Theorem: Oai227-A finite stationary–quench certificate

For the data in (eq:certificate-data), every real coupling $J$ satisfies

In particular,

The two functions are explicitly given by (eq:two-stat)–(eq:two-quench) in the frozen units. Under each of the Gaussian and Rademacher coupling laws, the displayed uniform comparison holds with probability one. No exceptional-disorder event, asymptotic theorem, numerical optimization, or empirical sample is used.

For every $J\ne 0$ and positive $s,t$, the difference is strictly positive. Thus the theorem certifies a tolerance, not exact finite-waiting-time stationarity.

Proof. Since $a\sinh (a\ell )\ge 0$, the difference is nonnegative. For $\ell \ge 0$,

Equation (eq:exact-difference) therefore gives, on $s\ge 4$,

All remaining comparisons have finite elementary certificates. Cubing positive quantities gives

because $T^3=4>27/8$ and $v^3=2<64/27$. The positive exponential series gives

Consequently,

The same strict upper bound applies to the supremum because it is a single bound independent of $J,s,t$.

The disorder conclusions follow from the statement for every real $J$. If $J\ne 0$, then $a\ne 0$, and $a\sinh (aTt)>0$ for $t>0$, proving strict positivity.