Proposition: Gate-block spectral radii do not certify contraction

There are GRU cells whose recurrent gate blocks all have spectral radius zero, but whose state map has Euclidean local expansion.

Proof. Take $H=2$, constant gates $r=\zeta =1/2$, and

All three spectral radii are zero. At $h=0$, the smooth cell Jacobian is

Its norm exceeds one for sufficiently large $L$. The same expansion holds near an interior positive state with external ReLU, where its gates are active, by continuity. The state can remain in the ordinary invariant cube.