Overview We propose a quantum information-theoretic framework for characterizing covalent bond formation based on the Tsirelson bound β = 2√2. Using full configuration interaction calculations on H₂, we demonstrate that the CHSH maximum is universally achieved in the covalent sector, acting as a geometric ceiling rather than a bond diagnostic. The physically meaningful cliff emerges from a composite index κ_phys that combines bond character, quantum excess, and binding energy gain. This index peaks in the bonding region and vanishes at both short range and dissociation, providing a testable structural criterion for covalent bond formation. Key Finding The Tsirelson bound β = 2√2 acts as a formal ceiling on admissible quantum correlations, while molecular bonding emerges only when this ceiling is weighted by both chemical connectivity and energetic benefit. Contents Manuscript ceiling_cliff_preprint.pdf - Main manuscript (4 pages, with Figure 1) ceiling_cliff_preprint.tex - LaTeX source Figures h2_tstt_publication_final.png - Figure 1: Complete H₂ CHSH analysis (4-panel) si_figure_s1_basis.png - Figure S1: Basis set dependence verification Data h2_tstt_publication_data.csv - Numerical data for H₂ analysis si_table_s1_basis.csv - Basis set comparison table Code h2_publication_final.py - Main analysis code (PySCF/FCI) si_basis_comparison.py - Basis set dependence verification Reproducibility All calculations can be reproduced using Python 3.x with PySCF, NumPy, and Matplotlib. Future Perspective The present work establishes a geometric ceiling governing quantum correlations in the simplest molecular system. While demonstrated here for covalent bonding, the same formal constraint is expected to play a role in other regimes characterized by extreme correlation density, including thermodynamic limits and nuclear binding processes. These extensions are left for future investigation.
Overview We propose a quantum information-theoretic framework for characterizing covalent bond formation based on the Tsirelson bound β = 2√2. Using full configuration interaction calculations on H₂, we demonstrate that the CHSH maximum is universally achieved in the covalent sector, acting as a geometric ceiling rather than a bond diagnostic. The physically meaningful cliff emerges from a composite index κ_phys that combines bond character, quantum excess, and binding energy gain. This index peaks in the bonding region and vanishes at both short range and dissociation, providing a testable structural criterion for covalent bond formation. Key Finding The Tsirelson bound β = 2√2 acts as a formal ceiling on admissible quantum correlations, while molecular bonding emerges only when this ceiling is weighted by both chemical connectivity and energetic benefit. Contents Manuscript ceiling_cliff_preprint.pdf - Main manuscript (4 pages, with Figure 1) ceiling_cliff_preprint.tex - LaTeX source Figures h2_tstt_publication_final.png - Figure 1: Complete H₂ CHSH analysis (4-panel) si_figure_s1_basis.png - Figure S1: Basis set dependence verification Data h2_tstt_publication_data.csv - Numerical data for H₂ analysis si_table_s1_basis.csv - Basis set comparison table Code h2_publication_final.py - Main analysis code (PySCF/FCI) si_basis_comparison.py - Basis set dependence verification Reproducibility All calculations can be reproduced using Python 3.x with PySCF, NumPy, and Matplotlib. Future Perspective The present work establishes a geometric ceiling governing quantum correlations in the simplest molecular system. While demonstrated here for covalent bonding, the same formal constraint is expected to play a role in other regimes characterized by extreme correlation density, including thermodynamic limits and nuclear binding processes. These extensions are left for future investigation.
