A Support Obstruction for Three-Cube Decompositions of Monomials

Let M be a complex monomial of degree 3d > 0, let N be the number of variables occurring in M, and let o be the number of odd exponents. We prove that the least number of cubes of degree-d forms summing to M is four whenever 2N > 3d and o > d. The proof restricts a primitive three-cube decomposition to coordinate hyperplanes. Rational maps to the Fermat elliptic curve give constant labels; distinct labels force coordinate divisibility, and degree counting forces linear dependence of the three forms. A dependent decomposition can produce only a cube or a monomial UV^2, with deg U = deg V = d. Combined with classical constructions, the obstruction determines the cube rank of every monomial of degree three, six or nine, and of every squarefree monomial of degree divisible by three. In particular, x_0^3 x_1 x_2 x_3 and x_0^2 x_1 x_2 x_3 x_4 both have cube rank four. The general monomial k-rank problem is not resolved. This is an unrefereed preprint; no absolute priority or formal verification is claimed.