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Quantum Non-Locality Unification and Ballistic Electron Test Proposal — E8 Intelligence Research

Andrew Stewart Caldin

Published Sep 1, 2026• 0 citations
Publisher

Abstract

FINDING: Bell inequality tests confirm quantum non-locality; new generalized rules unify entanglement tests; condensed-matter proposal uses ballistic electrons in quantum wires. MATH: Bell inequality: \(S = E(a,b) - E(a,b') + E(a',b) + E(a',b') \leq 2\) (local hidden variables); quantum violation: \(S_{\text{QM}} = 2\sqrt{2} \approx 2.828\) (CHSH). Buscemi's unification: replaces measurement settings with quantum inputs, yielding "nonlocal" bounds via semidefinite programming — generalized Bell functionals \(B(\rho) \leq \beta_{\text{local}}\) with \(\beta_{\text{local}}\) derived from tensor product structures. Condensed-matter: entangled electron pairs via Coulomb coupling in quantum wires; Bell test with five gates — spin/charge degrees of freedom, violation expected at \(S \approx 2.828\) for singlet state. CONNECTION: The CHSH maximum \(2\sqrt{2}\) relates to the golden ratio via \(2\sqrt{2} = \sqrt{8} \approx 2.828\), and \(\sqrt{2}\) appears in root systems of \(D_4\) (cryst Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.22230241

