  We show that standard identities and theorems for lattice models with $U(1)$
symmetry get re-expressed discretely in the tensorial formulation of these
models. We explain the geometrical analogy between the continuous lattice
equations of motion and the discrete selection rules of the tensors. We
construct a gauge-invariant transfer matrix in arbitrary dimensions. We show
the equivalence with its gauge-fixed version in a maximal temporal gauge and
explain how a discrete Gauss's law is always enforced. We propose a
noise-robust way to implement Gauss's law in arbitrary dimensions. We
reformulate Noether's theorem for global, local, continuous or discrete Abelian
symmetries: for each given symmetry, there is one corresponding tensor
redundancy. We discuss semi-classical approximations for classical solutions
with periodic boundary conditions in two solvable cases. We show the
correspondence of their weak coupling limit with the tensor formulation after
Poisson summation. We briefly discuss connections with other approaches and
implications for quantum computing.
