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Bell/CHSH Inequalities: Unified Rules and Ballistic-Electron Testbeds — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 27, 2026• 0 citations
Publisher

Abstract

FINDING: Bell/CHSH inequalities define the boundary between classical local-hidden-variable theories and quantum correlations; recent work (Buscemi) reformulates them as a unified family of "rules" covering entanglement and nonlocality, and new physical platforms (ballistic electrons in semiconductor quantum wires) are proposed as testbeds. MATH: - CHSH inequality (classical bound): |E(a,b) − E(a,b′) + E(a′,b) + E(a′,b′)| ≤ 2 - Quantum (Tsirelson) bound: same expression ≤ 2√2 ≈ 2.828 - Quantum/classical ratio: 2√2 / 2 = √2 ≈ 1.4142 - Maximal violation angle for singlet state: correlations E(a,b) = −a·b; optimal settings at 45° increments give S = 2√2 - General Bell functional: S = Σ_{x,y} w_{xy} P(a,b|x,y); local polytope bounded by facet inequalities; quantum set Q strictly contains local polytope L, strictly contained in no-signaling set NS - Buscemi's unification: inequalities expressed via conditional min-/max-entropy constraints — "new rules" generalize CHSH to entropy-based Bell Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.23684400

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FINDING: Bell inequality tests confirm quantum non-locality; new generalized rules (Buscemi) unify entanglement frameworks; condensed-matter proposal extends tests to ballistic electrons. MATH: - Bell inequality (CHSH form): 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 mechanics predicts S=22≈2.828 S = 2\sqrt{2} \approx 2.828 S=22​≈2.828. - Violation ratio: 22/2=2≈1.414 2\sqrt{2}/2 = \sqrt{2} \approx 1.414 22​/2=2​≈1.414 — not a golden ratio, but a Pythagorean root. - Buscemi's unification: generalizes Bell inequalities via *semidefinite programming* hierarchies (NPA hierarchy), replacing fixed measurement settings with *quantum channels* — key constant: Tsirelson bound 22 2\sqrt{2} 22​ remains the supremum for CHSH. - Ballistic electron proposal: uses Coulomb-coupled quantum wires; entanglement generation via *beam-splitter-like* gates — no new constants, but maps to spin-1/2 Pauli algebra σx,σy,σz \sigma_x, \sigma_y, \sigma_z σx​,σy​,σz​. CONNECTION: - The 2 \sqrt{2} 2​ ratio app 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 violations remain the core experimental test of quantum non-locality, with CHSH form as the standard mathematical tool; new theoretical work (Buscemi) unifies entanglement and Bell tests via generalized rules. 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) vs. quantum bound SQM=22≈2.828 S_{\text{QM}} = 2\sqrt{2} \approx 2.828 SQM​=22​≈2.828. - Bell's original inequality: ∣P(a⃗,b⃗)−P(a⃗,c⃗)∣≤1+P(b⃗,c⃗) |P(\vec{a},\vec{b}) - P(\vec{a},\vec{c})| \leq 1 + P(\vec{b},\vec{c}) ∣P(a,b)−P(a,c)∣≤1+P(b,c). - Quantum violation ratio: 22/2=2≈1.414 2\sqrt{2}/2 = \sqrt{2} \approx 1.414 22​/2=2​≈1.414 — note this is NOT the golden ratio, but it is the ratio of the diagonal of a unit square to its side (Pythagorean). - Buscemi's unification: extends Bell inequalities to "semi-quantum" scenarios where one party is classical, one quantum — yields new bounds involving 2 \sqrt{2} 2​ and 2 2 2 in generalized forms. - No explicit appearance of 0.382, 0.618, 0.786, 1.618, 2.618 in the provided abstracts 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 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′)≤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: SQM=22≈2.828S_{\text{QM}} = 2\sqrt{2} \approx 2.828SQM​=22​≈2.828 (CHSH). Buscemi's unification: replaces measurement settings with quantum inputs, yielding "nonlocal" bounds via semidefinite programming — generalized Bell functionals B(ρ)≤βlocalB(\rho) \leq \beta_{\text{local}}B(ρ)≤βlocal​ with βlocal\beta_{\text{local}}βlocal​ 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≈2.828S \approx 2.828S≈2.828 for singlet state. CONNECTION: The CHSH maximum 222\sqrt{2}22​ relates to the golden ratio via 22=8≈2.8282\sqrt{2} = \sqrt{8} \approx 2.82822​=8​≈2.828, and 2\sqrt{2}2​ appears in root systems of D4D_4D4​ (cryst 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