A Class of Arithmetica Programming under Zulfia Invexity

In this paper, we introduce a class of Arithmetica programming problems underZulfia invexity through the operator-generated mappingZ(x, u) = η(x, u) @ ∆(x, u), @ = (+, −, ×, ÷, hybrid),where η(x, u) and ∆(x, u) denote the directional and structural components,respectively. Based on the constrained Arithmetica programming problem, ZKKTconditions are constructed under the proposed framework.Within this setting, an induced sufficiency theorem is established under Zul-fia invexity together with its associated induced sufficiency mechanism. Further-more, an induced realization theorem is developed to support realization behaviorgenerated by the operator architecture. Realization models including equilibrium,entropy, and likelihood are presented as representative directions of the proposedframework.