Grand Unified Cosmology

Abstract & Overview This preprint presents the concise mathematical and physical core of Grand Unified Cosmology (GUC), focusing on the 5-dimensional spacetime bulk and 3-dimensional twin-universe membrane architecture. Rejecting unprovable high-dimensional complexities and zero-spatial-size point particles ($r=0$), this work defines a 5-dimensional bulk $\mathcal{M}^{(5)}$ (1 temporal and 4 spatial dimensions) hosting a floating 3D twin-universe membrane ($\Sigma^{(4)}_+, \Sigma^{(4)}_-$) governed by $\mathbb{Z}_2$ mirror symmetry and CPT invariance. Key Mathematical & Physical Highlights: 5D Einstein Field Equations & Singularity Cancellation: By introducing the geometric smoothing kernel $\mathcal{K}_{\text{smooth}}(w)$ constrained by membrane thickness $D_{\text{membrane}}$, ultraviolet (UV) loop divergences and spacetime central singularities cancel automatically without artificial cutoffs. We perform the exact step-by-step derivation of all 5D Einstein tensor components $G^M_N$ under a warped FLRW metric ansatz ($ds^2 = e^{-2A(w)}[-c^2 dt^2 + a(t)^2 d\mathbf{x}^2] + dw^2$), proving the autonomous cancellation of the delta-function derivative $A''(w)$ in the normal component $G^4_4$. Israel Junction Conditions & AdS Bulk: Using the Israel junction conditions, the membrane curvature scale $k = (\kappa_5^2 \sigma) / 6$ and the negative Anti-de Sitter (AdS) bulk cosmological constant $\Lambda_5 = -6k^2 / \kappa_5^2 < 0$ are uniquely determined. Autonomous Observer Scale Contraction & Hubble Tension: Driven by the geometric dissipative coupling constant $\kappa$ (representing minute gravitational/torsion wave dissipation into the 5D bulk), elementary particles and observer measuring rods undergo exponential spatial scale contraction $\lambda(t) = \lambda(0)e^{-\kappa t}$. This naturally yields the cosmological redshift $1+z = e^{\kappa t}$ and algebraically resolves the Hubble tension between the local universe ($H_0 \approx 73.0\text{ km/s/Mpc}$) and the distant universe ($H_0 \approx 67.4\text{ km/s/Mpc}$), predicting a smooth S-shaped transition curve $H(z)$ across intermediate redshifts ($z = 0.5 \sim 5.0$). Falsifiable Predictions: We articulate 4 distinct, testable predictions: (1) post-merger gravitational wave echoes ($\Theta_{\text{merger}} \ge 1.0$), (2) pre-echo inspiral phase modulations, (3) the $H(z)$ S-curve, and (4) the absolute non-existence of dark matter particles in underground direct detection experiments. --- Note & Monograph Reference: This preprint serves as a concentrated, journal-submission version focusing on the 5D field equations and cosmological scale contraction. For the full 7-chapter comprehensive mathematical foundation covering quantum non-locality, four fundamental interactions, black hole internal physics, and galaxy structure, please refer to the complete monograph: *Tanaka, A. (2026). Grand Unified Cosmology: Complete 7-Chapter Monograph (DOI:10.5281/zenodo.24051176).*