D2d-C3 covers the Bell inequality structure of the Triniton discrete framework. Results established: (I) R4 — CHSH ≤ 2 from local factorization, with explicit verification on the classical Triniton sector and the geometric note on spatial rotations; (II) dim=2 qubit from M2+PE; (III) su(2) algebra from step transitions in Im(Q); (IV) ∣Ψ𝑇+⟩from PE+unitarity+IV.N2+M1 — four-proposition derivation; (V) E(a,b) from su(2) — algebraiccomputation with declared convention; (VI) CHSH = 2√2 with explicit angle assignment; (VII) finite-N correction with conservative bound 4π/N. Zero free parameters. Zero ad hoc adjustments.
D2d-C3 covers the Bell inequality structure of the Triniton discrete framework. Results established: (I) R4 — CHSH ≤ 2 from local factorization, with explicit verification on the classical Triniton sector and the geometric note on spatial rotations; (II) dim=2 qubit from M2+PE; (III) su(2) algebra from step transitions in Im(Q); (IV) ∣Ψ𝑇+⟩from PE+unitarity+IV.N2+M1 — four-proposition derivation; (V) E(a,b) from su(2) — algebraiccomputation with declared convention; (VI) CHSH = 2√2 with explicit angle assignment; (VII) finite-N correction with conservative bound 4π/N. Zero free parameters. Zero ad hoc adjustments.
FINDING: The only substantive mathematical result is the arXiv paper on Tsirelson bounds derived via geometric algebra, linking Bell-CHSH violations to Clifford algebra structure. The other results are irrelevant (Python tutorials, quantum control sweeps, correlated-electron optics, sheaf-code QEC, R² statistics). MATH: - CHSH inequality: ⟨AB⟩+⟨AB′⟩+⟨A′B⟩−⟨A′B′⟩≤2 (local hidden variables) - Quantum mechanical bound (Tsirelson): ≤22≈2.828 - Geometric algebra formulation: Bell operators as bivectors in Cl(3,0), with {A,A′}={B,B′}=0 and [A,A′]=[B,B′]=0 (commuting spacelike separated) - Key ratio: 22/2=2≈1.414 — not directly a golden-ratio constant, but note 2=2sin(π/4), related to 45° symmetry. CONNECTION: - The Tsirelson bound 22 emerges from the algebraic structure of Cl(3,0) — s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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.
This work provides an explicit, reproducible demonstration of how theClauser-Horne-Shimony-Holt (CHSH) inequality depends jointly on thechoice of Bell state AND measurement configuration in the spin-1/2qubit convention used by gate-based quantum processors. Main result. The combination of CHSH Form I, S = | E(a,b) - E(a,b') + E(a',b) + E(a',b') |, with the commonly-cited angular configuration {0, pi/2, pi/4, -pi/4}yields S = 0 for ALL FOUR Bell states in the spin-1/2 Bloch-sphereconvention, not selectively for one. This frequently misleadspractitioners who prepare |Phi+> on gate-based hardware and applytextbook angles, obtaining S = 0 and misdiagnosing the issue as ahardware fault, code bug, or absence of entanglement. Corrections provided. For the two states whose correlation functionin the XZ plane has argument theta_a - theta_b (|Phi+> and |Psi->),two minimal one-parameter fixes recover the Tsirelson bound 2*sqrt(2): 1. Retain the angles and apply CHSH Form II: S = | E(a,b) + E(a,b') + E(a',b) - E(a',b') |. 2. Retain Form I and replace b' = -pi/4 with b' = 3*pi/4. For the remaining two Bell states (|Phi-> and |Psi+>, whose correlationargument is theta_a + theta_b), neither single-parameter fix suffices;the optimal angles b and b' must be exchanged. A complete optimal-angletable for all four states is given in Section 4 of the manuscript. Contents. - paper.pdf / paper.tex: 8-page manuscript with full derivation, tables, code listing and appendix. - chsh_verification.py: self-contained Python script (NumPy only) reproducing all numerical claims of the paper. Its standard output is the verbatim text reproduced in Section 5, and the script additionally cross-checks the closed-form correlation expressions against explicit 4x4 matrix calculations on random angles (azimuthal angles included) to machine precision. - CHANGELOG.md: complete history of corrections relative to earlier versions of this deposit. Reproducibility. Running 'python3 chsh_verification.py' produces theverification tables displayed in Section 5 of the paper, byte for byte.
