Lemma: Deterministic compact moduli

For every $n,J$, the finite autocorrelation functions are smooth. Uniformly over their nonnegative time arguments,

For ${\Delta }_{n,J}(s,t)=A_{n,J}(t)-B_{n,J}(s,t)$,

Proof. Finite matrix exponentials are smooth. By (eq:spin-action), ${\Vert L_J{f}_i\Vert }_{\infty }\le 2$. Since the semigroup is a supremum-norm contraction and commutes with its generator, differentiating a lag contributes at most

For the waiting derivative use

Every $G_{\ell }$ is bounded by one, and ${\Vert L_J{G}_{\ell }\Vert }_{\infty }\le 2n$. Thus the derivative is at most $v_n{T}_{n}2n=2n^2$. Subtracting the two lag derivatives proves the final bounds.