A crucial feature of primitive recursive functions is that the number of steps they take to evaluate is always limited, and can always in effect be determined in advance, since the basic operation of primitive recursion can be unwound simply as f[x_, y___] := Fold[h[#1, #2, y] &, g[y], Range[0, x - 1]] And what this means is that any computation that for example fundamentally involves a search that might not terminate cannot be implemented by a primitive recursive function.

These have led for example to the 1993 proof of Fermat's Last Theorem and to the 1983 Faltings theorem (Mordell conjecture) that the topology of the algebraic surface formed by allowing variables to take on complex values determines whether a Diophantine equation has only a finite number of rational solutions—and shows for example that this is the case for any equation of the form x n  a y n + 1 with n > 3 .

Instead, what has normally been done is to take the array of spins to be in thermal equilibrium with a heat bath, so that, following standard statistical mechanics, each possible spin configuration occurs with probability Exp[- β e[s]] , where β is inverse temperature.

For even if one takes these processes to be pure quantum ones, what I believe is that in almost all cases appropriate idealized limits of them will reproduce what are in effect the usual rules for observations in quantum theory.