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: $$\lambda_3 \approx 0.1482 \quad \Rightarrow \quad \hat{\pi} = \sqrt{\frac{3}{2\lambda_3}} \approx 3.181$$ with only 1.25% deviation from the true value. Key Equation $$\pi = \sqrt{\frac{3}{2\lambda_3^*}}$$ 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.
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⇒π^=2λ33≈3.181 with only 1.25% deviation from the true value. Key Equation π=2λ3∗3 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.
Quantum Structural Benchmark (QSB): Evaluating Quantum Devices via π-Reconstruction Fidelity Overview We propose a novel quantum device benchmark—the Quantum Structural Benchmark (QSB)—based on the reconstruction accuracy of π from GHZ state measurements. Unlike conventional benchmarks (T₁/T₂, gate error rates, etc.), QSB measures how faithfully a quantum device reproduces fundamental geometric structure constants embedded in quantum mechanics. Key Metric: Structural Constant Fidelity (SCF) SCF = 1 - |π̂ - π| / π Where π̂ = √(3/2λ₃) is derived from Mermin inequality measurements on GHZ states. SCF Value Interpretation 1.000 Perfect (ideal simulator) 0.99+ Excellent (best hardware) 0.95-0.99 Good (typical NISQ) < 0.95 Structural degradation Main Results Measurement Value Mermin violation >700σ above classical bound Structure constant λ₃ 0.1482 ± 0.0006 Reconstructed π 3.181 SCF 0.9875 Platform IBM Quantum ibm_torino Paradigm Shift Aspect Conventional QSB/SCF Evaluates Gate quality Structural fidelity Question "How fast/quiet?" "How faithful to reality?" Type Engineering metric Physics metric Companion Paper This work builds upon: Quantum Buffon's Needle: Deriving π from the GHZ Structure Constant λ₃*DOI: 10.5281/zenodo.17855223 Files Included quantum_structural_benchmark.pdf — Full manuscript qsb.py — Complete QSB library (Python) README.md — Documentation Significance SCF could become the "LINPACK of quantum computing" — a universal metric for comparing quantum devices by their ability to reproduce fundamental physical constants. This represents a paradigm shift from hardware-centric to physics-centric quantum device evaluation.
I present a perspective in which the mathematical constant π, traditionally considered a mere geometric ratio of a circle’s circumference to its diameter, emerges as a fundamental invariant in quantum physics. While π has long been recognized for its central role in mathematics and classical physics—appearing in trigonometry, wave phenomena, harmonic oscillators, and statistical distributions—its ubiquity in the quantum domain reveals a deeper structural significance. In this work, I demonstrate that π is indispensable for the normalization of quantum wavefunctions, appearing naturally in Gaussian integrals, and is equally essential in Fourier transforms, ensuring the unitarity of transformations between conjugate variables such as position and momentum. Beyond these mathematical appearances, π is embedded in the very definition of Planck’s reduced constant (ħ = h / 2π), suggesting that it underlies the quantization of action and the discrete structure of phase space. By framing π as a quantum invariant, I argue that it is not merely an artifact of geometry or algebra but a universal constant that governs coherence, symmetry, and continuity in physical systems. Its recurring presence in quantum statistics, probability distributions, and density-of-states calculations indicates that π plays a central role in maintaining internal consistency within the quantum framework. This article inaugurates a series aimed at exploring π as a fundamental building block of quantum reality, bridging concepts from classical physics, quantum mechanics, and statistical theory. By revealing π as a structural constant rather than a mathematical convenience, I offer new perspectives on the underlying architecture of nature, with implications ranging from fundamental theory and quantum computation to cosmology and the interpretation of physical laws.
This article presents a novel perspective on the mathematical constant π, exploring its role as a fundamental invariant in quantum physics. Traditionally viewed as a geometric ratio, π is shown to appear ubiquitously in classical and quantum systems, including wavefunction normalization, Fourier transforms, quantum statistics, and Planck’s reduced constant. I argue that π is not merely a numerical artifact, but a structural constant underlying coherence, symmetry, and periodicity across physical systems. The work bridges classical physics, quantum mechanics, and statistical theory, proposing that geometry itself may emerge as a consequence of π rather than the other way around. This manuscript includes conceptual illustrations, diagrams of Gaussian wavefunctions, Fourier transform symmetry, complex amplitude topologies, and cosmic representations of π, highlighting its universality from subatomic to cosmological scales. This research offers new insights into the foundations of quantum theory, quantum field theory, cosmology, and quantum computation, positioning π as a guiding principle of physical reality.
In this article, I explore the profound and recurrent emergence of the mathematical constant π within the foundations of quantum mechanics. Through detailed examination of three cornerstone systems—the quantum harmonic oscillator, the hydrogen atom, and the Fourier duality linking wave and particle domains—I demonstrate that π is not merely a geometrical artifact, but a structural necessity for the internal consistency of quantum theory. In the harmonic oscillator, π arises through Gaussian normalization, ensuring that the total probability of a quantum state remains unity. In the hydrogen atom, π governs the spherical harmonics that describe the spatial symmetry of atomic orbitals, embedding π directly into the fabric of atomic structure. In Fourier transformations, π regulates the conversion between conjugate variables, such as position and momentum, preserving the unitarity and coherence of quantum information. These manifestations point toward a unifying interpretation: π acts as the mathematical signature of equilibrium between the discrete and the continuous, between quantization and continuity, probability and certainty. Far from being an incidental constant, π represents the invisible regulator of quantum harmony—the silent measure that maintains coherence and symmetry throughout the quantum universe.
