Let T⁴ be a 4-dimensional torus where each dimension Dᵢ ∈ Z₄ = {0,1,2,3}
The state space S is the Cartesian product:
S = D₀ × D₁ × D₂ × D₃ = Z₄ × Z₄ × Z₄ × Z₄ = Z₄⁴
|S| = |Z₄|⁴ = 4⁴ = 256 ∎Yes, that's correct!
The 4-dimensional torus $T^4$ can be thought of as the manifold $(\mathbb{R}/\mathbb{Z})^4$, but when we discretize it with period 4 in each direction (i.e., identifying coordinates modulo 4), we get exactly the finite grid
$$S = \mathbb{Z}_4 \times \mathbb{Z}_4 \times \mathbb{Z}_4 \times \mathbb{Z}_4 = \mathbb{Z}_4^4$$
with
$$|S| = 4 \times 4 \times 4 \times 4 = 256$$
elements. This is the standard discrete model of a 4-torus with 4 points per dimension (or equivalently, a 4-dimensional hypercube lattice with periodic boundary conditions).