Residual Optimality and Classical Recovery under Zulfia Invexity

Classical optimization theory is formulated on feasibility, where admissible sets aredefined through constraint satisfaction and treated as given. Optimality conditionssuch as the Karush–Kuhn–Tucker (KKT) system are then derived on this predefineddomain.In this paper, we introduce a primitive admissibility framework under Zulfia Invex-ity, characterized by the structural relationf (x) − f (u) ≥ ∇f (u)T Z(x, u),together with the decompositionZ(x, u) = η(x, u) + ∆(x, u),and the residual representationR(x, u) = F (x, u) + Φ(x, u).Here F (x, u) denotes admissibility and Φ(x, u) represents the residual component gov-erning optimality. On this primitive layer, admissible states are determined throughthe residual structure rather than through classical feasibility alone. In the differen-tiable case, this framework yields a first-order condition of KKT type as a projectedrealization of the residual structure. Classical optimization, statistical inference, andphysical law formulations may therefore be interpreted as recovered realizations of thesame primitive residual-admissibility mechanism.