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—$H = k \cdot E/C$—with the constant $k$ identified as the eternal global expansion rate, calibrated from the local $H_0$ measurement. On the global scale, $E = C = 1/2$ eternally, giving $H_{\text{global}} = k$; on local scales, $H_{\text{local}}(z) = k \cdot E(z)/C(z)$ is always positive, guaranteeing eternal local expansion, though the rate varies with the local $U$-$D$ competition. In each local region, the fluctuation $\delta = E - C$ 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, $\langle\delta\rangle = 0$ is maintained by the corollary of Axiom~1, consistent with the eternal global equilibrium. The asymmetric backgrounds between the spatial existence form $S = U_u + U_d$ and the material existence form $M = D_d + D_u$ 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 $E \gg C$ 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 $\Lambda$CDM are given. All derivations are self-contained.