Exact Rank of Complex Hankel Tensors

Let $H_{i_1 \cdots i_m} = h_{i_1 + \cdots + i_m}$ for $0 \le i_j \le n-1$, over
$\mathbb{C}$. Write $D = m(n-1)$, and let $r$ be the rank of the middle Hankel
catalecticant. For $H \ne 0$, choose a homogeneous binary apolar polynomial of
smallest degree $r$, and let $s$ be its number of distinct projective zeros. We
prove the exact formula
$$R(H) = R_{\mathrm{sym}}(H) = \min\{D - r + 2,\ (m-1)r - (m-2)s\}.$$
In particular, ordinary and symmetric rank agree for every complex Hankel
tensor, including exceptional and mixed-multiplicity tensors. The lower bound uses a contextual product-space
inequality: an annihilating Frobenius functional excludes nilpotents from the
auxiliary stabilizer algebras, leaving finitely many support partitions. The
upper bounds are explicit symmetric interpolation constructions. We also prove
that all three border ranks equal $r$, and that the maximum ordinary and
symmetric Hankel ranks are $(m-1)(n-1)+1$.