The Speed Gap: A Scaling Framework from Atomic Constants to the Hubble Constant

We present a self-contained derivation of the Hubble constant H₀ = 70.05 km s⁻¹ Mpc⁻¹ from five fundamental constants (α, G, ℏ, c, mₑ) and one model-independent CMB observation (θ* = 0.5965°). The derivation requires no freely adjustable cosmological parameters. Starting from the Speed Gap principle (gravitational stability when field-demanded speed matches actual speed), we show that the same condition with scale-dependent coupling G_eff governs systems from nuclear binding to galactic rotation. The dark energy exponent p = e^(-3/4) = 0.4724 is derived from D = 3 sphere geometry through the identity 4/3 × 3/4 = 1. The matter fraction Ω_m = (1-p)² = 0.2784 and dark-energy fraction Ω_DE = p(2-p) = 0.7216 = OE(β_cosmic=0.6808) follow as virial complementarity. The differential form of A+B=1, namely ΔA+ΔB=0, constrains the background to standard Friedmann. Unmodified CLASS reproduces the CMB angular scale to 0.094%, S₈ = 0.775 matches DES Y3 to 0.07σ, and DESI DR1 BAO distances favour the framework (χ²/N = 1.51) over ΛCDM (1.79). H₀ = 70.05 matches TRGB (Freedman 2024) to 0.1σ and DESI + DES PEDE to 0.01σ. This V4 supersedes V3 (10.5281/zenodo.19244123): the modified Friedmann of V3 §5.7 violated A+B=1 and has been replaced by standard Friedmann, derived in the companion paper Singh 2026g (10.5281/zenodo.19383758).