Zulfia Optimality Conditions for Mathematical Programming Problems: Sufficiency Beyond KKT: Applicable Where KKT Fails,

Classical optimality theory for Mathematical Programming Problems (MPPs) rests onKarush-Kuhn-Tucker (KKT) conditions, derived as equations under constraint qualifications,convexity, and differentiability. In this paper, we establish the Zulfia Optimality Conditions(ZOC) as a theory derived from a single identity, the Zulfia IdentityM (C) = ∆,where M is the primitive mapping, C is the configuration of the problem, and ∆ is the bladeoperator measuring deviation from the Blade-Point.Unlike KKT, which verify a static point x∗, ZOC operate on ∆. The ZOC areZOC: M (C∗) = ∆(C∗) = 0.We prove sufficiency:∆(C∗) = 0 =⇒ C∗ is optimal,without LICQ/MFCQ, without convexity, and without differentiability. Moreover, ZOCremain applicable where KKT fail, are degenerate, or cannot be defined, including non-convex, non-differentiable, and degenerate cases. We show KKT emerge as a special example,∇L(x∗, λ∗) = 0, of the general Zulfia Identity theory. Consequently, ZOC extend optimalitybeyond KKT.