In 1964, John Bell showed that any local hidden‑variable theory must satisfy a specific inequality on correlations between distant measurements. In its CHSH form (Clauser, Horne, Shimony, Holt 1969), the inequality reads |S| = |E(a,b) + E(a,b′) + E(a′,b) − E(a′,b′)| ≤ 2, where E(a,b) is the correlation between measurement outcomes at two separated parties using settings a and b. Any local realist theory respects this bound. Quantum mechanics violates it. Experiments confirm the violation. The standard quantum prediction for a maximally entangled state with optimal measurement angles is |S| = 2√2 ≈ 2.828. In 1980, Boris Tsirelson proved that 2√2 is not just one quantum prediction — it is the *maximum* value any quantum correlation can produce. No quantum state, no measurement scheme, no clever protocol can push |S| beyond 2√2. The bound is sharp, the value is exact, and it has been measured in laboratories around the world to experimental precision. The Tsirelson bound is one of the deepest empirical structures in physics. It tells us that quantum correlations exceed classical correlations by a specific, finite amount — there is room to violate Bell’s inequality, but only so much room. The number 2√2 has no obvious physical interpretation. It does not appear in the formulation of quantum mechanics. It emerges from the mathematics of the bipartite Hilbert space when the CHSH operator is constructed and bounded. The question is where it comes from. In standard QM, the answer is that it falls out of the Hilbert‑space structure plus Cauchy–Schwarz plus an operator‑norm bound on the CHSH operator. Technically correct, and Tsirelson’s 1980 proof remains the canonical derivation — but it leaves the deeper structure unanswered. The Hilbert space is postulated. The inner product that makes Cauchy–Schwarz available is postulated. The bipartite tensor‑product structure on H_A ⊗ H_B is postulated. The 2√2 value is what those postulates produce when run through the CHSH operator. This walkthrough shows how the Event Density framework derives the Tsirelson bound from a smaller set of structural commitments. The Hilbert‑space structure, the inner product, and the bipartite tensor‑product structure all emerge as forced consequences of the substrate ontology rather than being postulated. With those structures forced rather than assumed, Tsirelson’s 1980 argument applies directly, and the 2√2 value is no longer the output of a chain of postulates but the output of a chain of derived theorems. Like Heisenberg, Bell–Tsirelson is a place where the framework’s contribution is upstream. The bound itself is mathematical physics already in the literature; what changes is the foundational status of the structures the argument requires. ED does not produce a new derivation step beyond standard QM here — it removes the postulational status of the structures the standard derivation already uses. The 2√2 value is now a derived consequence of the substrate ontology, all the way down.