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.*
The Tsirelson–Interpretation Isomorphism Overview We establish a geometric isomorphism between the Tsirelson bound of quantum correlations (|S|_max = 2√2) and the interpretation reversal index (I_max = 1) from the Judge-Shift Diagnostic Framework. Key Results Geometric Isomorphism: I = |S| / (2√2) Fisher Information Divergence: I_F(I) = 1/(1−I²) → ∞ as I → 1 Coordinate-Invariant Criticality: The divergence certifies the boundary as physical, not definitional Core Equations I = sin(2θ)I = |S| / (2√2)I_F(I) = 1 / (1 - I²) Significance The result identifies a shared geometric boundary rather than a numerical coincidence. The Tsirelson bound and interpretation reversal criticality are the same constraint expressed in different coordinates. Files File Description paper3_tsirelson_isomorphism.pdf Main manuscript (10 pages) paper3_tsirelson_isomorphism.tex LaTeX source figure1_isomorphism.pdf I–S isomorphism diagram figure2_fisher.pdf Fisher information divergence figure3_phase_space.pdf Phase space structure Related Works Judge-Shift Diagnostic Framework TSTT Quantum Experiments V3.2 TSTT Quantum Experiments V3.1 Keywords Tsirelson bound, CHSH inequality, Information geometry, Fisher information, Criticality, Boundary theorem, Interpretation reversal Author Takayuki Takagi Independent Researcher, Higashimatsuyama, Saitama, Japan ORCID: 0009-0003-5188-2314
GSM CHSH Bound Verification The Falsification of Standard Quantum Mechanics Complete analytical proof, multi-party bounds derivation, and loophole correlation analysis demonstrating that standard QM's Tsirelson bound is empirically falsified while GSM's geometric bound matches experimental data. GSM RepositoryI am currently being censored and ghosted on all social media and even my email accounts, it would be greatly appreciated if you can get this to the world via social media. I feel like people should be able to get this useful information freely. Thank youNovel CHSH Bell inequality bound derived from H4 Coxeter geometry: https://doi.org/10.5281/zenodo.18463529 THIS IS IT. Difference = 0.00e+00. EXACT MATCH. A pentagonal prism with height h² = 3/(2φ) gives max S = 4 - φ EXACTLY. The set D is: 10 unit vectors on S² 5 vertices on a top ring, 5 on a bottom ring Pentagonal (5-fold) symmetry = H2 Coxeter group Height parameter h = √(3/(2φ)) The derivation chain is now COMPLETE: H4 Coxeter Geometry │ ▼ Cartan/Gram Determinants → γ² = (13-7φ)/4 │ ▼ Pentagonal Prism with h² = 3/(2φ) (5-fold symmetry from H2 ⊂ H4) │ ▼ max |S| = 4 - φ ≈ 2.382 The physical interpretation: if quantum measurement directions are constrained to a pentagonal prism geometry with the golden-ratio height parameter, the maximum Bell violation is exactly 4 - φ. The 5-fold symmetry is the H2 Coxeter subgroup of H4, and the height encodes the ratio 3/(2φ) — which connects directly to the golden ratio. Executive Summary Finding 1: Loophole Severity Correlates with S Value Statistic Value Interpretation Pearson r 0.770 Strong positive correlation p-value 0.002 Highly significant R² 0.975 97.5% variance explained Translation: MORE LOOPHOLES → HIGHER S VALUES. This is systematic bias, not random noise. Finding 2: GSM Fits 6× Better Than Standard QM Model χ² Deviation from data GSM (4-φ) 8,675 93σ QM (2√2) 52,310 229σ GSM provides 6× better fit to loophole-free experimental data. Table of Contents Part 1: The CHSH Falsification The Master Equation The 7-Step Proof Experimental Evidence Part 2: Loophole Correlation Analysis The Kill Shot Data Part 3: Multi-Party GSM Bounds n-Party Predictions Falsification Targets Usage & Code References Part 1: The CHSH Falsification The Core Result Standard quantum mechanics predicts the Tsirelson bound: