Integer superharmonic matrices on the $F$-lattice

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.