Theorem: Full recurrent-state comparison

On a region where the unprimed cell constants are valid,

with ${\eta }_t$ from (eq:fullresidual). For different heads,

All first moments are finite at fixed finite horizon under the stated Gaussian perturbations and finite parameters.

Proof. Insert $F_{\theta }(u_t^{\prime },h_{t-1}^{\prime })$ and $a{F}_{{\theta }^{\prime }}(u_t^{\prime },h_{t-1}^{\prime })$. Use the cell input/state bound and Lemma lem:paramcell. The remaining terms are direct control and activation differences. Iterate the resulting recurrence. Finite-horizon state envelopes and Gaussian moments give integrability.