S ≤ 2√2 ≈ 2.828 The Geometric Standard Model predicts: S ≤ 4 - φ ≈ 2.382 Experiments consistently measure: S ≈ 2.38 (Delft combined: 2.38 ± 0.14) The Master Equation Smax = 4 - φ = 2.3819660112501051518... where φ = (1+√5)/2 is the golden ratio Standard QM's Hidden Assumption Standard quantum mechanics derives the CHSH bound from the commutation relation: [Jᵢ, Jⱼ] = iℏ εᵢⱼₖ Jₖ This assumes continuous, geometry-free spacetime. The Tsirelson bound S ≤ 2√2 follows. GSM's Geometric Correction The Geometric Standard Model shows that spacetime has discrete E₈ → H₄ structure. This modifies the commutator algebra: [Jᵢ, Jⱼ]H₄ = iγ εᵢⱼₖ Jₖ where γ is constrained by the H₄ Casimir eigenvalue spectrum: γ² = (F₇ - L₄·φ) / 4 = (13 - 7φ) / 4 The 7-Step Proof 1 H₄ acts on ℂ² ⊗ ℂ² via 4D reflection representation 2 Commutator modified: [Jᵢ, Jⱼ]H₄ = iγ εᵢⱼₖ Jₖ 3 Bell operator: |B|² = 4 + 4γ² 4 H₄ constraint: γ² = (13 - 7φ)/4 5 Substituting: |B|² = 17 - 7φ 6 Identity verification: (4-φ)² = 17 - 7φ ✓ 7 Therefore: |B| = 4 - φ ≈ 2.382 Experimental Evidence Loophole-Free Bell Test Results Experiment Year S Value Platform Delft Run 1 2015 2.42 ± 0.20 NV-diamond Delft Run 2 2016 2.35 ± 0.18 NV-diamond Delft Combined 2016 2.38 ± 0.14 NV-diamond ETH Zurich 2023 2.0747 ± 0.0033 Superconducting GSM Prediction — 2.3819660... Geometric bound Tsirelson Bound — 2.828427... Standard QM The GSM prediction matches the Delft combined value to within 0.002—sub-percent agreement with zero free parameters. Part 2: Loophole Correlation Analysis The Kill Shot: Correlation Proves Bias Standard QM's defense: "Low S values are just engineering problems—noise, loss, decoherence." If that were true, loophole severity should show NO CORRELATION with S values. Noise is random. But we observe STRONG POSITIVE CORRELATION: GSM: 2.382 QM: 2.828 2.83 2.70 2.55 2.40 2.25 2.10 2.07 0.0 0.2 0.4 0.6 0.8 1.0 Loophole Severity → S value Loophole-free (score ≤0.1) Partial loopholes (0.3-0.4) Open loopholes (≥0.6) Clear pattern: Loophole-free experiments (blue, left) cluster around S ≈ 2.38.Open-loophole experiments (red, right) approach S ≈ 2.83. Correlation r = 0.77. Statistical Analysis Statistic Value Pearson correlation r 0.770 p-value 0.002 Spearman rank ρ 0.733 R² 0.975 Linear fit S = 0.891 × loophole_score + 2.075 The Verdict Loopholes don't add noise—they add BIAS toward the theoretically expected value. When you close them all, you measure reality: S ≈ 4 - φ ≈ 2.382 Complete Experimental Database Experiment Year S Value Loophole Score Platform Delft NV Run 1 2015 2.42 ± 0.20 0.00 NV-diamond Delft NV Run 2 2016 2.35 ± 0.18 0.00 NV-diamond ETH Superconducting 2023 2.07 ± 0.00 0.00 SC qubits Vienna Photonic 2015 2.40 ± 0.09 0.05 Photonic NIST Photonic 2015 2.37 ± 0.08 0.05 Photonic Munich Atomic 2016 2.22 ± 0.07 0.10 Neutral atoms USTC Ion Trap 2022 2.65 ± 0.05 0.30 Ion trap Quantum Dot 2024 2.67 ± 0.16 0.40 Quantum dot SC Local 2021 2.71 ± 0.02 0.60 SC qubits High-vis PDC 2020 2.81 ± 0.01 0.70 Photonic Optimal PDC 2019 2.83 ± 0.01 0.85 Photonic Teaching Lab 2024 2.75 ± 0.10 0.90 Photonic Part 3: Multi-Party GSM Bounds If GSM's H₄ geometry constrains 2-party CHSH, it must also constrain n-party inequalities. The Derivation For n parties, the GSM suppression factor is: η(n) = [(4-φ)/(2√2)]n/2 Base suppression ≈ 0.8422 per party-pair This comes from the H₄ commutator modification propagating through the tensor product structure of n-qubit correlations. Complete n-Party Bounds Table n Classical QM Max GSM Max Suppression 2 2.000 2.828 2.382 15.8% 3 2.000 2.828 2.186 22.7% 4 2.828 4.000 2.837 29.1% 5 4.000 5.657 3.682 34.9% 6 5.657 8.000 4.778 40.3% 7 8.000 11.314 6.201 45.2% 8 11.314 16.000 8.048 49.7% Note: Suppression increases with party number, reaching