Based on the three axioms of Geometric Theory and all theorems in the 0.X series, this article reconstructs the theory of quantum entanglement with zero external assumptions. The work includes: (I) Within the framework of holographic screen pixel structure, strictly deriving the minimum resolvable angle Δξmin=π/(2√λ2); (II) Establishing the geometric definition of entanglement—two S-minima on the geometric constraint section Ση achieve geometric maximum entanglement through I-sector freezing and M-sector antipodal locking; (III) Proving the two-level theorem—the low-energy excitation space of a geometric maximum entanglement pair is exactly 2-dimensional, with its ground state spanned by the first zonal harmonic function of the M-sector; (IV) Under the physical mapping convention, deriving the correlation function C(→a,→b)=−→a⋅→b, where the readout directions →a,→b are unit vectors on the M-sector S3; (V) In the hypothetical limit λ2→∞, proving information field localization and that the algebraic upper limit of the CHSH combination is 2; (VI) With finite λ2 and accepting the correlation mapping, proving that the supremum of the CHSH combination is 2√2, with equality conditions →b1⊥→b2 and →a1∥(→b1+→b2), →a2∥(→b1−→b2); (VII) Under the external input assumption of double-slit topological defects, proving that a single S-minimum flowing around the double-slit topological defects produces a phase difference Δϕ=√λ1ΔL, with fringe spacing Δx=2πL/(√λ1d), and providing numerical verification after dimensional bridge conversion. The full text does not use wave function assumptions, does not use Bell's original derivation, and does not introduce Planck's constant as an external input.
Based on the three axioms of Geometric Theory and all theorems in the 0.X series, this article reconstructs the theory of quantum entanglement with zero external assumptions. The work includes: (I) Within the framework of holographic screen pixel structure, strictly deriving the minimum resolvable angle Δξmin=π/(2√λ2); (II) Establishing the geometric definition of entanglement—two S-minima on the geometric constraint section Ση achieve geometric maximum entanglement through I-sector freezing and M-sector antipodal locking; (III) Proving the two-level theorem—the low-energy excitation space of a geometric maximum entanglement pair is exactly 2-dimensional, with its ground state spanned by the first zonal harmonic function of the M-sector; (IV) Under the physical mapping convention, deriving the correlation function C(→a,→b)=−→a⋅→b, where the readout directions →a,→b are unit vectors on the M-sector S3; (V) In the hypothetical limit λ2→∞, proving information field localization and that the algebraic upper limit of the CHSH combination is 2; (VI) With finite λ2 and accepting the correlation mapping, proving that the supremum of the CHSH combination is 2√2, with equality conditions →b1⊥→b2 and →a1∥(→b1+→b2), →a2∥(→b1−→b2); (VII) Under the external input assumption of double-slit topological defects, proving that a single S-minimum flowing around the double-slit topological defects produces a phase difference Δϕ=√λ1ΔL, with fringe spacing Δx=2πL/(√λ1d), and providing numerical verification after dimensional bridge conversion. The full text does not use wave function assumptions, does not use Bell's original derivation, and does not introduce Planck's constant as an external input.
The quantum entanglement theory is reconstructed within the Encoded Orbital pixel structure framework. The geometric definition of entanglement is established — two S-minima on Σ_η realize geometric maximal entanglement via ℐ-Sector freezing and ℳ-Sector antipodal locking. The Two-Level Theorem is proved — the low-energy non-trivial excitation space of a geometrically maximally entangled pair is exactly 2-dimensional. Under the physical mapping conventions, the correlation function is derived. The CHSH combination supremum is proved to be . Consummate Reproduction serves as the unique physical mapping input.
The quantum entanglement theory is reconstructed within the Encoded Orbital pixel structure framework. The geometric definition of entanglement is established — two S-minima on Σ_η realize geometric maximal entanglement via ℐ-Sector freezing and ℳ-Sector antipodal locking. The Two-Level Theorem is proved — the low-energy non-trivial excitation space of a geometrically maximally entangled pair is exactly 2-dimensional. Under the physical mapping conventions, the correlation function is derived. The CHSH combination supremum is proved to be . Consummate Reproduction serves as the unique physical mapping input.
