Quantum Non-Locality Unification via Semiquantum Games and Ballistic-Electron Tests — E8 Intelligence Research

Andrew Stewart Caldin

Published Sep 15, 2026• 0 citations

Abstract

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') \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

DOI: 10.5281/zenodo.22762279

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Publisher

Abstract

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') \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

DOI: 10.5281/zenodo.22762279

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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.comFINDING: 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

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Andrew Stewart Caldin

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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

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 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

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

(CHSH form:

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

); quantum violation

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

(Tsirelson bound). Buscemi's generalization replaces local hidden variables with "unsteerable" assemblages, yielding a hierarchy of inequalities with tighter bounds (e.g.,

S_{\text{Buscemi}} \leq 2

for classical,

2\sqrt{2}

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 \approx 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 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

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