The Euler-Mascheroni Constant from First Principles: 66-Digit Precision from Icosahedral Geometry

We derive a closed-form expression for the Euler-Mascheroni constant γ achieving 66-digit precision through a multi-pass harmonic recurrence structure rooted in icosahedral geometry. The base formula γ = (e² + 5)/(9e − 3) encodes the icosahedral vertex count V = 12, with successive correction terms following a Fibonacci-like recurrence in denominators (ae + b)^p. Harmonic numbers Hₙ appear in coefficients, connecting to the classical definition γ = lim(Hₙ − ln n). A key insight is that γ√3 ≈ 1 (accurate to 0.02%), revealing γ as the Eisenstein dual of unity in D = 3 dimensions. The recurrence coefficient 19 = 2D² + 1 = E − V + 1 appears as a universal capacity threshold with connections to K3 moduli and the Hodge conjecture. This is the first derivation of γ from geometric first principles