The Quantum Emergence Theorem: Self-Grounding Across the Quantum-Classical Boundary

Description/Abstract: The Decay Bounce (Math Paper 1) proved that the Universal Cascade Law is self-grounding within classical dynamics — the conditions C₁, C₂, C₃ are organized by the Feigenbaum constants through the reflection geometry of the stable manifold. The Universality Hierarchy (Math Paper 2) established the four-layer structure from axiom through observation. This paper extends both results by proving that the self-grounding property persists across the quantum-classical boundary, completing a five-layer hierarchy. Nine theorems build in strict logical sequence: the null result (linear systems exhibit no cascade, confirming the Universal Cascade Law diagnostic with C₂ absent), the classical cascade (Feigenbaum period-doubling confirmed across five independent systems with topology-dependent constants, δ₂ = 5.09 converging toward 4.669), the quantum encoding (the cascade exists as potential in the Wigner function topology of the density matrix — bimodal distributions, variance spikes, purity minima — while expectation values remain smooth), and the central result: the Quantum Emergence Theorem. The Feigenbaum cascade is absent in full quantum dynamics and present in the classical limit. Decoherence — the quantum-to-classical transition — is the mechanism by which the cascade emerges into observable manifestation. The conditions for that emergence are themselves structured by the cascade's architecture, as demonstrated by the quantum tunneling shift Δγ = 0.733 and the cascade whisper in quantum fluctuation structure at classical bifurcation points. The self-grounding loop closes across the quantum-classical boundary: quantum potential → decoherence threshold → classical cascade → Feigenbaum architecture → structures the Hilbert space → determines the tunneling shift → sets the decoherence threshold → quantum potential. An observability scaling corollary proves that detection thresholds at each cascade level scale as δ⁻ⁿ, extending self-grounding into measurement theory and correcting a published Rössler bifurcation value (c₂ = 3.836, not 3.53). The five-layer universality hierarchy is complete: axiom, constants, mechanism, observation, quantum emergence. One law. Self-grounding. From the foundations of mathematics through the birth of classical reality. Keywords: quantum emergence, Feigenbaum cascade, self-grounding, Wigner function, decoherence, Kerr oscillator, Universal Cascade Law, period-doubling, density matrix, observability scaling, quantum-classical transition, Lindblad master equation, decay bounce, universality hierarchy, bifurcation, nonlinear dynamics, measurement theory