Relational Formation and the Bell Bound investigates an alternative structural reading of Bell-type correlations. Rather than assuming that alternative measurement settings reveal values attributable to a single pre-existing complete state, the paper introduces a minimal Relational Formation Algebra (RFA) in which a definite relational state is formed under the realized context. The framework is developed through four stages, R0–R3, describing a pre-relational possibility space, contextual relational formation, intrinsic compatibility through composition, and subsequent observational projection. A further condition, R4 (local substructure consistency), requires globally context-dependent relational states to retain locally consistent marginal structure. Two reproducible blind numerical experiments explore the resulting correlation domain. An unrestricted R0–R3 ensemble produces Bell violations but also permits CHSH values above the Tsirelson bound. In a structured bipartite realization satisfying R4 to numerical precision, Bell violations remain common while none of 30,000 randomized trials exceeds 2\sqrt{2}. The paper then examines the corresponding binary, cross-compatible operator algebra and the elementary origin of the 2\sqrt{2} norm bound. The proposal is not presented as a refutation of Bell’s theorem or CHSH, nor as an independent derivation of quantum mechanics. Instead, it asks whether the assumption of a pre-existing jointly attributable order can be replaced by a framework in which complete order is formed relationally while local consistency is retained. The conceptual distinction is summarized by the proposition: “Relation does not select an order. Relation forms the order.” The work includes the numerical parameters and fixed random seeds required to reproduce the reported simulations and references the original EPR (1935), Bell (1964), and CHSH (1969) papers. AI assistance: ChatGPT (OpenAI) was used as a collaborative tool for mathematical exploration, numerical simulation, structural analysis, and preparation of the manuscript. The conceptual direction, interpretation, selection of hypotheses, and responsibility for the published content remain with the human author.
The textbook CHSH measurement configuration {a, a', b, b'} = {0, pi/2, pi/4, -pi/4} with Form I evaluates to S = 0 on all four Bell states in the spin-1/2 convention used by gate-based quantum processors. This "failure mode", documented as a state-formula-angle mismatch in a companion pedagogical note (doi:10.5281/zenodo.20146219), is shown here to have a clean algebraic origin: the corresponding Bell operator reduces to sqrt(2) * (sigma_z (x) sigma_x + sigma_x (x) sigma_z), whose spectrum is {-2*sqrt(2), 0, 0, +2*sqrt(2)}, with the kernel spanned by {|Phi+>, |Psi->} and the non-kernel pair {|Phi->, |Psi+>} mapped to each other. Main results. The note establishes three properties of the failure-mode configuration: (1) The CHSH value S is preserved at exactly zero under arbitrary symmetric local Pauli channels (depolarizing, dephasing, bit-flip) and under local Rx and Rz rotations on either qubit, for every Bell state. Verified numerically with 1000 random Pauli channels per state and across the full range of standard noise channels. (2) Local Ry(theta) rotations break S = 0 only for the pair {|Phi->, |Psi+>}, with closed form S = 2*sqrt(2)*|sin(theta)|, reaching the Tsirelson bound at theta = pi/2. (3) The asymmetry has a direct algebraic origin in the correlation tensor: the kernel pair has equal XX and ZZ moments (t_1 = t_3), while the broken pair has opposite signs (t_1 = -t_3). This connects to and gives algebraic meaning to the Bell-state taxonomy of the companion paper. Diagnostic interpretation. The same CHSH measurement that fails in textbook form becomes a no-extra-resource diagnostic for coherent Ry-type errors restricted to a specific Bell-state family. If S > 0 in this configuration, then (i) the state is in the {|Phi->, |Psi+>} family, and (ii) a coherent Ry error of magnitude theta_err = arcsin(S/(2*sqrt(2))) is present. Incoherent Pauli noise alone cannot produce this signal. Calibration of novelty. The note explicitly searches the self-testing literature (Supic-Bowles 2020 review) and the CHSH-under-noise literature (Kofman 2012, Kofman-Korotkov 2008) and reports that this specific asymmetric breaking pattern was not located in those communities. Priority is explicitly disclaimed with respect to gate-set-tomography and randomized-benchmarking literatures, which were not exhaustively surveyed. The result is presented as either a novel observation or, if previously catalogued elsewhere, an independent algebraic rederivation with closed-form expression in terms of the correlation tensor. Reproducibility. The repository contains the LaTeX source, the bibliography, both figures, and seven Python scripts that reproduce every numerical claim from scratch using only NumPy and Matplotlib. Each script is self-contained and runs in seconds. Companion paper. This work extends the pedagogical note "State-dependent CHSH inequality on gate-based quantum processors" (Perez-Eugenio 2026, doi:10.5281/zenodo.20146219), which documents the original state-formula-angle compatibility issue and the corrected configurations. The present note can be read independently.