Zulfia Decomposition Mapping: Foundation of Zulfia Theory

We introduce the Zulfia Decomposition Mapping — hereafter the Zulfia Map-ping — defined asZ(x, u) = η(x, u) + ∆(x, u)where η(x, u) is the Directional Component and ∆(x, u) is the StructuralComponent. This single mapping, founded on no external assumptions, consti-tutes the seed of Zulfia Theory: the Theory of Emergence and Conver-gence.We establish three fundamental theorems: the Unavoidable DecompositionTheorem, the Unavoidable Separation Theorem, and the Theorem of In-trinsic Feasibility. We introduce the class of Zulfiable functions and the prop-erty of Zulfiability as the general framework from which all classical theoriesemerge as special limiting cases.The defining characteristic of the Zulfia Mapping is that feasibility is not as-sumed — it is constructed, generated, unavoidable, and inseparable fromthe mapping itself. This makes Zulfia Theory the first self-contained mathematicalframework in which feasibility, differentiability and optimality are all intrinsic —requiring no external assumption whatsoever