Legendre–Fenchel Duality as the Symmetry Linking Optimization, Geometry, and Self-Dual Functions — E8 Intelligence Research

FINDING: Legendre–Fenchel duality is the fundamental symmetry linking primal optimization, information geometry, and convex geometry — with self-dual functions forming a fixed-point class that mirrors harmonic ratios. MATH: - Legendre–Fenchel conjugate: \( f^*(p) = \sup_x \{ \langle p, x \rangle - f(x) \} \). - Self-duality condition: \( f = f^* \) — the fixed points of the transform. On \(\mathbb{R}^n\), these are exactly \( f(x) = \frac{1}{2} \|x\|^2 + c \) (up to orthogonal transformations) — the quadratic self-dual family. - Smoothness ↔ strong convexity duality: \( f \) is \( \beta \)-smooth iff \( f^* \) is \( 1/\beta \)-strongly convex. This is the exact reciprocal symmetry: \( \beta \cdot (1/\beta) = 1 \). - In information geometry, the Legendre transform maps between dual coordinate systems (e.g., natural vs. expectation parameters of exponential families), with the Fisher metric invariant under the transform. - Complex Legendre duality (Klartag): extends to comple Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com