Canonical integer lifts for quadratic-free quartic Boolean Hessian systems

We consider Boolean set functions whose discrete Hessian triangles never contain exactly two ones. For algebraic degree at most four with zero quadratic part in fixed coordinates, we prove that replacing each supported nonlinear algebraic-normal-form coefficient by the integer +1 gives a full integer flag-incidence lift. The ground set has arbitrary finite size and any compatible integer affine part is allowed. The proof classifies the cubic support, isolates every permitted quartic attachment, and treats the remaining quartic support by a three-colored matching argument. The resulting Hessians are nonnegative integer counts; the negative potential is a gross-substitutes set function in the unrestricted convention. Classical tetrad codes give an optional interpretation and are not a premise of the proof. The principal results retain candidate status; the general proof does not yet meet the established-status threshold.