The first spectator theorem

For Contract E, allow a classical configuration

where $g$ is a geometric label, $\Gamma$ the original accessible interface, $h$ a controller state, and $\mathrm{loc}$ assigns abstract cell labels to names. A geometric rule changes declared geometric data and location labels. A matter rule applies a declared channel. A coupling rule may inspect a declared state functional $\rho {|}_{{\mathcal{A}}_v}$ and update a classical label.

Such a coupling function need not be affine. Contract E's equal-input comparison does not by itself make it a quantum channel.