7 COS-GUT - Collapsing-Structure Grand Unification: SU(5)–SO(10)–E6 Embeddings Realized on Discrete Shell Graphs and Candidate Phenomenological Links

This work develops the COS–GUT module, a discrete GUT-compatible interface within the Collapsing-Structure (COS) program. It investigates how SU (5) SU (5) SU (5), SO (10) SO (10) SO (10), E6E₆E6, and related grand-unification structures can be represented on a quantized shell–filament background and connected to the COS–SM, COS–SUSY, and COS–QD modules. The manuscript organizes Standard Model multiplets, gauge connections, Higgs sectors, and Yukawa structures through Wilson-type discrete principal-bundle data on shell graphs. Chiral structure and anomaly conditions are treated using Ginsparg–Wilson/overlap-type constructions on admissible, gap-protected, reconstruction-compatible backgrounds. Shellwise embeddings, gauge-invariance conditions, anomaly checks, and operator-closure conditions are formulated with correction terms controlled under stated locality, admissibility, and refinement assumptions. The paper discusses SU (5) SU (5) SU (5), SO (10) SO (10) SO (10), and E6E₆E6 representation embeddings, hypercharge normalization, seesaw-related neutrino structures, E6-motivated Abelian mixing, shell-dependent symmetry breaking, doublet–triplet separation mechanisms, threshold matching, and proton-decay-sensitive operators. A discrete step-scaling RG scheme is outlined for gauge, Yukawa, and scalar couplings, including threshold effects and geometric O (ai2) O (aᵢ²) O (ai2) -type corrections. Candidate phenomenological outputs include coupling-unification diagnostics, proton-decay channels, neutrino-sector fits, Z'-related signatures, flavor/CP observables, and electroweak precision constraints. These are presented as model-dependent targets for further numerical fitting and COS–NUM implementation, not as closed predictions. The module provides a structured GUT-compatible formalism and numerical-interface roadmap rather than a fully closed predictive grand-unification model.