Balance Equilibrium Framework: A New Class of Equilibrium under Zulfiability

This paper introduces a new class of equilibrium framework under Zulfiable throughthe decomposition of the Zulfia mappingZ(x, u) = η(x, u) + ∆(x, u),where η(x, u) represents the directional component and ∆(x, u) represents the inducedstructural interaction component. Using this decomposition, the Balance EquilibriumFramework is developed through the balance lawsD(x, u) = ∇f (u)T Z(x, u),I(x, u) = Φ(x, u, ∆),andlimk→∞ Sk = 0,representing directional generation, induced balance, and stabilization, respectively.Using differentiable feasible structures, Balance KKT conditions are established toderive a sufficiency theorem for balance equilibrium systems under Zulfiability. Aninduced mechanism theorem is further developed to explain how decomposition-inducedinteraction preserves balance stabilization within the equilibrium structure.