Unified Invariant Optimization Theory (UI Optimization Theory): Force–Energy–Geometry–Curvature

We develop a unified invariant formulation of optimization in which optimality is ex-pressed as a structural identity preserved under admissible transformationsTη(x, y) = x + η(y, x).Under this transformation, the stationary identity∇f (x∗) = 0generates the invariant optimality conditionf (Tη(x∗, y)) ≥ f (x∗) ∀y.This invariant identity naturally defines the structural parameterΦ(y) = f (y) − f (x∗),which remains nonnegative and serves as the invariant identity parameter of optimizationsystems.From this invariant structure emerge four inseparable faces: force (equilibrium identity),energy (Φ), geometry (transformation structure), and curvature (admissibility preservation).Classical convex and invex optimization appear as special cases.This establishes the foundation of Unified Invariant Optimization Theory.