β = 2√2 as a Geometric Fixed Point: From Quantum Correlations to Galactic Dynamics Overview This work addresses a foundational question in quantum mechanics: Why does nature permit quantum entanglement up to the Tsirelson bound (β = 2√2), but no further? We demonstrate that β = 2√2 emerges as a geometric fixed point from three scale-independent interface constraints, requiring no reference to quantum postulates such as Hilbert spaces or superposition. This geometric derivation suggests that quantum correlations may reflect deeper structural necessities rather than being fundamental axioms. Key Results 1. Geometric Derivation (Theorem 3.5) Under three interface constraints—binary freezing (G1), interface separability (G2), and minimal-distortion lock-in (G3)—we prove that the maximal admissible CHSH correlator is uniquely: β = 2√2 ≈ 2.828 This is derived from Clifford algebra structure at a pre-quantum interface layer, without invoking quantum mechanics. 2. Experimental Validation Reanalyzing CHSH inequality measurements on IBM Quantum hardware (ibm_kyiv, 127-qubit processor), we obtain: S = 2.797 ± 0.012 This represents 69σ agreement with the geometric prediction, confirming that the interface constraints are physically realized. 3. Scale Invariance We prove that β = 2√2 remains invariant under coarse-graining maps, predicting its manifestation across: Quantum scale (10⁻¹⁵ m): CHSH correlations ✓ Confirmed Atomic scale (10⁻¹⁰ m): Mode coupling (Predicted) Galactic scale (10²⁰ m): Radial acceleration relation ✓ Derived Cosmological scale (10²⁶ m): CMB parity (Predicted) 4. Astrophysical Prediction Without fitting to galaxy data, we derive a radial acceleration response function Φ_β that: Reproduces the Newtonian limit with 0.037 dex precision Exhibits systematic deviations from empirical exponential forms Provides a falsifiable prediction through functional form differences (algebraic √ vs exponential) Methodology Theoretical Framework: Clifford geometric algebra, interface geometry, coarse-graining theory Experimental Validation: IBM Quantum Experience (GHZ states, CHSH inequality) Astrophysical Analysis: SPARC database comparison (limited validation) Statistical Significance: 69σ for quantum confirmation Significance This work proposes a paradigm shift from asking "what are the laws?" to asking "why these laws?" The derivation suggests: Quantum bounds as geometry: The Tsirelson bound may reflect geometric necessities rather than quantum axioms Cross-scale universality: Diverse phenomena (quantum correlations, galaxy dynamics) may probe the same underlying interface structure Falsifiable framework: Predicted functional forms enable empirical tests at multiple scales Data and Code Availability This deposit includes: Full manuscript (18 pages, peer-review submission format) All figures (8 publication-quality images) Experimental data documentation (IBM Quantum CHSH measurements) Analysis code (Python scripts for figure generation) Complete documentation (README, citation metadata, license) All materials are released under CC BY 4.0 license for maximum reproducibility and reuse. Publication Status Submitted to: Foundations of Physics (February 2026) Version History: v1.0: Initial deposit (February 2, 2026) v2.0: Author affiliation standardized (same day update) Limitations and Future Work Acknowledged limitations: Galactic-scale predictions require comprehensive SPARC dataset analysis (planned companion paper) Atomic-scale predictions await experimental verification Cosmological predictions require CMB higher-point function analysis Planned extensions: Full statistical analysis of 153 SPARC galaxies Detailed CMB selection rule derivation Experimental proposals for atomic-scale tests Contact Author: Takayuki TakagiORCID: 0009-0003-5188-2314Email: lemissio@gmail.comLocation: Higashimatsuyama, Saitama, Japan For questions about the theoretical framework, experimental validation, or collaboration opportunities, please contact via email. Citation If you use this work, please cite: @misc{takagi2026beta, title={β = 2√2 as a Geometric Fixed Point: From Quantum Correlations to Galactic Dynamics}, author={Takagi, Takayuki}, year={2026}, publisher={Zenodo}, doi={10.5281/zenodo.18454720}, url={https://doi.org/10.5281/zenodo.18454720}, note={Submitted to Foundations of Physics} } Keywords: Tsirelson bound, geometric fixed point, interface geometry, quantum foundations, Bell inequalities, CHSH inequality, radial acceleration relation, dark matter, galactic dynamics, scale invariance, Clifford algebra License: Creative Commons Attribution 4.0 International (CC BY 4.0) *This work was conducted independently with no institutional funding or conflicts of interest.*
In this article, I explore the fundamental role of the mathematical constant π in quantum mechanics, beyond its traditional geometric interpretation. I demonstrate that π naturally emerges in the normalization of wavefunctions, the structure of quantum probability distributions, and the statistical frameworks of Bose–Einstein and Fermi–Dirac systems. Through analytical reasoning and numerical simulations, π is shown to act as a structural constant of probabilistic space, linking geometry, quantization, and the coherence of quantum states. This work highlights that π is not merely a geometric ratio but a universal invariant underpinning the architecture of quantum reality, ensuring that probability, information, and symmetry remain consistent across scales.