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Unifying Quantum Non-Locality Tests via Device-Independent Frameworks — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 1, 2026• 0 citations
Publisher

Abstract

FINDING: Bell inequality tests confirm quantum non-locality; new generalized rules (Buscemi) unify entanglement tests via device-independent frameworks. MATH: CHSH inequality: \(S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2\) (local hidden variables); quantum violation max \(S = 2\sqrt{2} \approx 2.828\). Bell's original inequality: \(P(\vec{a},\vec{b}) - P(\vec{a},\vec{c}) \leq 1 + P(\vec{b},\vec{c})\). Buscemi's unification: replaces measurement settings with quantum inputs, yielding semi-device-independent bounds—generalizes to \(S_q \leq 2\sqrt{2}\) for qubits, but for higher-dimensional systems, Tsirelson bounds scale as \(2\sqrt{d}\) for maximally entangled states. CONNECTION: The maximal quantum violation \(2\sqrt{2}\) relates to the golden ratio via \(2\sqrt{2} = 2 \times 1.414\), and \(1.414^2 = 2\)—a Pythagorean doubling. More directly, the CHSH bound \(2\sqrt{2}\) appears in the geometry of the square (diagonal/side ratio), linking to 45° rotations in 2D lattice symme Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.23075501

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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
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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 inequalities quantify the maximum classical correlation limit; quantum entanglement violates this bound, and new generalized rules (Buscemi) unify entanglement tests across operational scenarios. MATH: - 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 maximum: SQM=22≈2.828 S_{\text{QM}} = 2\sqrt{2} \approx 2.828 SQM​=22​≈2.828 (Tsirelson bound) - 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 note 2 \sqrt{2} 2​ is the diagonal of unit square (crystallographic root system A1×A1 A_1 \times A_1 A1​×A1​) - Buscemi's extension: replaces measurement choices with quantum inputs, yielding *nonlocal* inequalities with no classical analogue — the bound becomes state-dependent, often expressed via semidefinite programming (SDP) hierarchies (NPA hierarchy). CONNECTION: - The Tsirelson bound 22 2\sqrt{2} 22​ is the maximal algebraic value of a Clifford algebra generator pair — linked to the octonionic/split-octonionic s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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