β = 2√2 as a Geometric Fixed Point: From Quantum Correlations to Galactic Dynamics

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 (ibmₖyiv, 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: @misctakagi2026beta, 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. *