Lemniscatic Elliptic Integral Identity: K(1/√2) = Γ(1/4)²/(4√π) — E8 Intelligence Research

FINDING: The search results are dominated by pedagogical videos on elliptic integrals, with the only substantive research item being an LIGO pulsar glitch paper — no direct theta-function/Gamma(1/4) closed-form derivations surfaced. The core mathematical content is the classical identity K(1/√2) = Γ(1/4)² / (4√π), which is the lemniscatic case. MATH: - Complete elliptic integral of first kind: K(k) = ∫₀π/2 (1 − k² sin²θ)^(−1/2) dθ - Singular value k = 1/√2 (lemniscatic case): K(1/√2) = Γ(1/4)² / (4√π) ≈ 1.854074677... - Related theta constant: θ₃(e−π) = Γ(1/4) / (√2 π3/4) — this is the classical Ramanujan/Weber result, though not explicitly in the search results. - The LIGO paper (arXiv:2104.14417) contains no elliptic integral content — it is gravitational-wave astronomy; the only "constant" is the glitch rate of PSR J0537-6910. CONNECTION: - The lemniscatic constant K(1/√2) is deeply tied to the ratio 1.854... which is NOT a golden-ratio multiple, but it does rel Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com