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.*
How does a purely geometric projection — without Hilbert space, tensor products, or the Born rule — recover the exact quantum correlation E(a,b) = −cos θ and saturate Tsirelson's bound 2√2?We derive both results from the involution φ(x) = 9 − x on the digit alphabet {1, …, 9}, which generates four inversion pairs summing to 9 with barycenter μ = 4.5 and projection constant k_arc = 1/2. A hidden variable λ emitted by the source is projected sequentially onto detector axes a and b, each projection costing k_arc = 1/2. The product of two such projections yields E(a,b) = −cos θ, matching the quantum prediction exactly. The CHSH sum of four correlations at optimal angles gives S = 2√2, saturating Tsirelson's bound.The projection constant k_arc = 1/2 is established by seven independent paths: (I) global — barycenter over closure, 4.5/9; (II) local — half-span of the maximal pair, 3.5/7; (III) per-pair — every inversion pair individually encodes half-span/span = 1/2; (IV) positional — exact center of the sequence 1–8, with four elements on each side; (V) cadastral — the barycenters of the three row-pairs {2,4}, {5,7}, {8,1} average to 4.5; (VI) geometric — the interface mirror 11 divided by π = 22/7 gives 3.5, then 3.5/7 = 1/2; (VII) linker — 5.5/11 = 1/2, the barycenter of the phase cycle over the interface operator. Seven paths, sharing neither data nor method, converging to the same constant. This excludes numerical coincidence.No Hilbert space is assumed. No tensor product is constructed. No Born rule is invoked. The correlation is geometric, not probabilistic.New in v4.0: We add the complete arithmetic foundation — the Universal Fold Theorem proving that every multiplication table folds into inversion pairs summing to 9, the visual multiplication table mirrors, and the proof that base 10 is not arbitrary. We introduce entanglement as bipolar inversion: if x + φ(x) = 9, knowing x determines φ(x) instantaneously at any distance — not by causal signal but by structural constraint. This is quantum entanglement without mystery. Einstein's objection to "spooky action at a distance" is resolved: the correlation is not action, it is structure. The minimum of relation is 2 = DR(11), the interface operator — the smallest cost of knowing the other.Keywords: Tsirelson's bound, quantum correlations, Bell inequality, CHSH, geometric projection, hidden variables, digital roots, inversion symmetry, entanglement, EPR, sevenfold convergence.
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.
We examine the claim, made in the corpus of the 5D Solitonic Theory (TS5D), to derivequantum mechanics the Schrödinger equation, the Born rule, spin, entanglement, andthe Tsirelson bound 2√2 from a deterministic geometric substrate whose compact fthdimension e = S1/Z6 represents energy. We separate two levels of structure: (a) the discretequantum numbers (KaluzaKlein tower, topological sectors, spin-statistics sign), which thediscrete/abelian geometry provides; and (b) the Hilbert-space structure (complex superposition,SU(2) qubit, entanglement, 2√2), which it does not provide without further input.We establish, through four independent and complete demonstrations (local CHSH bound;Tsirelson bound; geometric quantization of S2; two-soliton correlation), that: (i) the qubit isrealizable without any new eld, as a topological sector of a KaluzaKlein monopole (Hopfbration, c1 = 1), but that no internal principle of TS5D forces it; (ii) the value 2√2 isnot derivable from a deterministic local substrate, because Bell's theorem is a theorem, nota circumventable empirical constraint. The defensible form of TS5D is a geometrized non-local Bohm: a deterministic, non-local theory whose 4D projection is quantum mechanics,the quantum content entering at two named, minimal points. The fth dimension e givesnon-locality a deterministic geometric mechanism, without rendering it local.Nature of the text. This is a foundational / clarication paper: it establishes the minimalconditions under which a deterministic 5D geometry can reproduce quantum mechanics, andthe theorem-level obstacles (Bell, Tsirelson, qubit topology, chirality, Born rule) that boundthem. It claims no new complete derivation.