Chaotic Origin of Tsirelson Bound

We report a rigorous, deterministic origin for quantum non-locality and the Tsirelson bound (2 2) by revealing profound macroscopic emergent properties hidden within exactly solvable chaos towers under a strictly fixed invariant Cauchy measure (γ = 1). While evaluating joint correlations via conventional real-axis micro-sampling is universally imprisoned within a clas- sical L1 metric space—where the lack of exponential mode-damping (e0 = 1) forces an infinite cascade of high-harmonic square-wave noises to remain unattenuated, collapsing the CHSH sum to the classical limit (|S| ≤ 1.5 < 2)—we demonstrate that this fundamental restriction is completely bypassed at the macroscopic layer. By utilizing unimodular real-axis observa- tion functions satisfying |gA(x)| = 1, the physical joint correlation is strictly bounded by unity (|E| ≤ 1), naturally preserving physical unitariness. By generalizing these observations as expan- x−insions over the complete orthonormal Cauchy-Szego ̋ basis en(x) = x+i of the Hardy space H2(H+), we establish a rigorous signal characterization of emergent non-locality. While real-axis discrete point-sampling retains all odd harmonic modes, the continuous parameter tracking of the ergodic ensemble acts as a topological Szeg ̋o projection. Under this projection, the residue theorem filters out high-order complex poles (n ≥ 2), isolating the fundamental mode n = 1 and instantaneously recovering the smooth quantum correlation cos(2∆θ). Without invoking any quantum-mechanical postulates, this geometric phase transition transforms the optimization of the CHSH sum into a constraint of geodesic quadrilaterals within an emergent hyperbolic bulk, providing a novel realization of the Anti-de Sitter / Conformal Field Theory (AdS/CFT) corre- spondence and the Ryu-Takayanagi holographic paradigm. Crucially, by deriving the Tsirelson limit as a purely geometric consequence of tracking statistical metadata over the complex topol- ogy, this framework demands no superluminal signal propagation across spacetime, thereby harmonizing emergent quantum non-locality with strictly local relativistic causality.