Theorem: Temporal perturbation and forgetting certificate

Suppose coupled trajectories satisfy

Then

For common weights and controls, valid local constants give $c_t=a{L}_{h,t}$.

If $c_t=c<1$ and ${\eta }_t\le \eta$, then

For $c=1$, the bound is $d_0+T\eta$. For $c>1$, it is

The case $c>1$ permits amplification in the upper bound; it does not prove actual instability.

Proof. Induct on the scalar recurrence. The constant-coefficient cases are geometric sums.