Beyond Three Centuries of Assumption: Global Sufficiency via Zulfiability and the Zulfia Optimality Condition

Since Leibniz and Newton introduced calculus in the seventeenth century, every suffi-ciency condition in optimization has rested on three assumptions: differentiability,convexity (or invexity), and constraint qualification. These assumptions persistedunquestioned through Lagrange multipliers (1755), the Karush-Kuhn-Tucker condi-tions (1939), and invexity theory (Hanson, 1981). For three centuries, no result askedwhether they were truly necessary.This paper shows they are not. Within the framework of , founded on the ZulfiaDecomposition Mapping Z(x, u) = η(x, u) + ∆(x, u) established in Paper I [1], weintroduce Zulfiability as the function class and the Zulfia Optimality Condition(ZOC) as the single sufficiency criterion:κZ < 1where κZ ∈ [0, 1) is the Zulfia contraction constant. We prove that any Zulfiablefunction satisfying the ZOC has a global minimizer — without differentiability, withoutconvexity, and without constraint qualification.Five concrete advantages of ZOC over KKT are established. We show that KKT isa special case of ZOC, not an alternative. Three examples are presented where KKTfails entirely but ZOC identifies the global minimizer.