Proposition: The observable and the worst-start rate differ

For the two-spin chain, the eigenvalues of $-L$ are

Its spectral gap is $1-|a|$, which can be arbitrarily small as $|J|\to \infty$. Nevertheless Theorem thm:finite-certificate is uniform in $J$.

Proof. The four independent eigenfunctions can be taken as

Their eigenvalues follow from (eq:two-actions). Since $a\to \pm 1$ as $J\to \pm \infty$, the gap tends to zero.

The uniform-start product expectation relaxes at rate two, as used in Lemma lem:two-functions. The finite certificate depends on that initialized observable, not on a disorder-uniform lower bound for the full gap.