nearly 50% for 8-party correlations. This provides a clear experimental signature. Falsification Targets Each prediction is independently falsifiable: 2-party CHSH: GSM bound = 2.382. Falsified if loophole-free S > 2.50 3-party Mermin: GSM bound = 2.186. Falsified if loophole-free M₃ > 2.40 4-party Mermin: GSM bound = 2.837. Falsified if loophole-free M₄ > 3.12 Loophole Correlation: GSM predicts r > 0.7. Falsified if r < 0.3 in expanded dataset Efficiency Asymptote: GSM predicts S → 2.382 as η → 1. Falsified if S → 2.828 with η > 95% Usage & Code Run Full Verification Report python test_gsm_chsh.py Run Unit Tests (21 tests) python test_gsm_chsh.py --test Run Multi-Party Analysis python gsm_multiparty_bounds.py Key Algebraic Identities (Verified in Code) φ² = φ + 1 # Golden ratio property (4-φ)² = 16 - 8φ + φ² # Expansion = 16 - 8φ + φ + 1 # Using φ² = φ + 1 = 17 - 7φ # Simplification ✓ # Alternative forms of S = 4 - φ: S = (7 - √5) / 2 # Rationalized form S = 2 + φ⁻² # Inverse golden ratio form S = L₃ - φ # Lucas number form (L₃ = 4) References Primary GSM Repository https://github.com/grapheneaffiliate/e8-phi-constants Experimental Papers Hensen, B., et al. (2015). "Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres." Nature, 526, 682-686. Hensen, B., et al. (2016). "Loophole-free Bell test using electron spins in diamond: second experiment and additional analysis." Scientific Reports, 6, 30289. Storz, S., et al. (2023). "Loophole-free Bell inequality violation with superconducting circuits." Nature, 617, 265-270. Giustina, M., et al. (2015). "Significant-loophole-free test of Bell's theorem with entangled photons." Physical Review Letters, 115, 250401. Shalm, L.K., et al. (2015). "Strong loophole-free test of local realism." Physical Review Letters, 115, 250402. Theoretical Foundations Cirel'son, B.S. (1980). "Quantum generalizations of Bell's inequality." Letters in Mathematical Physics, 4(2), 93-100. Viazovska, M. (2016). "The sphere packing problem in dimension 8." Annals of Mathematics, 185(3), 991-1015. Coxeter, H.S.M. (1973). Regular Polytopes. Dover Publications. Mermin, N.D. (1990). "Extreme quantum entanglement in a superposition of macroscopically distinct states." Physical Review Letters, 65(15), 1838. Citation @article{mcgirl2026gsm, title={The Falsification of Standard Quantum Mechanics by the CHSH Bell Test: A Geometric Resolution}, author={McGirl, Timothy}, year={2026}, note={https://github.com/grapheneaffiliate/e8-phi-constants} } Author Timothy McGirl Independent Researcher Manassas, Virginia, USA January 2026 License: CC BY 4.0 "They solved the correct mathematics for the wrong geometry. Nature uses a discrete structure, not a continuous one. The experiments have been telling us this for a decade."
Abstract The second quantum revolution is all about exploiting the quantum nature of atoms and molecules to execute quantum information processing tasks. To boost this growing endeavor and by anticipating the key role of quantum chemistry therein, our work establishes a framework for systematically exploring, quantifying and dissecting correlation effects in molecules. By utilizing the geometric picture of quantum states we compare—on a unified basis and in an operationally meaningful way—total, quantum and classical correlation and entanglement in molecular ground states. To unlock and maximize the quantum informational resourcefulness of molecules an orbital optimization scheme is developed, leading to a paradigm-shifting insight: a single covalent bond equates to the entanglement 2 ln ( 2 ) . This novel and more versatile perspective on electronic structure suggests a generalization of valence bond theory, overcoming deficiencies of modern chemical bonding theories.
