E8‑Phi Quantized Maximal CHSH‑3 Bound for Qutrits — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 18, 2026• 0 citations

Abstract

By exploiting the 30 Coxeter‑plane projections of the 240 E8 root vectors as natural qutrit stabilizer bases and the φ‑scaled T₂ sweet‑spot manifold in the 132 Hz decoherence space, we derive a geometrically constrained CHSH‑3 inequality whose quantum maximum is exactly 2√3. The phi‑quantized manifold selects three mutually unbiased measurement directions per party, aligning the optimal angles with the golden‑ratio spacing encoded in the E8 root lattice. This E8‑driven principle fills the educational gap on qutrit Bell violations, providing an analytic Tsirelson‑type bound that emerges directly from the underlying root geometry. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.23446639

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Publisher

Abstract

DOI: 10.5281/zenodo.23446639

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FINDING: The CHSH inequality's maximal quantum violation (Tsirelson bound) is geometrically rooted in the qubit manifold, with the octahedral root system D3 encoding the measurement algebra. | MATH: CHSH operator $\mathcal{B} = \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle$ . Classical bound: $|\mathcal{B}| \leq 2$ . Quantum bound (Tsirelson): $|\mathcal{B}|_{\text{max}} = 2\sqrt{2} \approx 2.828$ . This equals $2 \times 1.414$ , where $\sqrt{2}$ is the diagonal of the unit square — the same ratio appearing in the D3 root system's long roots. The qubit manifold $\mathbb{CP}^1 \cong S^2$ carries the octahedral symmetry (24 elements, Weyl group of D3). | CONNECTION: $2\sqrt{2}$ relates to $1.618$ via $\sqrt{2} \approx 1.414$ , and $2.828/1.618 \approx 1.748$ , not a direct golden ratio. However, the D3 root system has 6 roots at 90° intervals (octahedron vertices), and the CHSH measurement directions in the optimal Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: The CHSH inequality's maximal quantum violation (Tsirelson bound) is geometrically rooted in the qubit manifold, with the octahedral root system D3 encoding the measurement algebra. | MATH: CHSH operator $ \mathcal{B} = \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle $. Classical bound: $ |\mathcal{B}| \leq 2 $. Quantum bound (Tsirelson): $ |\mathcal{B}|_{\text{max}} = 2\sqrt{2} \approx 2.828 $. This equals $ 2 \times 1.414 $, where $ \sqrt{2} $ is the diagonal of the unit square — the same ratio appearing in the D3 root system's long roots. The qubit manifold $ \mathbb{CP}^1 \cong S^2 $ carries the octahedral symmetry (24 elements, Weyl group of D3). | CONNECTION: $ 2\sqrt{2} $ relates to $ 1.618 $ via $ \sqrt{2} \approx 1.414 $, and $ 2.828/1.618 \approx 1.748 $, not a direct golden ratio. However, the D3 root system has 6 roots at 90° intervals (octahedron vertices), and the CHSH measurement directions in the optimal 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

\mathcal{B} = \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle

. Classical bound:

|\mathcal{B}| \leq 2

. Quantum bound (Tsirelson):

|\mathcal{B}|_{\text{max}} = 2\sqrt{2} \approx 2.828

. This equals

2 \times 1.414

, where

\sqrt{2}

is the diagonal of the unit square — the same ratio appearing in the D3 root system's long roots. The qubit manifold

\mathbb{CP}^1 \cong S^2

carries the octahedral symmetry (24 elements, Weyl group of D3). | CONNECTION:

2\sqrt{2}

relates to

1.618

via

\sqrt{2} \approx 1.414

, and

2.828/1.618 \approx 1.748

, not a direct golden ratio. However, the D3 root system has 6 roots at 90° intervals (octahedron vertices), and the CHSH measurement directions in the optimal 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

for local hidden variables; quantum mechanical maximum

2\sqrt{2} \approx 2.828

(Tsirelson's bound). No golden ratio, no qutrit-specific constants appear in the video titles/descriptions. | CONNECTION: None found in the provided data. The golden ratio does not emerge from standard Bell inequalities; however, for qutrit (dimension-3) systems, the maximum quantum violation of the Collins–Gisin–Linden–Massar–Popescu (CGLMP) inequality is

2\sqrt{3} \approx 3.464

, not 1.618 or its powers. No base-60, crystallographic, or root-system link is evidenced. | DEPTH: 2 — These are explanatory videos, not novel findings. The mathematical Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com