Satellite Bell Tests Confirm Distance-Independent Quantum Nonlocality at Tsirelson Bound 2√2 — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 30, 2026• 0 citations

Abstract

FINDING: Satellite Bell tests confirm Tsirelson bound 2√2 as distance-independent — quantum nonlocality persists across 1200 km with no attenuation of correlation strength | MATH: CHSH inequality |S| ≤ 2 (classical local hidden variables); quantum maximum |S| = 2√2 ≈ 2.8284; measured S values in satellite experiments ~2.5–2.7 (limited by detector efficiency, not distance); no distance-dependent decay term observed | CONNECTION: 2√2 = 2 × √2 is the operator norm of the tensor product of two Pauli observables; it arises from the geometry of the Bloch sphere (real projective plane RP²) — specifically from the maximum of ⟨A₁B₁⟩ + ⟨A₁B₂⟩ + ⟨A₂B₁⟩ − ⟨A₂B₂⟩ over projective measurements. The ratio 2√2/2 = √2 ≈ 1.414 is the quantum-to-classical gap, not a Fibonacci/φ ratio. However, √2 is the diagonal of the unit square and appears in the octahedral/cubic root system B₃ normalization. No direct link to 0.618, 1.618, or base-60 in the Tsirelson bound itself | DEPTH: 7 --- FINDING: Bell nonloca Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.23796752

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Satellite Bell Tests Confirm Distance-Independent Quantum Nonlocality at Tsirelson Bound 2√2 — E8 Intelligence Research

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Publisher

Abstract

DOI: 10.5281/zenodo.23796752

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Satellite Bell Test: Distance-Independent Quantum Nonlocality at 1200 km — E8 Intelligence Research

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FINDING: Quantum nonlocality (Bell violation) is distance-independent — satellite Bell tests confirm violation of the CHSH inequality at 1200 km, saturating toward the Tsirelson bound 2√2, with no decay attributable to separation. | MATH: CHSH parameter S ≤ 2 (local hidden variables) vs. S_max = 2√2 ≈ 2.828 (quantum/Tsirelson). Quantum correlation E(a,b) = -cos(a-b) for singlet state; S = |E(a,b) - E(a,b') + E(a',b) + E(a',b')|. Maximal violation at measurement angles a=0°, a'=45°, b=22.5°, b'=67.5°. | CONNECTION: 2√2 = 2·√2 is the diagonal of a 2×2 square (side 2), i.e. the space diagonal of a unit cube scaled — a root-system constant of the B₂/C₂ lattice. The angle set {0°, 22.5°, 45°, 67.5°} partitions 90° into four equal 22.5° steps = 90/4, a base-4 (dyadic) subdivision, not golden. No 0.618/1.618 ratio appears in the optimal CHSH angles; the geometry is square/cubic (√2), not pentagonal (φ). | DEPTH: 7 Notes on the remaining findings (lower relevance, extracted for completeness): Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Quantum nonlocality (Bell violation) is distance-independent — satellite Bell tests confirm violation of the CHSH inequality at 1200 km, saturating toward the Tsirelson bound 2√2, with no decay attributable to separation. | MATH: CHSH parameter S ≤ 2 (local hidden variables) vs. S_max = 2√2 ≈ 2.828 (quantum/Tsirelson). Quantum correlation E(a,b) = -cos(a-b) for singlet state; S = |E(a,b) - E(a,b') + E(a',b) + E(a',b')|. Maximal violation at measurement angles a=0°, a'=45°, b=22.5°, b'=67.5°. | CONNECTION: 2√2 = 2·√2 is the diagonal of a 2×2 square (side 2), i.e. the space diagonal of a unit cube scaled — a root-system constant of the B₂/C₂ lattice. The angle set {0°, 22.5°, 45°, 67.5°} partitions 90° into four equal 22.5° steps = 90/4, a base-4 (dyadic) subdivision, not golden. No 0.618/1.618 ratio appears in the optimal CHSH angles; the geometry is square/cubic (√2), not pentagonal (φ). | DEPTH: 7 Notes on the remaining findings (lower relevance, extracted for completeness): Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Andrew Stewart Caldin

FINDING: CHSH inequality violation at 2√2 is the maximal quantum bound, separating quantum correlations from classical (≤2) and signaling a fundamental limit tied to Hilbert-space geometry. | MATH: Classical bound: ⟨CHSH⟩ ≤ 2. Quantum bound (Tsirelson): ⟨CHSH⟩ ≤ 2√2 ≈ 2.828. Violation ratio: 2√2 / 2 = √2 ≈ 1.414. The gap (2√2 − 2) ≈ 0.828. In base-60: 2√2 ≈ 2°49'41" (2.8284 × 60 = 169.704′ → 2′49.7″). | CONNECTION: 2√2 = √(8) = 2 × 1.414. Note 1.414² = 2. The ratio 2/2√2 = 1/√2 ≈ 0.7071, which is sin(45°) — the diagonal of a unit square. The golden ratio φ = 1.618 appears in the *ratio* (2√2)/φ ≈ 1.748, not directly. However, √2 is the root of the square lattice (crystallographic point group D₄), and 2√2 is the length of the diagonal of a 2×2 square — a lattice vector in the square lattice. The CHSH-3 inequality (arXiv:2003.00124) extends to ternary outcomes, with violations computed via semidefinite programming — the geometry of the set of quantum correlations is a convex body whose e Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: CHSH inequality violation at 2√2 is the maximal quantum bound, separating quantum correlations from classical (≤2) and signaling a fundamental limit tied to Hilbert-space geometry. | MATH: Classical bound: ⟨CHSH⟩ ≤ 2. Quantum bound (Tsirelson): ⟨CHSH⟩ ≤ 2√2 ≈ 2.828. Violation ratio: 2√2 / 2 = √2 ≈ 1.414. The gap (2√2 − 2) ≈ 0.828. In base-60: 2√2 ≈ 2°49'41" (2.8284 × 60 = 169.704′ → 2′49.7″). | CONNECTION: 2√2 = √(8) = 2 × 1.414. Note 1.414² = 2. The ratio 2/2√2 = 1/√2 ≈ 0.7071, which is sin(45°) — the diagonal of a unit square. The golden ratio φ = 1.618 appears in the *ratio* (2√2)/φ ≈ 1.748, not directly. However, √2 is the root of the square lattice (crystallographic point group D₄), and 2√2 is the length of the diagonal of a 2×2 square — a lattice vector in the square lattice. The CHSH-3 inequality (arXiv:2003.00124) extends to ternary outcomes, with violations computed via semidefinite programming — the geometry of the set of quantum correlations is a convex body whose e Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 13, 20260 citations

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

FINDING: Tsirelson bound is the maximal quantum violation of the CHSH inequality, equal to

2\sqrt{2}

, and recent work explores its algebraic structure via C*-algebra tensor products (Tsirelson pairs). MATH: CHSH inequality:

\langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle \leq 2

(classical). Quantum bound:

2\sqrt{2} \approx 2.828

. Tsirelson's theorem: for any quantum observables

A, A', B, B'

with

\pm1

eigenvalues, the maximum is

2\sqrt{2}

. The ratio

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

. CONNECTION:

\sqrt{2}

is the diagonal of a unit square, linking to 2D Euclidean geometry and the root system

A_1 \times A_1

(or

D_2

). The ratio

1.414

is not a golden ratio (1.618) but appears in octonion norm and Clifford algebra structures. No direct link to 0.382, 0.618, 0.786, 2.618, or base-60. Crystallographic:

\sqrt{2}

appears in square lattice and body-centered tetragonal symmetries. DEPTH: 8 — Tsirels Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com