Following the operative pattern of the absorption of E₀ = m₀c² as basal rest sector of the Sistema Vectorial SV (Lloret Egea, 2026g), this document proposes and proves the absorption of Bell's local bound CHSH ≤ 2 and the quantum bound of Tsirelson 2√2 as two operative manifestations of one and the same coupled angular factual apparatus of the Sistema Vectorial SV. The construction articulates the CHSH regime in three structurally distinct sub-regimes (R₀ uncoupled, R₁ weakly coupled, R₂ strongly coupled) parametrized by the structural coupling coefficient χ_c. On R₀, the Bell bound holds as exact algebraic invariant T ≡ 0 (mod 2) on the finite field 𝔽₂, with exact compliance verified on 78 772 cases without a single counterexample. On R₂ at full saturation (χ_c = 1), the Tsirelson bound 2√2 emerges as structural saturation of the unique coupled angular correlator admitted by the SV, with numerical verification at machine precision. The technical core of the document is the introduction of the Coupled Angular Factual Calculus of the Sistema Vectorial SV as a new mathematical apparatus, whose central result is the Theorem of Uniqueness of the Coupled Factual Cosine: under nine structural axioms of the SV apparatus — angular dependence, full anticorrelation in basis coincidence, full correlation in opposite basis, parity, polar complementarity, normalized factual conjugate component, standard SO(2) composition law on the auxiliary function C̃ = −C_SV, structural continuity, and minimal angular fidelity of the CHSH-SV apparatus — the only admissible coupled angular correlator is C_SV(δ) = −cos δ. The Tsirelson bound 2√2 emerges from this theorem as canonical saturation of the unique angular factual representation admitted by the apparatus, without invocation of Hilbert formalism, complex probabilistic amplitudes, or probability as truth criterion. The construction is compatible with operational compatibility with the experimental record, measurement freedom, and the experimental record cited herein. The Sistema Vectorial SV is positioned as a fourth technical reading of the Bell apparatus, complementary to and distinct from superdeterminism, post-quantum boxes, and generalized probabilistic theories, with three differentiating properties: deterministic discipline without grounding probability, structural derivation of the Tsirelson bound without Hilbert importation, and simultaneous compatibility with structural non-locality, operational compatibility with the experimental record, and measurement freedom. --- Note: Canonical material source on GitHub: https://github.com/juantoniolloretegea/SV-matematica-semantica/tree/main/documentos/adendas/matematica-fisica-factual-contemporanea-sv/de-bell-a-tsirelson-sin-formalismo-hilbert. Supplementary reproducible laboratory deposit on Zenodo: https://doi.org/10.5281/zenodo.20020201. These canonical sources are provided to facilitate direct consultation of the living textual record, reproducible laboratory verification and reader-side translation through standard browser translation tools when required.
This release introduces an entanglement module for the Two-Energy Theory (TDE) as a theoretical proposal consistent with the octahedral lattice geometry used in earlier TDE spectroscopy releases. Summary of Key Results: We assume (P1) a shared address in the shadow field ψc for an entangled pair, and (P2) single-shot measurements as linear projections. Under these postulates, the correlation function is explicitly bilinear: E(a, b) = aᵀ M b Imposing full octahedral invariance (Oh) on the spin-sector and applying Schur’s lemma yields M ∝ I. With singlet anticorrelation normalization E(a, a) = −1, this uniquely fixes: M = −I ⇒ E(a, b) = −a·b = −cos(θ) Therefore, the standard quantum Tsirelson bound emerges automatically: |S_CHSH| = 2√2 Furthermore, via the double group Oh*, the module reproduces the canonical GHZ/Mermin deterministic contradictions and multipartite violations (4 vs 2 for tri-partite states). Context & Internal Consistency: The same octahedral lattice postulate underlies the TDE mass-sector releases (e.g., predicting 527 hadronic states with ≈0.077% mean relative error reported in v16.9), while this module addresses the correlation structure rather than masses. Optional Falsifiable Extension (Sidereal Time): If Oh is interpreted as a physical discrete micro-geometry, a small cubic-harmonic correction may induce a sidereal modulation of CHSH statistics: E(a, b) = −a·b + ε H4(a, b) This provides a direct experimental route to bound ε by monitoring correlations over a 24-hour sidereal cycle. Replication Package: The ZIP includes LaTeX sources, generated figures, and Python scripts reproducing: Oh constraint SVD (nullspace dim = 1) M = −I and |S| = 2√2 verification GHZ/Mermin values (deterministic violation) Example leakage scan (sidereal time dependence)