Fibonacci Lattice Condensates as FCCL Simulators: Born Rule Measurement and Golden-Ratio Revivals in Bose-Einstein Condensates

We show that a Bose-Einstein condensate (BEC) loaded into a Fibonacci-modulated optical lattice constitutes a direct experimental implementation of the Fibonacci Cascade Conformal Layer (FCCL). The FCCL Born rule 𝑃(𝑛) = 𝜑^{−2𝑛}/√5 manifests as the timeof-flight momentum distribution after ballistic expansion: peaks at momenta 𝑘_𝑛 = 𝑛𝑘₀ carry amplitudes decaying as 𝜑−𝑛/5^{1/4}, with consecutive peak ratios exactly equal to 𝜑. Three additional predictions follow from the FCCL cascade structure: (i) quantum revival times spaced by the golden ratio, 𝑡_𝑛 = 𝑡₀𝜑^𝑛; (ii) critical temperatures for successive subbands scaling as 𝑇_𝑐(𝑛)/𝑇_𝑐(0) = 𝜑^{−𝑛/3}, with 𝑇_𝑐(3)/𝑇_𝑐(0) = 1/𝜑 exactly; (iii) a spectral bandwidth exactly equal to 2𝐽₀ (independent of the ratio 𝐽₁/𝐽₂), from the Fibonacci identity 𝜑 −1 + 𝜑−2 = 1. These predictions are accessible with current cold-atom technology and constitute the first direct laboratory test of the FCCL Born rule that does not require a particle collider.