Classification of translation invariant topological Pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices

We prove that on any two-dimensional lattice of qudits a prime dimension, every translation invariant Pauli stabilizer group with local generators and the code distance being linear system size is decomposed by Clifford circuit constant depth into finite number copies toric (Abelian discrete gauge theory). This means under circuits, complete topological codes. Previously, same conclusion was obtained assumption nonchirality for qubit codes or Calderbank–Shor–Steane structure qudit codes; we do not assume these.