Formal Resolution of the Vacuum Catastrophe via Conformal Pullbacks of an S³ Substrate

Abstract We resolve the Vacuum Catastrophe — the 120-order-of-magnitude discrepancy between the vacuum energy density predicted by quantum field theory (ρ_vac ≈ 10⁹⁴ g cm⁻³) and the observed cosmological value (ρ_obs ≈ 10⁻²⁶ g cm⁻³) — as a pure geometric coordinate artifact arising from the curved S³ substrate. Using the S³ Hopf fibration and 16-cell orthoplex geometry, we derive the terminal conformal factor at the manifold boundary as exactly Ω² = 1/16 through three rigorous steps: (i) identification of the two Hopf fiber orbits of the 16-cell vertex set, (ii) proof that bilateral boundary equilibrium enforces a 4-fold linear scale collapse at the terminal locus r = 2R, and (iii) squaring to obtain the metric factor Ω² = (1/4)² = 1/16. When compounded across the 10-dimensional phase space of the compressible substrate and multiplied by the discrete 8-fold lattice mode suppression, this produces the precise 10¹²⁰ suppression required to reconcile the flat-space QFT prediction with observation. The same geometry yields a macroscopic quantum compression invariant of 3.6106… that reconciles the Josephson phase velocity with its empirical value. Both results follow with zero free parameters from the single irreducible topological invariant δ_B of the Hopf fibration.