Heptagonal Nested Radicals via Chebyshev T7 and Cubic Field Link — E8 Intelligence Research

FINDING: Nested radical \(2cos(2π/7)\) encodes the heptagon's cubic field via Chebyshev \(T_7\), linking a 7×7 determinant to \(cos(7θ)\) and enabling periodic nested radical solutions of certain polynomials. | MATH: Chebyshev identity: \(T_7(x) = 64x^7 - 112x^5 + 56x^3 - 7x = cos(7 x)\). For \(x = 2cos(2π/7)\), \(T_7(x/2) = cos(2π) = 1\) → \(x\) satisfies \(x^3 - x^2 - 2x + 1 = 0\) (minimal polynomial, degree 3, discriminant 49). Nested radical: \(2cos(2π/7) = √[3]{{7 + 21√-3}{2}} + √[3]{{7 - 21√-3}{2}}\) (Cardano form). Ramanujan nested radical: \(√[3]{7/2 + 3/2√-3} + √[3]{7/2 - 3/2√-3}\). The 7×7 determinant: \((I - 2x A)\) where \(A\) is adjacency of cycle \(C_7\) yields \(T_7(x/2) - 1\), linking to Chebyshev. | CONNECTION: Heptagon central angle \(2π/7\) → \(2cos(2π/7) ≈ 1.24698\). Golden ratio cousin: \(2cos(π/7) ≈ 1.80194\), \(2cos(3π/7) ≈ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com