The Cayley-Dickson Cascade: Spectral Fingerprints of the Pre-Geometric Vacuum Filtration

We extend the Axiomatic Physical Homeostasis (APH) cosmology to the full hierarchy of Cayley-Dickson algebras An. We model the Big Bang not as a singular event, but as a sequential AlgebraicCascade A∞ → · · · → T → S → O, driven by the thermodynamic imperative to minimize the Asso-ciator Hazard. By generalizing the Geometric Closure principle, we derive the discrete gravitationalwave spectrum associated with the collapse of the hyperspheres S2n−1 inherent to each algebra. Wepredict a Geometric Comb of background radiation frequencies scaling as fn ≈ ν0 · 2n−2, an-chored to the Hydrogen line. This establishes the “Sedenion Scream” (1.8 GHz) as merely the n = 2fundamental of a harmonic series that maps the cooling of the 16D, 32D, and 64D vacuum lattices.