Cube Ranks of Monomials

Let \(M=\prod_i x_i^{a_i}\) have degree \(3d>0\), and let o be the number of odd exponents. We determine the least number of cubes of homogeneous degree-d forms whose sum is M. Over C, this number is one if every exponent is divisible by three, three if M is not a cube and \(o\le d\), and four if \(o>d\). The same formula holds over R. The new lower bound uses an exponent-one coordinate, which must exist when \(o>d\). Restricting a hypothetical three-cube expression to its coordinate hyperplane gives proportional forms, because the Fermat plane cubic has no nonconstant rational parametrization. Differentiation then forces a degree-2d square divisor in a monomial whose square capacity is smaller than d. Classical constructions give the matching upper bounds. Version 2 removes the support restriction of version 1 (DOI 10.5281/zenodo.23425441); it is a strengthening of the same manuscript, not a new paper. This completes the \(k=3\) case of Problem C in Algebraic Stories, but does not determine general monomial k-ranks for \(k\ge4\). This is an AI-assisted, self-audited, unrefereed preprint; no independent human review, formal verification or absolute-priority certification is claimed.