  We complete the program indicated by the Ansatz of D'Hoker and Phong in genus
~4 by proving the uniqueness of the restriction to Jacobians of the weight 8
Siegel cusp forms satisfying the Anstaz. We prove $\dim [\Gamma_4(1,2),8]_0=2$
and $\dim [\Gamma_4(1,2),8]=7$. In each genus, we classify the linear relations
among the self-dual lattices of rank {16}. We extend the program to genus ~5 by
constructing the unique linear combination of theta series that satisfies the
Ansatz.
