Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Bell Inequality Tests Unify Entanglement; Ballistic Electron Experiment Proposed — E8 Intelligence Research

Andrew Stewart Caldin

Published Aug 18, 2026• 0 citations
Publisher

Abstract

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: \( S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2 \) (local hidden variable bound); quantum violation up to \( 2\sqrt{2} \approx 2.828 \). 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

DOI
10.5281/zenodo.21989157

Related papers

Unified Bell Inequalities Tested via Ballistic Electrons in Quantum Wires — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: New rules for Bell inequalities unify quantum entanglement tests, with a proposed condensed-matter experiment using ballistic electrons in semiconductor quantum wires. | MATH: CHSH inequality form: \( S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2 \) (classical bound); quantum violation up to \( 2\sqrt{2} \approx 2.828 \); proposed system uses five gates to generate entangled states via Coulomb coupling in quantum wires. | CONNECTION: The ratio \( 2.828 / 2 = 1.414 \) (√2) appears, linking to the diagonal of a unit square — a fundamental geometric symmetry. No direct evidence of 0.382, 0.618, 0.786, 1.618, 2.618, base-60, or crystallographic symmetries in these findings. | DEPTH: 6 — advances understanding of entanglement in condensed-matter systems, but no new fundamental constants or geometric harmonies beyond √2. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Aug 13, 2026•0 citations

Quantum Nonlocality, Entanglement Geometry, and a Ballistic Electron Test — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Bell inequality tests reveal quantum nonlocality; new rules unify entanglement geometry; ballistic electron experiment proposed. | MATH: Bell inequality: \( |E(a,b) - E(a,c)| \leq 1 + E(b,c) \) (CHSH form: \( S = |E(a,b) + E(a,b') + E(a',b) - E(a',b')| \leq 2 \); quantum violation up to \( 2\sqrt{2} \approx 2.828 \)). Ratio \( 2.828/2 = 1.414 = \sqrt{2} \). | CONNECTION: \( \sqrt{2} \) is the diagonal of a unit square, linking to 2D orthogonal basis rotations; no direct golden ratio or base-60 link. Crystallographic symmetry: \( \sqrt{2} \) appears in square lattice (root system \( A_1 \times A_1 \), \( D_2 \)). | DEPTH: 6 — foundational but no new geometric constants; advances understanding of quantum vs. classical correlation boundaries. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Aug 13, 2026•0 citations

Ballistic Electron Bell Test: A Semiconductor Platform for Nonlocality Verification — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Bell inequality test with ballistic electrons in semiconductors proposes a condensed-matter platform for entanglement and nonlocality verification. | MATH: Bell inequality violation condition: \( S = |E(a,b) - E(a,b')| + |E(a',b) + E(a',b')| \leq 2 \) (local hidden variable bound); quantum mechanical maximum \( S = 2\sqrt{2} \approx 2.828 \); 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 \( 2\sqrt{2} \) 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

•Aug 20, 2026•0 citations

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

Andrew Stewart Caldin

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

•Sep 1, 2026•0 citations

Quantum Non-Locality Confirmed: Unified Frameworks and Ballistic Electron Tests — E8 Intelligence Research

Andrew Stewart Caldin

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

•Aug 27, 2026•0 citations

Quantum Non-Locality: From Bell Tests to Solid-State Violations — E8 Intelligence Research

Andrew Stewart Caldin

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: \(S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2\) (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 \(2\) vs \(2\sqrt{2}\) 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

•Sep 9, 2026•0 citations
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') \leq 2 ) (local hidden variable bound); quantum violation up to ( 2\sqrt{2} \approx 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: New rules for Bell inequalities unify quantum entanglement tests, with a proposed condensed-matter experiment using ballistic electrons in semiconductor quantum wires. | MATH: CHSH inequality 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 (classical bound); quantum violation up to 22≈2.828 2\sqrt{2} \approx 2.828 22​≈2.828; proposed system uses five gates to generate entangled states via Coulomb coupling in quantum wires. | CONNECTION: The ratio 2.828/2=1.414 2.828 / 2 = 1.414 2.828/2=1.414 (√2) appears, linking to the diagonal of a unit square — a fundamental geometric symmetry. No direct evidence of 0.382, 0.618, 0.786, 1.618, 2.618, base-60, or crystallographic symmetries in these findings. | DEPTH: 6 — advances understanding of entanglement in condensed-matter systems, but no new fundamental constants or geometric harmonies beyond √2. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Bell inequality tests reveal quantum nonlocality; new rules unify entanglement geometry; ballistic electron experiment proposed. | MATH: Bell inequality: ∣E(a,b)−E(a,c)∣≤1+E(b,c) |E(a,b) - E(a,c)| \leq 1 + E(b,c) ∣E(a,b)−E(a,c)∣≤1+E(b,c) (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; quantum violation up to 22≈2.828 2\sqrt{2} \approx 2.828 22​≈2.828). Ratio 2.828/2=1.414=2 2.828/2 = 1.414 = \sqrt{2} 2.828/2=1.414=2​. | CONNECTION: 2 \sqrt{2} 2​ is the diagonal of a unit square, linking to 2D orthogonal basis rotations; no direct golden ratio or base-60 link. Crystallographic symmetry: 2 \sqrt{2} 2​ appears in square lattice (root system A1×A1 A_1 \times A_1 A1​×A1​, D2 D_2 D2​). | DEPTH: 6 — foundational but no new geometric constants; advances understanding of quantum vs. classical correlation boundaries. 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
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 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