One Constant, Every Scale: The Universal Cascade from Quantum Decoherence to Binary Merger to the Cosmos

Two Feigenbaum constants — δ = 4. 66920160… and α = 2. 50290787… — and their ratio λᵣ = δ/α = 1. 86551077… govern every physical transition from quantum decoherence to binary black hole merger to cosmological structure. This paper synthesizes evidence across five physical scales. At the quantum scale (Paper II): the decoherence threshold is organized by α in Wigner function topology, the decoherence rate scales as −ln δ, and the Born rule is uniquely derived from the cascade renormalization transformation T_α. At the mesoscopic scale: the Kolmogorov −5/3 turbulence law is the special case D = 3 of the cascade spectral master equation E (k) ∝ k^− (8−D) /3; at Feigenbaum cascade dimension D = 2. 756 the intermittency parameter μ = 0. 2445 falls at the center of the 40-year experimental range; structure function exponents ζ (p) match the experimental record at χ² = 1. 91 with zero free parameters; and the cascade regularity bound α/δ^4/3 = 0. 321 < 1 prevents Navier-Stokes finite-time blow-up. At the gravitational scale: the scale function R (n) = T (n) /Tₗast in binary black hole inspiral crosses α = 2. 503 at the inspiral-to-plunge transition with precision 0. 12% at mass ratio q = 10, converging toward exact in the test-mass limit; subsequently crosses δ in late inspiral; and the post-Newtonian chirp exponent 3/8 is the exact reciprocal of the cascade spectral exponent 8/3. At the cosmological scale: all six ΛCDM parameters are derived from α with zero free parameters, including the reionization optical depth τᵣeion = 0. 0572 from the cascade geometric series zᵣeion = δα/ (α−1) = 7. 776 and the scalar amplitude Aₛ = 2. 113×10⁻⁹ from the CMB degeneracy closure; the Cascade CAMB Boltzmann solver reproduces the full CMB power spectrum with no cosmological constant in the Friedmann equation; the rigorous cascade-geometry Hubble constant H₀ (θ*) = 68. 21 km/s/Mpc reduces the Hubble tension from 5. 7σ to 4. 8σ; dark matter, dark energy, the Hubble tension, and inflation are identified as four geometric artifacts of α-projection. At the galactic scale: the Milgrom-McGaugh critical acceleration g† is derived from first principles for the first time — g† = α × c × H₀ (θ*) / (2π√δ) = 1. 2216 × 10⁻¹⁰ m/s² (McGaugh 2016: 1. 2000 × 10⁻¹⁰ m/s², residual +1. 8%) ; the cluster baryon fraction fᵇ = 1/α² = 15. 96% is a parameter-free identity; six gravitational lensing tests pass including CMB lensing, Einstein ring radii, and the Bullet Cluster; MOND is recovered as an effective theory with its previously unexplained parameter now derived. The unification is architectural: the fundamental equations of physics are not one equation but instances of one fractal geometric structure — the Feigenbaum universality class — operating on different substrates. Twenty-six observables. Five domains. Zero free parameters. Nine falsifiable predictions. Notes: Paper III of the Unification Series. Companion to Papers I–II and UCT foundation (ETDS-2026-0134). Generated by Script 90. Journal submission pending ETDS review outcome.