Zulfiability: Structural Origin, Residual Optimality and the Unification-Reduction Closure C = |R| + ∥∆∥

Classical invex inequality f (x) − f (u) ≥ ∇f (u)T η(x, u) is sign-based and non-measurable. We establish Zulfiability as structural origin:• Structural mapping Z(x, u) = η(x, u) + ∆(x, u), where η is kernel directionand ∆ is deviation field,• Residual R(x, u) = f (x) − f (u) − ∇f (u)T Z(x, u), and• Closure scale C(x, u) = |R| + ∥∆∥.We show: if ∆ = 0, Z reduces to invexity; if η = x − u, it reduces to convexity.Hence convexity and invexity are not primitive but emergent realizations of Z. Weprove C = 0 ⇐⇒ R = 0, ∆ = 0 ⇐⇒ f (x) = f (u). The criterion |C| < ϵ gives aComputable Universal Optimality (CUO) Certificate serving:(i) Computability from local data (f, ∇f, Z),(ii) Universality across Optimization, Physics, Game, and Statistics,(iii) Rigorous ϵ-optimality guarantee.