The Transition Constant: Feigenbaum's α Governs the Onset of Nonlinear Dynamics from Quantum Decoherence to Gravitational Merger

Description (Abstract): The Feigenbaum spatial scaling constant α = 2.502907875… is proven to govern not only the internal structure of cascades in nonlinear systems but the transition INTO cascade dynamics at every physical scale tested. Three independent domains are examined. At the quantum scale, α organizes the decoherence threshold where quantum superposition gives way to classical cascade behavior (established in the Lucian Law, Theorems L10–L15). At the fluid scale, the critical Reynolds numbers for five flow geometry families are organized by combinatoric functions of the Feigenbaum constants, with inter-family ratios matching α-dependent predictions at 0.7% to 5.4% (established in the companion Navier-Stokes papers). At the gravitational scale, analysis of SXS numerical relativity binary merger waveforms reveals that the scale function R(n) = T(n)/T(last cycle) crosses α at the inspiral-to-plunge transition with precision 1.04% (mass ratio q = 1), 0.43% (q = 6), and 0.12% (q = 10) — with accuracy increasing toward exact in the test-mass limit. In all three domains, α marks the moment when perturbative or linear dynamics give way to full nonlinear behavior. The compound ratio λᵣ = δ/α = 1.86551077…, identified in Theorem L4 of the Lucian Law, is established as the universal transition velocity — the rate at which systems move between the α threshold and the δ destination across cascade levels. The post-Newtonian chirp exponent 3/8 is identified as the reciprocal of the Feigenbaum spectral exponent 8/3, connecting gravitational inspiral to turbulent cascade through a single mathematical structure. The unification of quantum mechanics and general relativity does not require merging the theories into one equation. It requires recognizing that both are expressions of the same cascade architecture, connected by a shared transition constant. Keywords: Feigenbaum constants, transition constant, alpha, spatial scaling, quantum decoherence, turbulence, gravitational merger, numerical relativity, SXS, Lucian Law, Resonance Theory, cascade architecture, period-doubling, universality, compound ratio, transition velocity, Navier-Stokes, Reynolds number, inspiral, ISCO, unification