Parity lifting of Boolean Hessian systems: A smallest obstruction and a cubic lifting theorem

We study prescribed coordinate parities of integer functions satisfying the tropical incidence-Pluecker relations on the complete Boolean cube. The input class consists of Boolean functions whose Hessian graph is a disjoint union of cliques at every context. An explicit eight-point function in this class has no full integer lift, while a complete retained census and positive certificates establish liftability through seven points. A symbolic integrability argument explains the obstruction. For Boolean algebraic degree at most three, an analytic structure theorem restricts every nontrivial interaction component to at most two classes of interchangeable variables and yields an explicit lift for arbitrary ground-set size. Equivalently, the problem concerns prescribed parities of integer gross-substitutes set functions in the unrestricted convention. The principal results are presented with candidate status; the support provides the proofs, certificates, verification records and reproduction instructions.