Wings of Functions: Classical to Generalization

The evolution of generalized convexity from 1940 to 2026 has been driven by definitions,not theorems. Convexity is characterized by η(x, u) = x − u and invexity by a generalmap η(x, u) (Hanson, 1981) via f (x) − f (u) ≥ ∇f (u)T η(x, u). Both are sign-basedinequalities and do not provide a computable measure of optimality.In this paper, we introduce the Wings of Functions framework. For f : Rn → Rdifferentiable at u, define structural:1. Map Z(x, u) = η(x, u) + ∆(x, u) with η(u, u) = 0,2. Residual R(x, u) = f (x) − f (u) − ∇f (u)T Z(x, u),3. Closure C(x, u) = |R(x, u)| + ∥∆(x, u)∥.The quadruple (η, ∆, R, C) is called the wings of f at u. A function is said to be Zulfiable[1] if such Z exists. We prove:• If ∆ = 0 and η(x, u) = x − u, then Zulfiability reduces to convexity; if ∆ = 0,it reduces to invexity; thus invexity and convexity are special realizations with∆ = 0,• C(x, u) = 0 ⇐⇒ R(x, u) = 0 and ∆(x, u) = 0,• For ϵ > 0, |C(x, u)| < ϵ provides a Computable Universal Optimality Certificate(CUO).The framework extends to Generalized Zulfiability via Lp-closures Cp(x, u) = (|R(x, u)|p+∥∆(x, u)∥p)1/p for 1 ≤ p < ∞, nested structures η = η0 + ∆η, and operator norms ∥C∥Xin Banach spaces.