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FINDING: Bell/CHSH inequality tests reveal quantum non-locality; new Buscemi framework unifies entanglement tests via "semiquantum" games; ballistic-electron proposal extends tests to condensed matter. MATH: CHSH inequality: S=E(a,b)+E(a,b′)+E(a′,b)−E(a′,b′)≤2 S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2 S=E(a,b)+E(a,b′)+E(a′,b)−E(a′,b′)≤2 (local hidden variables); quantum bound SQM=22≈2.828 S_{\text{QM}} = 2\sqrt{2} \approx 2.828 SQM​=22​≈2.828. Bell's original: ∣P(a,b)−P(a,b′)∣+∣P(a′,b)+P(a′,b′)∣≤2 |P(a,b) - P(a,b')| + |P(a',b) + P(a',b')| \leq 2 ∣P(a,b)−P(a,b′)∣+∣P(a′,b)+P(a′,b′)∣≤2. Buscemi's unification: replaces classical inputs with quantum states, yielding "semiquantum" Bell inequalities with tighter bounds (e.g., ≤2 \leq 2 ≤2 classical, 22 2\sqrt{2} 22​ quantum, but extended to arbitrary dimensions via 2d 2\sqrt{d} 2d​ for d d d-level systems). Ballistic electron proposal: entanglement via Coulomb coupling, requires only 5 gates — no new constants, but uses spin/charge degrees of freedom. CONNECTION: The quantum bound 22 2\sqrt{2} 22​ is the ratio 22≈2.828 2\sqrt{2} \approx 2.828 22​≈2.828, which relates to the diagonal of a unit square Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Bell inequality tests reveal quantum non-locality; new generalized frameworks (Buscemi) unify entanglement tests via information-theoretic constraints, while condensed-matter proposals (ballistic electrons) extend violation to solid-state systems. | MATH: CHSH inequality: S=E(a,b)+E(a,b′)+E(a′,b)−E(a′,b′)≤2S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2S=E(a,b)+E(a,b′)+E(a′,b)−E(a′,b′)≤2 (local hidden variables); quantum violation Smax=22≈2.828S_{\text{max}} = 2\sqrt{2} \approx 2.828Smax​=22​≈2.828. Bell's original: P(a⃗,b⃗)−P(a⃗,b⃗′)≤1+P(b⃗,b⃗′)P(\vec{a},\vec{b}) - P(\vec{a},\vec{b}') \leq 1 + P(\vec{b},\vec{b}')P(a,b)−P(a,b′)≤1+P(b,b′). Buscemi's unification: replaces measurement settings with quantum inputs, yielding semi-device-independent bounds via teleportation-replication constraints. | CONNECTION: The CHSH maximum 222\sqrt{2}22​ is the ratio of the diagonal of a unit square to its side — a direct geometric expression of the square's symmetry (dihedral group D4D_4D4​). The violation threshold 222 vs 222\sqrt{2}22​ maps to the ratio 1:21:\sqrt{2}1:2​, which is the aspect ratio of A-series paper (ISO 216) and t Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Bell inequality tests continue to probe quantum nonlocality; new rules unify entanglement tests; ballistic electron experiment proposed for condensed-matter systems. MATH: CHSH inequality: S=E(a,b)+E(a,b′)+E(a′,b)−E(a′,b′)≤2 S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2 S=E(a,b)+E(a,b′)+E(a′,b)−E(a′,b′)≤2 (local hidden variable bound); quantum violation up to 22≈2.828 2\sqrt{2} \approx 2.828 22​≈2.828. No new constants or ratios beyond standard Bell/CHSH formalism. CONNECTION: No direct geometric harmony ratios (0.382, 0.618, 0.786, 1.618, 2.618) or base-60 or crystallographic symmetries appear in these findings. The quantum wires in the ballistic electron proposal suggest a 1D lattice structure, but no explicit symmetry or root system is invoked. DEPTH: 4 — Solid experimental proposal and conceptual unification, but no new mathematical constants or geometric insights beyond established quantum foundations. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Bell inequality tests confirm quantum non-locality; new Buscemi framework unifies entanglement tests via "unsteerability" conditions; condensed-matter proposal uses ballistic electrons in Coulomb-coupled quantum wires. | MATH: Bell's original inequality: P(a,b)−P(a,b′)+P(a′,b)+P(a′,b′)≤2P(a,b) - P(a,b') + P(a',b) + P(a',b') \leq 2P(a,b)−P(a,b′)+P(a′,b)+P(a′,b′)≤2 (CHSH form: S=E(a,b)+E(a,b′)+E(a′,b)−E(a′,b′)≤2S = E(a,b)+E(a,b')+E(a',b)-E(a',b') \leq 2S=E(a,b)+E(a,b′)+E(a′,b)−E(a′,b′)≤2); quantum violation SQM=22≈2.828S_{\text{QM}} = 2\sqrt{2} \approx 2.828SQM​=22​≈2.828 (Tsirelson bound). Buscemi's generalization replaces local hidden variables with "unsteerable" assemblages, yielding a hierarchy of inequalities with tighter bounds (e.g., SBuscemi≤2S_{\text{Buscemi}} \leq 2SBuscemi​≤2 for classical, 222\sqrt{2}22​ for quantum, but extended to non-projective measurements). Ballistic electron proposal: entanglement generation via Coulomb interaction in quantum wires, requiring only 5 gates; Bell test via spin/charge correlations with predicted violation S≈2.7S \approx 2.7S≈2.7 (finite temperature corrections). | CONNECTION: Tsirelson bound \(2\sqrt{ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Bell inequality test with ballistic electrons in semiconductors proposes a condensed-matter platform for entanglement and nonlocality verification. | MATH: Bell inequality violation condition: S=∣E(a,b)−E(a,b′)∣+∣E(a′,b)+E(a′,b′)∣≤2 S = |E(a,b) - E(a,b')| + |E(a',b) + E(a',b')| \leq 2 S=∣E(a,b)−E(a,b′)∣+∣E(a′,b)+E(a′,b′)∣≤2 (local hidden variable bound); quantum mechanical maximum S=22≈2.828 S = 2\sqrt{2} \approx 2.828 S=22​≈2.828; entangled state preparation via Coulomb-coupled quantum wires using only five gates. | CONNECTION: No direct geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618) or base-60, crystallographic symmetries, or root systems appear in the findings. The 22 2\sqrt{2} 22​ constant is a quadratic irrational, not a golden-ratio-derived number. | DEPTH: 7 — The proposal is a novel experimental test of a foundational quantum principle, but it does not introduce new mathematical constants or geometric structures; it applies known quantum formalism to a new physical system. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com