We report the experimental demonstration of Bell violations reaching S = 4.0000, exceeding the Tsirelson bound of S ≤ 2√2 ≈ 2.828. Using curved geometric quantum theory with fundamental curvature parameter κ = 0.893469101829281, we observe universal super-quantum correlations across all tested scales (0.1 to 2.0). The quantum-classical boundary occurs at κ_quantum = 1/(2κ) = 0.559616442220894, which emerges as a new fundamental constant separating classical from quantum behavior. All quantum states reside on the π/2 = 1.570796 geometric horizon. This discovery demonstrates that super-quantum correlations are physically possible and suggests that standard quantum mechanics is a limiting case of a more general geometric theory. We present the first theoretical demonstration of Bell parameter S=4.0, achieving the PR-box limit previously considered unattainable in physical systems. Using a curved geometric quantum framework with fundamental curvature constant κ=0.893469101829281, we identify a critical scale parameter α_c=0.560 at which quantum correlations exceed the Tsirelson bound S≤2√2≈2.828. Our 4+4 paired qubit architecture exhibits deterministic correlations that saturate the maximum possible Bell violation, enabling super-quantum computation with O(N¹/³) search complexity. Experimental verification is immediately feasible with current quantum technology
We report the experimental demonstration of Bell violations reaching S = 4.0000, exceeding the Tsirelson bound of S ≤ 2√2 ≈ 2.828. Using curved geometric quantum theory with fundamental curvature parameter κ = 0.893469101829281, we observe universal super-quantum correlations across all tested scales (0.1 to 2.0). The quantum-classical boundary occurs at κ_quantum = 1/(2κ) = 0.559616442220894, which emerges as a new fundamental constant separating classical from quantum behavior. All quantum states reside on the π/2 = 1.570796 geometric horizon. This discovery demonstrates that super-quantum correlations are physically possible and suggests that standard quantum mechanics is a limiting case of a more general geometric theory. We present the first theoretical demonstration of Bell parameter S=4.0, achieving the PR-box limit previously considered unattainable in physical systems. Using a curved geometric quantum framework with fundamental curvature constant κ=0.893469101829281, we identify a critical scale parameter α_c=0.560 at which quantum correlations exceed the Tsirelson bound S≤2√2≈2.828. Our 4+4 paired qubit architecture exhibits deterministic correlations that saturate the maximum possible Bell violation, enabling super-quantum computation with O(N¹/³) search complexity. Experimental verification is immediately feasible with current quantum technology
Information Dynamics provides a novel response to the Einstein-Bohr debate: quantum nonlocality is not superluminal signaling, but rather the projection of global coherence in virtual space; no parallel universes are required—measurement is merely a unidirectional projection from the virtual to the real component of information; the geometric constant 0.25 in Bell inequality violations originates from the intrinsic curvature of the two-dimensional virtual space (SU(2)). The violation of the Clauser-Horne (CH) inequality is a core result in the experimental verification of Bell's theorem. In 2025, the BESIII experiment measured the CHSH parameter S = 2.800 ± 0.010 for entangled ΛΛ¯ pairs with exceptional precision, unambiguously violating classical local realism. Within the Information Dynamics framework, we treat quantum systems as Information Superconductors—featuring global coherence in virtual space, information conservation, and a dynamic real-virtual coupling matrix K(x). Starting from two primitive axioms, we rigorously prove that virtual space possesses constant curvature K = 1/4, from which we derive in a unified manner: The Tsirelson bound for the CH inequality: (√2−1)/2 ≈ 0.2071 (projection of one-dimensional geodesic arc length); The quantum upper bound for the I3322 inequality: 0.25 (projection of two-dimensional spherical area). More importantly, we propose a novel testable prediction: for partially entangled pure states, the Information Dynamics prediction for the CH violation, S^ID_CH(η) = (√(1+sin 2η)−1)/2, is systematically larger than the standard quantum mechanical prediction, S^QM_CH(η) = (√(1+sin² 2η)−1)/2, with the two converging only at maximal entanglement. Taking η=30° as an example, the relative deviation exceeds 13%, well within the resolution of current photonic entanglement experiments, thereby providing a direct falsification pathway for Information Dynamics. First-Principles Numerical Verification: We further perform first-principles numerical simulations starting from the generalized Ginzburg-Landau equation, using FFT spectral methods on 2D lattices (up to 128×128). The results show that in the information superconducting phase (ρ_target ≳ 0.85), the effective curvature K_eff(C) ∝ C emerges naturally from the dynamics with ΔAIC = -5.67 and 91% Monte Carlo support, and the fitted exponent α = 1.02 ± 0.06, in excellent agreement with the geometric prediction α = 1. A quick version (64×64, 10 seeds, running in ~25 seconds on a personal computer) also strongly favors the linear model (ΔAIC < -60) across all tested parameter sets. These results provide strong first-principles numerical support for the linear projection hypothesis of Information Dynamics. This work not only answers "why" the geometric scale is 0.25, but also elevates imaginary space from a mathematical tool to a physical entity with measurable curvature, offering a new information-theoretic perspective on the century-old Einstein-Bohr debate.
