Four-Dimensional Bulk Inflation: A Closed Geometric Framework for Dark Energy, the Hubble Tension, and Spectral Dimension Suppression

We present a unified, mathematically closed geometric framework in which dark energy arises from the pressure of an inflating fourth spatial dimension acting on our three-dimensional brane. Starting from the 5D Einstein-scalar system, we derive — without free parameters — that the potential slope satisfies λ² = 2/n from the G55 consistency condition, yielding the geometric baseline w(0) = −0.75. The Israel junction conditions are solved explicitly to derive the (1+τ)⁻¹ suppression factor from first principles. We derive V0 from the bulk self-tuning condition Λ4 = 0, closing the second observational input functionally. The geometric coupling ξ_geom = 1/33 is obtained in closed form from the exact warp-factor integrals ∫A⁴dy = 1/31 and ∫A⁴(φ')²dy = 8/33. The meV-scale of the boundary coupling σ is explained as a geometric consequence of the dark energy scale through σ = τρ_φ/n. The structural prediction wa = −(1+w0)², testable by Euclid in October 2026, is the primary falsifiable test. With one observational input (τ = 0.445 from DESI) the framework is overconstrained: two genuine independent predictions (Δχ² < 1 vs ΛCDM; σ ≈ (1.30 meV)⁴) are satisfied without adjustment.