The Noether charge, corresponding to lattice gauge theory is similar to the one derived for the motion of a point particle on the unit circle, with the however important difference that this time a rotation, i.e. a multiplication with a unitary matrix G has to be done either at the beginning or at the end of the link influencing naturally some further links attached to that site. This kind of local unitary rotation is a gauge transformation.
The total change of a link variable U by such infinitesimal rotations
at its beginning and end independently can be disantangled to a sum
of transformation generators located at the lattice sites. Since such
transformations leave the Hamiltonian
( 50 ) due to its construction
using traces of group elements multiplied over oriented closed link
doubles --- like <P,P> --- or elementary squares (plaquettes)
--- like 
--- invariant, the above mentioned infinitesimal
generators of the transformation define a Noether charge.
In the continous time limit it is
a quaternion (for SU(2) ) defined on each lattice site summing up
contributions from outgoing (
) and subtarcting reverse contributions
from incoming (
) links,
Its rate of change, upon using the continuum time equation of motion
and the correct value of the Lagrange multiplier 
obtained from
the orthogonality condition becomes
The corrections steming from the use of the Lagrange multiplier 
manage to make each incoming and outgoing link contribution to the
change rate of the Noether charge --- to the non commutivity of the
Gauss' law with the equations of motion --- traceless.
It is now relatively easy to convince ourselves that this total change
is zero if using a little bit of geometrical imagination.
Denoting each group element multiplication by oriented lines sweeping
through the corresponding links --- in positive x- y- or z- direction
if encountering a U and in the reverse direction if encountering a
--- the complement variable 
can be viewed as oriented
pathes matching to the link of U. The products 
and
consist then of complete plaquette pathes oriented
correspondingly. (These pathes are not closed, because in calculating
the Gauss law we do not take the trace of these products.)
Summing up now outgoing and subtracting incoming plaquette contributions
one immediately realizes that half of these plaquette pathes
(in two dimensions the right upper and the left lower plaquettes
attached to the site) directly cancel while the other half
(the left upper and right lower corner plaquettes)
consists of pairwise oppositely oriented
sums. An oppositely oriented sum for any path --- i.e. for any product
of link variables --- means, however, an expression like 
,
a result equal to the unit element multiplied by 
.
But exactly this has been substracted by the corresponding <U,V>1 terms
steming from the Lagrange multiplier (in other words the Noether charge
change rate is traceless if orthogonality is conserved)!
Finally the respective definitions of the finite time step recursion equations and the Noether charge, norm and orthogonality at half integer time points ensure that the same proof holds for the numerical recursion for a general Type III algorithm.