary couples a fermionic cluster to the geometry through the resolvent of the same operator and exhibits attraction, curvature generation, and finite collapse thresholds bounded by the angular budget of the winding. A boundary lemma shows that under maximal compression the shift value of both worlds converges to the rational, Galois-fixed value $-5/2$. The polytopal Regge setting itself is classical; claimed as new are the representation-theoretic decomposition, the operator identity, and the golden ratios. 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.