Quantum Buffon's Needle: Deriving π from the GHZ Structure Constant λ₃*

Quantum Buffon's Needle: Deriving π from the GHZ Structure Constant Overview We demonstrate that the fundamental mathematical constant π can be derived from the quantum structure constant λ₃* = 3/(2π²), which characterizes the phase-space capacity of three-qubit entangled systems. By measuring the structural coherence of GHZ states on IBM Quantum hardware (ibm_torino), we obtained: λ₃ ≈ 0.1482 ⇒ π̂ = √3/2λ₃ ≈ 3.181 with only 1.25% deviation from the true value. Key Equation π = √3/2λ₃^* This "Quantum Buffon's Needle" experiment establishes π as a measurable property of quantum geometry, paralleling Buffon's classical probability-based derivation while revealing deeper connections between quantum information theory, spherical geometry, and physical reality. Main Results Measurement Value Mermin inequality violation >700σ above classical bound Measured structure constant λ₃ = 0.1482 ± 0.0006 Reconstructed π 3.181 Relative error 1.25% Platform IBM Quantum ibm_torino (Heron r2) Files Included quantum_buffons_needle_pi_derivation.pdf — Full manuscript quantum_pi_scan.py — Python script for multi-backend π derivation experiments README.md — Documentation Significance This result establishes a novel paradigm: π is not merely an abstract mathematical entity but a measurable physical constant embedded in the geometric fabric of quantum state space. The method enables evaluation of quantum devices by their ability to reproduce fundamental geometric constants—a new direction in quantum benchmarking.