This Part presents a computational investigation of discrete four-dimensional structures based on H₄ (600-cell) geometry. Using spectral graph theory, we analyze the Laplacian spectrum of the lattice and examine whether the geometric signatures predicted in earlier Origin Geometry Parts arise directly from computation. The results reveal naturally quantized eigenvalue spectra, strong spectral gaps, high degeneracy patterns, and irrational scaling relations associated with the golden ratio ϕ. These features emerge directly from the combinatorial and metric organization of the 600-cell and are absent in degree-matched control systems including random graphs and hypercubic lattices. Importantly, the analysis introduces no physical assumptions, no phenomenological fitting, and no external parameters. The purpose of the present work is not to derive physical constants, but to test whether the geometric structures predicted in earlier Parts possess objectively measurable computational signatures. The results provide computational evidence that H₄ geometry possesses a distinctive spectral identity and supports the existence of intrinsic geometric invariants predicted by the broader Origin Geometry framework.