Separate name storage and spectator-independent geometry

Apply Lemma lem:synthesis to $P_{\mathrm{geo}}$ with

This geometric circuit is independent of the number of names.

For a supplied list of $n$ original names, append six location and six history bits per name. On the forward domain the location transport is the identity. After the geometric circuit, copy each location bit to its initially zero history bit, controlled by the outgoing mode bit. This uses six Toffoli gates per name and never feeds a name bit into a geometric output.

A sufficient total bit count, including clean work, is

A sufficient gate count is

The geometric budget is independent of $n$. The statement concerns a family of finite circuits, one for each fixed finite name list, not a single bounded circuit accepting unboundedly many names.