Generalized Zulfia Mappings: Structural Admissibility and Realization under an Auxiliary State

We extend the Zulfia kernel framework to mappings of the form Z(x, u) = η(x, u, E)+∆(x, u, S), where η takes values in a fixed inner product space E with an orthogonaldecomposition E = V1 ⊕· · ·⊕Vm, and ∆ is constrained to take values in a fixed subspaceS ⊆ E, the structural admissibility subspace, itself decomposed as S = S1 ⊕ · · · ⊕ Sk.This lets us separate the directional signature of η from the structural signature of ∆,and to define a single scalar residual, ρ(x, u) = ∥∆(x, u, S)∥, measuring how far a pairis from admissibility. We prove:(i) an exact decomposition of the Zulfia kernel into a directional part and a structuralpart (Theorem 3.1);(ii) a partial order on pairs by joint signature, together with a monotonicity result forthe residual along this order (Theorem 3.4); and(iii) that fixing S = {0} recovers the Zulfia kernel of Khan [2], and that furtherspecializing η(x, u, E) → x − u recovers the classical kernel of a linear map.We give a worked example with dim E = 4.