Exponential Reduction of Detection Efficiency Loophole in Bell Tests via General Method — E8 Intelligence Research

Andrew Stewart Caldin

Published Aug 18, 2026• 0 citations

Abstract

FINDING: Detection efficiency loophole in Bell tests can be exponentially reduced by using any Bell inequality, with critical efficiency η scaling down via a general method. | MATH: Critical detection efficiency η must be surpassed to avoid loophole; method reduces η exponentially (no explicit equation given, but implies η ∝ e^{-k} for some factor k). | CONNECTION: No direct geometric ratio or crystallographic symmetry found. The exponential reduction hints at iterative or recursive structure, but no explicit link to 0.382, 0.618, 0.786, 1.618, 2.618, base-60, or root systems. | DEPTH: 6 — Important for practical quantum nonlocality, but mathematical essence is algorithmic rather than revealing new constants or symmetries. FINDING: Subtraction of "accidentals" in Aspect-type Bell tests can inflate violation statistics by up to 60%, questioning validity of some claimed violations. | MATH: Increase in test statistic up to 60% (i.e., factor 1.6). | CONNECTION: No geometric ratio or symme Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.21989383

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

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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

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Exponential Reduction of Detection Efficiency Loophole in Bell Tests via General Method — E8 Intelligence Research

Andrew Stewart Caldin

Published Aug 18, 2026• 0 citations

Publisher

Abstract

DOI: 10.5281/zenodo.21989383

Related papers

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.comFINDING: 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

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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.comFINDING: 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

Aug 18, 20260 citations

Quantum Entanglement Explained: No New Math, Just Bell's Inequality and Known Physics — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: No single mathematical discovery emerges; the corpus is popular-science exposition of quantum entanglement and nonlocality, with no new equations, constants, or experimental data beyond the known 2019 first image of entanglement. | MATH: No novel equations. Implicitly references Bell's inequality violation: $P(a,b|x,y) \neq \int d\lambda \rho(\lambda) P(a|x,\lambda)P(b|y,\lambda)$ for local hidden variables; entanglement entropy $S = -\mathrm{Tr}(\rho \log \rho)$ ; correlation function $E(a,b) = \langle \sigma_a \otimes \sigma_b \rangle$ with Tsirelson bound $E \leq 2\sqrt{2}$ for quantum correlations vs. $2$ for local. | CONNECTION: None explicit. However, the Tsirelson bound $2\sqrt{2} \approx 2.828$ relates to the golden ratio via $2\sqrt{2} = 2 \times 1.414$ , and $1.414^2 = 2$ — a Pythagorean/root-2 symmetry, not golden. No 0.382, 0.618, 0.786, 1.618, 2.618, base-60, or crystallographic structures appear in the provided text. | DEPTH: 2 — The findings are pe Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: No single mathematical discovery emerges; the corpus is popular-science exposition of quantum entanglement and nonlocality, with no new equations, constants, or experimental data beyond the known 2019 first image of entanglement. | MATH: No novel equations. Implicitly references Bell's inequality violation: $P(a,b|x,y) \neq \int d\lambda \rho(\lambda) P(a|x,\lambda)P(b|y,\lambda)$ for local hidden variables; entanglement entropy $S = -\mathrm{Tr}(\rho \log \rho)$; correlation function $E(a,b) = \langle \sigma_a \otimes \sigma_b \rangle$ with Tsirelson bound $E \leq 2\sqrt{2}$ for quantum correlations vs. $2$ for local. | CONNECTION: None explicit. However, the Tsirelson bound $2\sqrt{2} \approx 2.828$ relates to the golden ratio via $2\sqrt{2} = 2 \times 1.414$, and $1.414^2 = 2$ — a Pythagorean/root-2 symmetry, not golden. No 0.382, 0.618, 0.786, 1.618, 2.618, base-60, or crystallographic structures appear in the provided text. | DEPTH: 2 — The findings are pe Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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No Source Addresses Detection-Loophole Threshold 2(√2−1) or Its Pell-Equation Structure — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: The search returned only pedagogical/experimental Bell-test material (NIST loophole-free test, IBM-Q implementations, 3Blue1Brown explainers) — no source in the result set actually addresses the detection-loophole threshold 2(√2−1) or its Pell-equation/metallic-mean structure. | MATH: The relevant quantity is the critical detection efficiency η* = 2(√2−1) ≈ 0.8284, the threshold below which a local-hidden-variable (LHV) model can reproduce observed correlations in a two-outcome, two-setting Bell test (CHSH). Derivation context: CHSH bound S ≤ 2 for LHV; quantum max S = 2√2; the "fair-sampling" assumption fails when η < η*, allowing an LHV model to fake S > 2. Note 2(√2−1) = 2/φ²·(√2−1)/(...) is not a clean golden-ratio expression; it is the positive root of x² + 4x − 4 = 0, i.e. x = −2 + 2√2. | CONNECTION: The claimed link to the "silver mean" δ_S = 1 + √2 ≈ 2.414 (Pell equation x² − 2y² = ±1) is not supported by the retrieved sources. The threshold 2(√2−1) ≈ 0.8284 is the rec Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: The search returned only pedagogical/experimental Bell-test material (NIST loophole-free test, IBM-Q implementations, 3Blue1Brown explainers) — no source in the result set actually addresses the detection-loophole threshold 2(√2−1) or its Pell-equation/metallic-mean structure. | MATH: The relevant quantity is the critical detection efficiency η* = 2(√2−1) ≈ 0.8284, the threshold below which a local-hidden-variable (LHV) model can reproduce observed correlations in a two-outcome, two-setting Bell test (CHSH). Derivation context: CHSH bound S ≤ 2 for LHV; quantum max S = 2√2; the "fair-sampling" assumption fails when η < η*, allowing an LHV model to fake S > 2. Note 2(√2−1) = 2/φ²·(√2−1)/(...) is not a clean golden-ratio expression; it is the positive root of x² + 4x − 4 = 0, i.e. x = −2 + 2√2. | CONNECTION: The claimed link to the "silver mean" δ_S = 1 + √2 ≈ 2.414 (Pell equation x² − 2y² = ±1) is not supported by the retrieved sources. The threshold 2(√2−1) ≈ 0.8284 is the rec Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 31, 20260 citations

