Rank reduction for quartic Boolean Hessian systems with variable twins

We study Boolean set functions whose discrete Hessian triangles never contain exactly two ones. For connected systems of algebraic degree at most four with a prescribed variable transposition-twin pair, write the twin normal form F = u + (a+b)D + abC with D nonconstant. We prove integer full flag-incidence lifting whenever the alternating quadratic coefficient matrix of C has rank other than two. The proof uses low-degree annihilator rigidity, persistent cores in original coordinates, isolation, four specified finite classification or exclusion premises, and explicit or imported integer potentials. Rank-zero controls reduce to cubic, small, collapsed or matroid-controlled components. The result identifies a rank-two boundary within this twin class; rank-two lifting and arbitrary-size twin existence remain open. The principal results retain candidate status; the general proof does not meet the established-status threshold.