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