Convergence: The Lucian Law in the Infinite-Dimensional Limit

Description/Abstract: The Quantum Emergence Theorem (Math Paper 4) proved that the Lucian Law's self-grounding property extends across the quantum-classical boundary, completing a five-layer universality hierarchy. That paper noted three technical convergence questions in its open remarks: stability of quantum observables in the infinite-dimensional Hilbert space limit, the progressive emergence of the Feigenbaum cascade as the system transitions from quantum to classical, and quantification of the cascade whisper in quantum fluctuations. This paper answers all three. The quantum phase transition at γq = 1. 1838 converges to six decimal places by N = 30 Fock states and is unchanged through N = 50 — the infinite-dimensional limit is established with 0. 000% variation (Theorem 10). The semiclassical correspondence is demonstrated across five photon number regimes, showing the Feigenbaum cascade progressively crystallizing from quantum fog as mean photon number increases from ⟨n⟩ = 0. 49 (pure quantum silence) through ⟨n⟩ = 16. 91 (cascade emerging, σ increase sixfold) to the full classical Feigenbaum cascade (Theorem 11). The whisper amplitude at classical bifurcation points decays with a measured scaling exponent of −4. 657 — within 0. 26% of the Feigenbaum constant δ = 4. 669 (Theorems 12-13). The whisper does not follow naive quantum scaling (1/√⟨n⟩). It follows the cascade's own architecture. Self-grounding extends to the quantum noise floor: the Lucian Law governs the amplitude of its own quantum fingerprint. The self-grounding chain is now complete: prerequisites → constants → mechanism → detectability → quantum emergence → quantum fingerprint amplitude. Each level governed by the same δ. No technical gaps remain. The quintet is complete. Keywords: convergence, infinite-dimensional limit, Hilbert space truncation, semiclassical correspondence, quantum whisper, Feigenbaum constant, self-grounding, Kerr oscillator, decoherence, cascade emergence, quantum noise floor, density matrix, stroboscopic sampling, Lucian Law, universality hierarchy Communities: Mathematics, Mathematical Physics, Quantum Physics, Nonlinear Dynamics Access Right: Restricted Notes: Fifth and final paper in the mathematical proof quintet. Closes all technical convergence questions identified in Math Paper 4. Companion to The Quantum Emergence Theorem. Computational scripts 82-84 available upon request. Contains an unexpected result: the quantum whisper scaling exponent (−4. 657) matches the Feigenbaum constant δ = 4. 669 to within 0. 26%, extending the self-grounding property into the quantum noise floor. Withheld pending formal mathematical verification. Not for public distribution until verification is complete.