\[def:cube\]). For each cube $X_n$ we are interested to know whether:

1.  there exists a unique point $\hat{x} \in X_n$ for which $ F(\hat{x})=0$, or

2.  there does not exist any points $\hat{x} \in X_n$ for which $F(\hat{x})=0$.

If we can show that $(a)$ holds for one cube, and $(b)$ holds for all the other cubes, then we will have shown that $ F=0$ has a unique solution.

This approach requires some additional preparation. Since periodic orbits to form a 2-manifold in phase space [@regala1989periodic], the functional $F$ we construct in Section \[sec:FunctionDomain\] will not have isolated zeros. The numerical techniques we employ are suited to finding isolated zeros, so it is necessary to reduce the dimension of the kernel by two. Along the principal branch $\alpha$ can be taken as one of the coordinate dimensions. We reduce this dimension by treating $\alpha$ as a parameter and performing our estimates uniformly in $\alpha$. The other dimension can be attributed to time translation; if $y(t)$ is a periodic orbit, then so is $ y(t+\tau)$ for any $\tau \in {\mathbb{R}}$. We reduce this dimension by imposing a phase condition; we may assume without loss of generality that the first Fourier coefficient is 