Zulfiability as a Structural Origin of Generalized Function Classes

AbstractThis paper introduces Zulfiability as a structural origin arising from a prim-itive transformation mapping Z(x, u), rather than as a generalization of convexlike functions. The central object is a mapping that governs admissible structuraltransitions between points. Under this formulation, the inequalityf (x) − f (u) ≥ ∇f (u)T Z(x, u)is interpreted as a fundamental structural relation from which known functionclasses emerge.The mapping Z(x, u) admits a decompositionZ(x, u) = η(x, u) + ∆(x, u),which reveals an internal hierarchy of structure: invexity appears as a lower projec-tion (∆ = 0), while convexity arises as a further reduction η(x, u) = x − u. Nonzerostructural components generate richer function classes beyond classical frameworks.This perspective establishes Zulfiability as an origin layer connecting directionalreductions and structural closures, providing a unified transformation-based foun-dation for optimality, sufficiency, and duality