Tensor Formulation of Four-Dimensional Bulk Inflation: Exact Resolution of the 5D Bulk Equations and Geometric Consolidation

This final version provides the exact analytical resolution for the 5D Einstein-scalar field equations governing a warped Friedmann-Robertson-Walker (FRW) brane-world. Moving beyond the phenomenological fits of previous versions, this work derives the Consolidation Rate (β≈0.173) as a fundamental geometric constant from the Israel junction conditions and brane tension (τ). By closing the mathematical loop between the 5D bulk dynamics and 4D brane expansion, the framework yields a definitive, first-principles prediction for the Dark Energy equation of state: wa≈−0.03. This value directly addresses the "thawing" signature identified in the DESI 2024/2026 data releases and offers a unique, falsifiable solution to the H0 Hubble Tension. This document serves as the theoretical conclusion to the 4D Bulk Inflation framework, linking prior work on Spectral Dimension Running (DOI: 10.5281/zenodo.19368601) and the initial Bulk Inflation hypothesis (DOI: 10.5281/zenodo.19285798). The model is now a closed system, awaiting final observational verification from the Euclid and Roman Space Telescope missions.