Unifying Entanglement and Bell Tests via Generalized Non-Locality Rules — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 21, 2026• 0 citations

Abstract

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') \leq 2 $ (local hidden variables) vs. quantum bound $ S_{\text{QM}} = 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}) $. - Quantum violation ratio: $ 2\sqrt{2}/2 = \sqrt{2} \approx 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 $ \sqrt{2} $ and $ 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

DOI: 10.5281/zenodo.23526081

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Unifying Entanglement and Bell Tests via Generalized Non-Locality Rules — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 21, 2026• 0 citations

Publisher

Abstract

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') \leq 2 ) (local hidden variables) vs. quantum bound ( S_{\text{QM}} = 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}) ). - Quantum violation ratio: ( 2\sqrt{2}/2 = \sqrt{2} \approx 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 ( \sqrt{2} ) and ( 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

DOI: 10.5281/zenodo.23526081

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Andrew Stewart Caldin

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Andrew Stewart Caldin

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Sep 4, 20260 citations

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

S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2

(local hidden variables); quantum violation

S_{\text{max}} = 2\sqrt{2} \approx 2.828

. Bell's original:

P(\vec{a},\vec{b}) - P(\vec{a},\vec{b}') \leq 1 + P(\vec{b},\vec{b}')

. Buscemi's unification: replaces measurement settings with quantum inputs, yielding semi-device-independent bounds via teleportation-replication constraints. | CONNECTION: The CHSH maximum

2\sqrt{2}

is the ratio of the diagonal of a unit square to its side — a direct geometric expression of the square's symmetry (dihedral group

D_4

). The violation threshold

2

2\sqrt{2}

maps to the ratio

1:\sqrt{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

S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2

(local hidden variables); quantum mechanics predicts

S = 2\sqrt{2} \approx 2.828

. - Violation ratio:

2\sqrt{2}/2 = \sqrt{2} \approx 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

2\sqrt{2}

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

\sigma_x, \sigma_y, \sigma_z

. CONNECTION: - The

\sqrt{2}

ratio app Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2

(local hidden variables); quantum bound

S_{\text{QM}} = 2\sqrt{2} \approx 2.828

. Bell's original:

|P(a,b) - P(a,b')| + |P(a',b) + P(a',b')| \leq 2

. Buscemi's unification: replaces classical inputs with quantum states, yielding "semiquantum" Bell inequalities with tighter bounds (e.g.,

\leq 2

classical,

2\sqrt{2}

quantum, but extended to arbitrary dimensions via

2\sqrt{d}

for

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

2\sqrt{2}

is the ratio

2\sqrt{2} \approx 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

S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2

(local hidden variables) - Quantum maximum:

S_{\text{QM}} = 2\sqrt{2} \approx 2.828

(Tsirelson bound) - Ratio:

2\sqrt{2}/2 = \sqrt{2} \approx 1.414

— not a golden ratio, but note

\sqrt{2}

is the diagonal of unit square (crystallographic root system

A_1 \times A_1

) - 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

2\sqrt{2}

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

S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2

(local hidden variables), quantum max

S = 2\sqrt{2} \approx 2.828

. - Ratio

2\sqrt{2}/2 = \sqrt{2} \approx 1.414

— not directly in golden set, but note

\sqrt{2} \approx 1.414

and

1.618

are related via

\phi = (1+\sqrt{5})/2

, and

\sqrt{2}

appears in octahedral/cubic symmetry (root system

B_3

). - Buscemi's unification: Bell inequalities as linear constraints on probability polytopes; violation corresponds to points outside the local polytope — the facet structure of these polytopes encodes all possible Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com