Why co-located gates do not settle arbitrary macro locality

Suppose a macro channel satisfies a physical endpoint no-signalling condition. This says what its outputs can depend upon, not how to generate them. A realization may require resource states, communication, or a decomposition whose ordinary circuit paths overestimate its actual influence.

The gather–operate–distribute proof avoids this gap by imposing a common station and explicit routes. The QCA localizability theorem ArrighiNesmeWerner and quantum causal graph dynamics ArrighiMartiel address broader settings under additional hypotheses. Their graph notion of causality does not automatically supply the geometric port timings of this paper.

Consequently:

Our Target B claim is for the printed compiler architecture and routes, not every representation of the same endpoint channel.