Odd symmetry removes even center-manifold jets

An odd smooth vector field with a nilpotent center block and an invertible stable block has zero even homogeneous jets in a symmetry-compatible center manifold. At degree two its invariance equation has the form $(N-S)h_2=0$, where $N$ is the nilpotent differentiation operator generated by the center block and $S$ the invertible stable linear operator. Since their spectra are disjoint, this operator is invertible, so $h_2=0$. The same argument applies inductively to even orders after lower even jets vanish. This is a statement about local Taylor jets; it does not assert uniqueness of a global center manifold.