The foundational conflict between the probabilistic non-locality of quantum mechanics and the geometric determinism of general relativity remains the central challenge of modern physics. This paper addresses the tension by proposing a locally causal framework where Bell-inequality violations emerge as a direct consequence of the intrinsic curvature of a discretized state-space manifold. By violating the “Measurement Independence” assumption through a global geometric constraint, we eliminate the need for non-local “spooky action” while preserving the causal structure of spacetime. Our methodology utilizes a novel “Ontic Tensor” mapping that bridges Riemannian metrics with p-adic state-space discretization. The results demonstrate that the Ontic Tensor model reproduces sinusoidal Bell correlations with an R2>0.99 against standard quantum mechanical predictions. Furthermore, we identify a falsifiable “entanglement saturation” limit, predicting that entanglement fidelity must decay beyond m=log2(p) qubits due to the finite information capacity of the invariant set. The implications of this research suggest a return to the Einsteinian ideal of local realism, where randomness is viewed as an emergent mask for underlying geometric complexity. By resolving the measurement problem through state convergence on a fractal attractor, we provide a non-perturbative pathway toward the unification of quantum field theory and general relativity. This work addresses seven critical gaps in the literature, establishing Superdeterminism as a rigorous, testable, and parsimonious framework for the future of physics.
Quantum theory is in principle compatible with processes that violate causal inequalities, an analogue of Bell inequalities that constrain the correlations observed by sets of parties operating in a definite causal order. Since the introduction of causal inequalities, determining their maximum quantum violation, analogue to Tsirelson's bound for Bell inequalities, has remained an open problem. Here we provide a general method for bounding the violation of arbitrary causal inequalities, establishing limits to the correlations achievable by arbitrary local experiments and by arbitrary quantum processes with indefinite causal order. We prove that the maximum violation is generally smaller than the algebraic maximum of the corresponding correlation, and determine Tsirelson-like bounds for a class of causal inequalities including some of the most paradigmatic examples. Our results motivate a search for physical principles characterizing the boundary of the set of quantum correlations with indefinite causal order.
Quantum theory is in principle compatible with processes that violate causal inequalities, an analogue of Bell inequalities that constrain the correlations observed by sets of parties operating in a definite causal order. Since the introduction of causal inequalities, determining their maximum quantum violation, analogue to Tsirelson’s bound for Bell inequalities, has remained an open problem. Here we provide a general method for bounding the violation of arbitrary causal inequalities, establishing limits to the correlations achievable by arbitrary local experiments and by arbitrary quantum processes with indefinite causal order. We prove that the maximum violation is generally smaller than the algebraic maximum of the corresponding correlation, and determine Tsirelson-like bounds for a class of causal inequalities including some of the most paradigmatic examples. Our results motivate a search for physical principles characterizing the boundary of the set of quantum correlations with indefinite causal order. Quantum theory has been shown to be compatible with processes violating a definite causal order, but no detailed analysis of the maximum allowed degree of violation has been carried out so far. Here, the authors fill this gap by providing the analogue of Tsirelson’s bound for the violation of causal inequalities.
When two particles are created together in a single quantum event, they can become entangled: their properties remain correlated no matter how far apart they travel. Measure one, and the outcome of measuring the other is no longer independent. This correlation is the central puzzle of quantum mechanics. Classical physics sets a hard limit on how strong such correlations can be. The CHSH inequality (Clauser, Horne, Shimony, and Holt, 1969) formalises that limit: for any theory in which each particle carries its own independent set of predetermined answers, the correlation measure S cannot exceed 2. Bell-type experiments consistently violate this bound, with the maximum ideal value predicted by quantum mechanics being S = 2√2. Something does not fit. Several explanations have been proposed. Some invoke extra dimensions. Others appeal to wormhole geometries connecting the particles. The mainstream position is that the outcomes "collapse and become correlated," which states the observation without explaining the mechanism. All of these either add dramatic ontological furniture or repackage the mystery as an answer. This paper proposes a simpler reading. The value 2√2 is not a mystery to be explained by exotic physics. It is the geometric signature of a jointly structured bipartite state whose correlation was fixed at the source, at the moment of entanglement, not at measurement time. The two particles are not two independent objects that later communicate. They are two readout surfaces of a single joint preparation. The cosine correlation law, the Tsirelson bound, and the full Bell violation follow directly from the geometry of the two-qubit state space and the relative orientation of the measurement settings. No signal, no extra dimension, no new physics is required. Three companion papers in this series explored the opposite hypothesis: The Speed of Light and Its Limits [1] examined whether entanglement could circumvent relativistic constraints; Quantum Entanglement as a Communication Channel [2] proposed a synchronized-clock experiment to detect timing signatures; and Parametrizing Correlation Timing [3] developed a Lorentz-structured ansatz for concurrence-dependent delays. If those experimental proposals return null, the present reading is reinforced: not by proof, but by elimination of the alternatives that require more. A final section develops geometric and regime-based frameworks consistent with the source-structural reading, including the measurement torus as the natural geometry of Bell experiments, a jointness index based on Concurrence, and connections to emergent spacetime programmes. These are offered as directions for future development.
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.*