  We prove that the set of quadratic growths achievable by integer
superharmonic functions on the $F$-lattice, a periodic subgraph of the square
lattice with oriented edges, has the structure of an overlapping circle
packing. The proof recursively constructs a distinct pair of recurrent
functions for each rational point on a hyperbola. This proves a conjecture of
Smart (2013) and completely describes the scaling limit of the Abelian sandpile
on the $F$-lattice.
