Proposition: Positive bias need not grow without bound over time

There is a source-compatible cell for which, at $a=1,p=0$, external ReLU, zero hidden noise, and constant $\beta >0$,

Thus positive bias can accumulate to a finite limit rather than cause unbounded temporal growth.

Proof. Set all cell weights and biases to zero. Then $r=\zeta =1/2$, $n=0$, and $F_{\theta }(u,h)=h/2$. The biased recurrence is $h_t=h_{t-1}/2+\beta 1$. Solve it from zero.