  We rigorously define superselection sectors in the 3d (spatial dimensions)
Toric Code Model on the infinite lattice $\mathbb{Z}^3$. We begin by
constructing automorphisms that correspond to infinite flux strings, a
phenomenon that's only possible in open manifolds. We then classify all ground
state superselection sectors containing infinite flux strings, and find a rich
structure that depends on the geometry and number of strings in the
configuration. In particular, for a single infinite flux string configuration
to be a ground state, it must be monotonic. For configurations containing
multiple infinite flux strings, we define "infinity directions" and use that to
establish a necessary and sufficient condition for a state to be in a ground
state superselection sector. Notably, we also find that if a state contains
more than 3 infinite flux strings, then it is not in a ground state
superselection sector.
