  The dynamics-from-permutations of classical Ising spins is generalized here
for an arbitrarily long chain. This serves as an ontological model with
discrete dynamics generated by pairwise exchange interactions defining the
unitary update operator. The model incorporates a finite signal velocity and
resembles in many aspects a discrete free field theory. We deduce the
corresponding Hamiltonian operator and show that it generates an exact
terminating Baker-Campbell-Hausdorff formula. Motivation for this study is
provided by the Cellular Automaton Interpretation of Quantum Mechanics. We find
that our ontological model, which is classical and deterministic, appears as if
of quantum mechanical kind in an appropriate formal description. However, it is
striking that (in principle arbitrarily) small deformations of the model turn
it into a genuine quantum theory. This supports the view that quantum mechanics
stems from an epistemic approach handling physical phenomena.
