The Hubble Theorem and the Local-Relative Nature of Inflation: A Rigorous Derivation from UD Theory

We derive the foundations of cosmology rigorously from the UD axioms. The cosmological principle is a derived consequence of the nature of U, not an independent assumption. The Hubble parameter is proven as a theorem—H = k E/C—with the constant k identified as the eternal global expansion rate, calibrated from the local H₀ measurement. On the global scale, E = C = 1/2 eternally, giving H₆₋₎₁₀₋ = k; on local scales, H₋₎₂₀₋ (z) = k E (z) /C (z) is always positive, guaranteeing eternal local expansion, though the rate varies with the local U-D competition. In each local region, the fluctuation = E - C evolves freely according to the unrestricted local field dynamics, and the local structure—whether U-dominated or D-dominated—is determined solely by the local U-D attribute ratio. Globally, = 0 is maintained by the corollary of Axiom~1, consistent with the eternal global equilibrium. The asymmetric backgrounds between the spatial existence form S = Uᵤ + Ud and the material existence form M = Dd + Dᵤ directly imply that spatial expansion (U-driven) is unconditional while material condensation (D-driven) is conditional. The apparent inflationary expansion is reinterpreted as a local and relative phenomenon: it occurs wherever E C locally, and persists in present-day cosmic voids, while regions of high matter density expand slowly or even contract locally. The universe originates from the breaking of a critical state devoid of space, time, and matter, not from a primordial singularity. Quantitative predictions that differ from standard CDM are given. All derivations are self-contained.