Bell Inequalities and the Classical-Quantum Boundary in the Fibonacci Cascade

We derive the Bell correlations of the Fibonacci Cascade Conformal Layer (FCCL) bilateral state — the quantum state whose Born rule probability distribution follows inverse powers of the golden ratio, normalized by the square root of five. The central result is the Fibonacci Correlation Theorem: the FCCL correlation function, evaluated at angles that are simple fractions of a full rotation (one-sixth, one-quarter, one-half), takes exact reciprocal Fibonacci values (one-half, one-third, one-fifth). These identities follow algebraically from the Fibonacci recursion and hold without approximation. The CHSH Bell inequality analysis reveals a sharp classical-quantum boundary determined by the measurement type. With diagonal measurements — functions of the cascade level, such as energy or position — the CHSH value is exactly 2 (the classical bound). The bilateral cascade state has perfect classical correlations in the cascade position basis: the cascade level acts as a local hidden variable for diagonal observations. With off-diagonal measurements — hopping operators that create coherence between adjacent cascade levels — the CHSH value reaches the quantum bound of 2 times the square root of 2 (Tsirelson). This boundary is not an approximation: classical physics corresponds to cascade-diagonal measurements; quantum mechanics to cascade-off-diagonal ones. The quantum-classical transition is a geometric property of the Fibonacci cascade, not a scale-dependent phenomenon. In Fibonacci-lattice Bose-Einstein condensates, the two measurement types correspond to time-of-flight imaging (diagonal, classical correlations) and Ramsey spectroscopy (off-diagonal, quantum correlations). The ratio of consecutive Fibonacci correlations is itself a Fibonacci ratio, providing a precision spectroscopic test accessible with current cold-atom technology.