Nathan O. Schmidt, Klee Irwin, Natasha Urakhchina (2026)
We construct radial dual lattice graphs for the Eisenstein, Hurwitz, and E8 lattices using admissible hyperspherical inversion. The inversion induces exact bijections between outer zone vertices and rational inner zone representatives, and it gives transported-edge graph isomorphisms once the radial-dual edge relation is defined. We verify the norm relation, involution, shell compression, and finite-shell adjacency identities using exact arithmetic. Composing the radial inversion with the Moxness E8-to-H4 folding matrix H4fold gives a candidate golden linear-radial dual compressor Υr for Cycle Clock Theory (CCT) workflows; on the E8 root shell, the top 4 × 8 projection block Π of H4fold maps the 240 roots into two 120-point layers, the regular 600-cell H4 and its golden-ratio scaled copy H4Φ, with radius ratio Φ, equivalently squared-norm ratio Φ2. For larger shells, this compressor is validated on finite domains and proposed as a proof target for full cycle-clock enumeration. The construction offers a practical exact-arithmetic method for shelling and scaling calculations, while full global injectivity of Υr on L8, 8D-to-4D graph-isomorphism preservation across arbitrary shells, and end-to-end CCT simulation integration remain proof obligations for future work.
