Law-Based Optimization under Zulfia Mapping

We develop a law-based optimization framework under the Zulfia Mapping structure. Themapping decomposes as:Z(x, u) = η(x, u) + ∆(x, u),where η(x, u) is the directional component and ∆(x, u) is the induced component. Thestructure establishes three governing optimization laws:1. Law of Directional Generation: D(x, u) = ∇f (u)⊤Z(x, u);2. Law of Inducement: I(x, u) = Φ(x, u, ∆(x, u));3. Law of Stability: limk→∞ ∥Z(x, uk)∥ = 0.These laws generate optimization directions, induced feasibility-constraint communica-tion, and stabilization behavior directly from the mapping decomposition. Classical opti-mality conditions emerge as limiting realizations of this structure.