Silver Ratio and Pell Equation in the Bell Test Detection-Loophole Threshold — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: The detection-loophole efficiency threshold in Bell tests is η ≥ 2(√2−1) ≈ 0.8284, the reciprocal of the silver ratio δ_S = 1+√2 ≈ 2.4142, arising from the geometry of the CHSH inequality and connected to the Pell equation x²−2y²=±1. MATH: - Silver ratio: δ_S = 1 + √2 = 2.41421356… ; 1/δ_S = √2 − 1 = 0.41421356… - Threshold: η_crit = 2(√2 − 1) = 2√2 − 2 = 0.82842712… - Equivalent form: η_crit = 2/δ_S = 2/(1+√2) - CHSH bound: |S| ≤ 2√2 (Tsirelson); classical |S| ≤ 2 - Pell equation: x² − 2y² = ±1, solutions (x,y) = (1,0),(1,1),(3,2),(7,5),(17,12),(41,29)… - Convergents of √2: 1/1, 3/2, 7/5, 17/12, 41/29 → ratios approach δ_S - Continued fraction: √2 = [1; 2,2,2,…] (all-2 periodicity = "silver" metallic mean) - Metallic means: δ_n = (n + √(n²+4))/2 ; n=1 → φ=1.618, n=2 → δ_S=2.414 CONNECTION: - 0.8284 = 2(√2−1) is the silver-ratio conjugate doubled. Silver ratio δ_S = 1+√2 is the n=2 metallic mean, sibling of golden ratio φ = (1+√5)/2 (n=1). - Pell equation x²−2y²=±1 is the in Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: The detection-loophole efficiency threshold in Bell tests is η ≥ 2(√2−1) ≈ 0.8284, the reciprocal of the silver ratio δ_S = 1+√2 ≈ 2.4142, arising from the geometry of the CHSH inequality and connected to the Pell equation x²−2y²=±1. MATH: - Silver ratio: δ_S = 1 + √2 = 2.41421356… ; 1/δ_S = √2 − 1 = 0.41421356… - Threshold: η_crit = 2(√2 − 1) = 2√2 − 2 = 0.82842712… - Equivalent form: η_crit = 2/δ_S = 2/(1+√2) - CHSH bound: |S| ≤ 2√2 (Tsirelson); classical |S| ≤ 2 - Pell equation: x² − 2y² = ±1, solutions (x,y) = (1,0),(1,1),(3,2),(7,5),(17,12),(41,29)… - Convergents of √2: 1/1, 3/2, 7/5, 17/12, 41/29 → ratios approach δ_S - Continued fraction: √2 = [1; 2,2,2,…] (all-2 periodicity = "silver" metallic mean) - Metallic means: δ_n = (n + √(n²+4))/2 ; n=1 → φ=1.618, n=2 → δ_S=2.414 CONNECTION: - 0.8284 = 2(√2−1) is the silver-ratio conjugate doubled. Silver ratio δ_S = 1+√2 is the n=2 metallic mean, sibling of golden ratio φ = (1+√5)/2 (n=1). - Pell equation x²−2y²=±1 is the in Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Nov 1, 20260 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.comFINDING: 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, 20260 citations

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

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

P(a,b|x,y) \neq \int d\lambda \rho(\lambda) P(a|x,\lambda)P(b|y,\lambda)

for local hidden variables; entanglement entropy

S = -\mathrm{Tr}(\rho \log \rho)

; correlation function

E(a,b) = \langle \sigma_a \otimes \sigma_b \rangle

with Tsirelson bound

E \leq 2\sqrt{2}

for quantum correlations vs.

2

for local. | CONNECTION: None explicit. However, the Tsirelson bound

2\sqrt{2} \approx 2.828

relates to the golden ratio via

2\sqrt{2} = 2 \times 1.414

, and

1.414^2 = 2

— a Pythagorean/root-2 symmetry, not golden. No 0.382, 0.618, 0.786, 1.618, 2.618, base-60, or crystallographic structures appear in the provided text. | DEPTH: 2 — The findings are pe 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 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