Metallic Means from Pell's Equation Bound Quantum Contextual Correlations — E8 Intelligence Research

FINDING: Pell's equation solutions via continued fractions generate the metallic means (silver ratio δ_S = 1+√2, etc.) as fundamental units in ℤ[√d], and these same algebraic structures bound quantum contextual correlations — the Tsirelson bound √2 emerges from the same √2 that defines the silver ratio's continued fraction [1;2,2,2,...]. MATH: - Pell's equation: x² − dy² = 1, d ∈ ℤ⁺ non-square - Continued fraction of √d: √d = [a₀; a₁,a₂,...,aₙ, 2a₀, a₁,...] (periodic, Lagrange) - Convergents pₖ/qₖ satisfy pₖ² − dqₖ² = ±1 - Metallic means: M_n = (n + √(n²+4))/2, continued fraction [n; n,n,n,...] - n=1: golden ratio φ = (1+√5)/2 = 1.6180339887... - n=2: silver ratio δ_S = 1+√2 = 2.4142135623... - n=3: bronze ratio = (3+√13)/2 = 3.3027756377... - Silver ratio identity: δ_S² = 2δ_S + 1, so δ_S = 1+√2 - Tsirelson bound: S_quantum ≤ 2√2 ≈ 2.828427... - CHSH inequality: |S| ≤ 2 (classical), |S| ≤ 2√2 (quantum) - Quantum maximum violation: S = 2√2, achieved by singlet state with optimal Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com