An elementary causal realization

Use the apex as the controller and rendezvous station. The model admits local finite quantum registers on its timelike axis under Contract contract:routing; no quantum-field boundary condition at the singular axis is asserted.

The qubit starting at $e$ travels radially inward at unit spacetime speed. It can arrive at the apex at

The other qubit is already there. Both wait until $T=0$.

Let $L$ be the finite number of elementary gates in the event-code circuit plus three CNOT gates implementing SWAP. Execute them at the apex, each with assigned spacetime duration

All gates finish before $T=h_{\ast }/2$. Route the appropriate output qubit to $e$ and retain the other at the apex. Since

both output ports are reached in time, with waiting as needed. The code remains at the apex.