Proposition: A pointwise branch is not an additional state loop

In (eq:modular), if $k_t,{\gamma }_t,m_t$ are not fed into the next recurrent update and have no separately persistent state, their branch introduces no additional recurrent state. Their influence on future predictions can occur only through an explicitly added feedback or training operation.

Proof. The next recurrent state is computed from the GRU state and next input. The displayed pointwise branch does not appear among those arguments.