  We consider two discrete completely integrable evolutions: the Toda Lattice
and the Ablowitz-Ladik system. The principal thrust of the paper is the
development of microscopic conservation laws that witness the conservation of
the perturbation determinant under these dynamics. In this way, we obtain
discrete analogues of objects that we found essential in our recent analyses of
KdV, NLS, and mKdV.
  In concert with this, we revisit the classical topic of microscopic
conservation laws attendant to the (renormalized) trace of the Green's
function.
