  A rigorous implementation of the Wheeler-Dewitt equations was derived in the
context of Loop Quantum Gravity (LQG) and was coined Quantum Spin Dynamics
(QSD). The Hamiltonian constraint of QSD was criticised as being too local and
to prevent "propagation" in canonical LQG. That criticism was based on an
algorithm developed for QSD for generating solutions to the Wheeler-DeWitt
equations. The fine details of that algorithm could not be worked out because
the QSD Hamiltonian constraint makes crucial use of the volume operator which
cannot be diagonalised analytically.
  In this paper, we consider the U(1)$^3$ model for Euclidean vacuum LQG which
consists in replacing the structure group SU(2) by U(1)$^3$ and otherwise keeps
all properties of the SU(2) theory intact. This enables analytical calculations
and the fine details of the algorithm {\it can} be worked out.
  By considering one of the simplest possible non-trivial class of solutions
based on very small graphs, we show that 1. an infinite number of solutions
{\it exist} which are 2. generically {\it not normalisable} with respect to the
inner product on the space of spatially diffeomorphism invariant distributions
and 3. generically display {\it propagation}.
  Due to the closeness of the U(1)$^3$ model to Euclidean LQG, it is extremely
likely that all three properties hold also in the SU(2) case and even more so
in physical Lorentzian LQG. These arguments can in principle be made water
tight using modern numerical (e.g. ML or QC) methods combined with the
techniques developed in this paper which we reserve for future work.
