Non-triviality (parsimony) of the 2D Ising spin model as a universal simulator

Determine whether the two-dimensional Ising spin model, viewed as a universal spin model within the categorical framework of simulators and parsimony, is a non-trivial universal simulator; specifically, establish whether it is strictly more parsimonious than the trivial simulator under the parsimony preorder defined via simulator morphisms. More generally, ascertain whether other universal spin models satisfy this non-triviality property in the same framework.

Background

The paper develops a categorical framework for universality and introduces a parsimony preorder on universal simulators via simulator morphisms. The trivial simulator is always universal, and a universal simulator is called non-trivial if it is strictly more parsimonious than the trivial simulator.

Within this framework, the authors prove that universality of Turing machines is non-trivial, i.e., there exists a universal simulator (a universal Turing machine) that is strictly more parsimonious than the trivial simulator. However, for spin models such as the two-dimensional Ising model, although universality as a spin model is known, it has not been established within this framework whether such spin models are non-trivial universal simulators in the parsimony sense.