Part 8: Computational Evidence for Geometric Eigenmodes, Spectral Gaps, and Golden-Ratio Invariants in Discrete H4 Geometry A Graph-Laplacian Validation of the 600-Cell Spectral Signature

Overview Earlier Parts of the Origin Geometry (OG) program proposed that discrete aperiodic four-dimensional geometry may possess intrinsic mode structure, geometric attractors, hierarchy baselines, and golden-ratio scaling prior to physical interpretation. Part 4 introduced geometric eigenmodes and attractor stability. Part 5 introduced screening and deviation from geometric baselines. Parts 6, 6A, and 6B identified hierarchy candidates arising from H4 geometry, including the interface baseline Iᵢnt = 20φ⁴ and the algebraic bulk–boundary baseline Ralgbb = 120√5φ⁴. Computational Methodology The present Part performs a computational validation task. It constructs the 600-cell vertex graph associated with H4 geometry, builds the combinatorial graph Laplacian L = D − A, and studies its spectral structure. No physical Hamiltonian, no empirical constant, no phenomenological fitting, and no particle-specific assumption are introduced. The graph Laplacian is used only as a coordinate-independent probe of connectivity, symmetry, and discrete mode organization. Spectral Analysis and Control Comparisons The computational analysis tests whether the H4 graph exhibits objectively measurable signatures compatible with earlier OG claims. The observed signatures include discrete spectral organization, symmetry-induced degeneracy patterns, spectral gaps, and repeated algebraic structure involving the golden ratio. Control comparisons with non-H4 geometries, including degree-matched random graphs and hypercubic-type lattices, are used to distinguish H4-specific organization from generic finite-graph behavior. Robustness tests under coordinate perturbation are introduced to determine whether the observed features depend on fine-tuned coordinate placement. Conclusion and Limitations The results support a limited conclusion: the 600-cell graph possesses a distinctive and reproducible spectral identity compatible with the existence of intrinsic geometric eigenmodes and invariants proposed in earlier OG Parts. This does not prove physical constants, particle masses, gauge theory, or dynamical laws. It provides computational evidence that the geometric structures used by OG are not merely symbolic assumptions but correspond to measurable spectral features of the H4 graph.