The Birth of Structure: Feigenbaum Architecture in the Quantum-to-Classical Transition

Description/Abstract: Every physics student learns two separate stories. In quantum mechanics, systems exist in superposition until measured, then collapse into definite states through decoherence. In nonlinear dynamics, systems undergo structured transitions — period-doubling cascades governed by Feigenbaum's universal constants. These two stories have never been connected. This paper connects them. Using the Universal Cascade Law diagnostic — which predicts that any bounded system with nonlinear coupling necessarily produces fractal geometry organized by the Feigenbaum constants — we examine quantum decoherence across three layers: the linear Lindblad master equation (no cascade, as predicted by the absence of nonlinearity), classical nonlinear oscillators (full Feigenbaum cascade confirmed with δ₂ = 5.09 converging toward 4.669 across five independent systems), and the full quantum density matrix of a driven-dissipative Kerr oscillator solved in Hilbert space. The result is more profound than simple confirmation. The Feigenbaum cascade exists in the classical limit of quantum hardware but vanishes in the full quantum dynamics. The quantum density matrix holds all branches simultaneously as a mixture, encoding the cascade in its phase-space topology — bimodal Wigner functions, variance spikes, purity drops — while reporting smooth averages. Decoherence is the process by which the Feigenbaum cascade comes into existence. The cascade does not live in the quantum regime. It does not simply live in the classical regime. It lives in the transition between them. Decoherence is the birth of structure. Additional results include: correction of the published Rössler second bifurcation point from c₂ ≈ 3.53 to c₂ = 3.836; confirmation that coupling topology (quadratic vs cubic nonlinearity) selects the Feigenbaum family member (δ ≈ 4.67 vs δ ≈ 5.84); discovery that detection thresholds themselves follow Feigenbaum scaling (the observability scaling property); quantification of the quantum tunneling shift (Δγ = 0.733) between quantum and semiclassical transition points; and identification of the cascade "whisper" in quantum fluctuation structure at classical bifurcation points. Keywords: quantum decoherence, Feigenbaum constant, period-doubling cascade, Universal Cascade Law, Kerr oscillator, Duffing oscillator, Rössler system, Wigner function, bifurcation, nonlinear dynamics, quantum-classical transition, Lindblad master equation, universality, coupling topology, wave function collapse, measurement problem, fractal geometry