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In this paper we show that there is a unique slowly oscillating periodic orbit to for all $ \alpha \in ( {\tfrac{\pi}{2}}, 1.9]$. Like in [@BergJaquette; @lessard2010recent], we recast the problem of studying the periodic orbits of as the problem of finding the zeros of a functional $F$ defined in a space of Fourier coefficients (see Section \[sec:FunctionDomain\]). Since periodic solutions to must have a high degree of smoothness, in particular real analyticity [@wright1955non; @nussbaum-analytic], their Fourier coefficients will decay very rapidly. That is to say, the functional we are interested in can be well approximated by a Galerkin projection onto a finite number of Fourier modes.

In finite dimensions, there are efficacious techniques for rigorously locating and enumerating the solutions to a system of nonlinear equations by way of interval arithmetic [@neumaier1990interval; @hansen2003global; @moore2009introduction]. We apply these techniques in infinite dimensions, specifically the *branch and bound* method, also referred to as a *branch and prune* method. That is, we first construct a bounded set $X$ of Fourier coefficients which contains all the zeros of $F$ (see Section \[sec:SolutionSpace\]). Then we partition $X$ into a finite number of pieces $\{X_n \}$ which we refer to as *cubes* (see Definition 