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.
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.
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.
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.*
Paper #45 gave the foam two propagation channels: the wall channel (the face Laplacian L, local, causal) and the void channel (ηV, antipodal, non-local, no signalling). This paper supplies the two pieces of quantum mechanics that the corpus had until now imported rather than derived: the Born rule and the entangled pair state. First, the Born rule is reduced to two structural premises about the void-pair imprint event. Phase-blindness (the imprint probability depends only on a channel's amplitude modulus) and fine-graining additivity (a local refinement of detector channels cannot change coarse probabilities) force p_k = |c_k|² uniquely, by the Cauchy functional equation. Both premises are then discharged in turn. Fine-graining additivity is equivalent to no-signalling, which Paper #45 §8 derives from bulk incompressibility. Phase-blindness follows from substrate stationarity: a channel's phase is a time offset of its carrier, and no threshold functional driven by stationary substrate fluctuations can depend on a time offset; the same argument kills the linear response identically, so the trigger rate is quadratic in the amplitude at leading order. The Born exponent thus traces to two equation-of-state properties of the foam, incompressibility of the bulk and equilibration of the substrate. On a discrete substrate the counting measure is literal: channel k holds exactly N|c_k|² micro-quanta (Parseval). Second, the pair state of a displacement event is derived rather than selected. The twin map Θ = V∘K (antipodal map composed with conjugation, both existing corpus operators) is antiunitary and flips the torsion charge exactly. The pair state Σ_k |k⟩⊗Θ|k⟩ is proved basis-independent if and only if the twin map is antiunitary, which is the operator content of "opposite in every direction simultaneously". The resulting state is unique, maximally entangled, saturates the Tsirelson bound (CHSH = 2√2), carries exactly zero total torsion charge, and is one local unitary from the singlet of Paper #2. Third, the laboratory qubit is unique by symmetry: orientation-averaged observables commute with the cell's O_h action, Schur's lemma confines two-state structure to the two multiplicity-two sectors, the gauge pair is excluded, and the chiral doublet with routing observable C is all that remains. The η-tension between Papers #2 and #45 is also resolved: the void coupling scales the formation probability of entangled pairs (∝ η²) and never the strength of their correlations.
L’intrication quantique est un des phénomènes les plus intéressants et intriguant en Mécanique Quantique, et de surcroît en Théorie de l’Information Quantique. Ressource fondamentale pour le calcul quantique, son rôle dans l’efficacité et la fiabilité des protocoles ou algorithmes quantiques n’est toujours pas totalement compris. Dans cette thèse, nous étudions l’intrication quantique des états multipartites, et notamment la nature de sa présence dans les algorithmes quantiques. L’étude de l’intrication se fait d’un point de vue théorique, en utilisant principalement des outils issus de la géométrie algébrique.Nous nous intéressons alors aux algorithmes de Grover et de Shor et déterminons quelles sont les classes d’intrication présentes (ou non) dans ces algorithmes, et ceci constitue donc une étude qualitative de l’intrication. De plus, nous mesurerons l’intrication quantitativement, à l’aide de mesures algébriques et géométriques, et étudions son évolution tout au long des différentes étapes de ces algorithmes. Nous proposons également des interprétations géométriques originales de ces résultats numériques.D’autre part, nous cherchons également à développer et exploiter de nouveaux outils pour mesurer, caractériser et classifier l’intrication quantique. Ceci se fait dans un premier temps d’un point de vue mathématique en étudiant les singularités des hypersurfaces liées aux systèmes quantiques pour caractériser différentes classes d’intrication. Dans un second temps, nous proposons des candidats pour les états maximalement intriqués, notamment pour les états symétriques et fermioniques, en utilisant des polynômes invariants et une mesure géométrique de l’intrication pour quantifier l’intrication. Enfin, nous avons également adopté une approche de type Machine Learning, notamment en entraînant des réseaux de neurones artificiels de manière supervisée, afin de reconnaitre certaines variétés algébriques modélisant certains types d’intrication précis.
The Axiom Zero decomposition B + V = D (Bubble + Void = Displacement) is shown to map directly onto a two-channel lattice Hamiltonian H = L + η(I − V), where L is the face Laplacian (wall-mediated propagation) and V is the antipodal partner-face map (void-mediated correlation). The bubble propagates on the walls at speed c. The void appears instantaneously on the antipodal face through the incompressible bulk, with coupling η = exp(−d_bulk). The antipodal map V is an involution (V² = I) whose eigenvalues partition the face representation into even (+1) and odd (−1) sectors. The partition aligns exactly with the boson/fermion distinction: A₁g (photon), Eg (weak bosons), and T₂g (gluons) are EVEN; T₁u (fermion generations) and A₂u (Higgs) are ODD. In the operative (I − V) form the void term annihilates the even sector at the zone centre and shifts the odd sector by 2η there; at the zone face-centres and corner the single-cell odd eigenvalues are exact for any η (void-protection theorem). The earlier statement that odd modes are pushed down and that this reinforces symmetry breaking is withdrawn in v3.0. The combined propagator G(j,i;t) = ⟨j|exp(−Ht)|i⟩ sums over two kinds of paths: wall walks through shared edges (36 channels, local, causal, giving QFT) and void jumps through the bulk (7 antipodal pairs, non-local, acausal, giving entanglement). The complete path integral is the sum over both. The void coupling is taken from the bulk geometry (Tier 2, physically motivated form): η_sq = exp(−2√2) = 0.059 for sq-sq pairs and η_hx = exp(−√6) = 0.086 for hx-hx pairs. The void corrections are O(η) ≈ 6−9%, consistent with known sub-leading corrections to the coupling constants. The trace of L is 72; the void term adds 2η per odd mode at the zone centre and nothing at the protected points. v3.0 (October 2026): sign of the void term corrected to the L + η(I − V) form used by every lattice verification script; the claim that the void correction improves the Higgs-to-Z mass ratio from 0.14% to 0.06% is withdrawn (arithmetic error, and the A₂u level is void-protected); the void-protection theorem is recorded in §4.3; the SSB statement is revised.