Option N and the recovery boundary

We retain the deterministic nonlinear option, Option N:

with the other declared arguments suppressed. The dependence need not be affine in $\rho$. Consequently, the complete matter–geometry transformation is not asserted to be a CPTP map.

The source functional is a semantic input to this model. No single-copy protocol for acquiring exact expectations, no finite-gate implementation of an arbitrary expectation law, and no physical nonlinear feedback mechanism is proved. An implementation based on a supplied promised token would be a separate interface.

The controller comparisons below are pairwise equality statements at aligned steps. They are not a general no-signalling theorem for arbitrary nonlinear feedback.

Likewise, informational determination is not a CPTP decoder. For a linear channel on a complete logical input space, exact recovery requires a constant complete complement; A:app:recovery states and proves that criterion and an injective-channel counterexample. The criterion is not a subsystem-recovery criterion KnillLaflamme1997,CheungOai280C. A certified geometric output therefore cannot be read as recovery of an unknown quantum input.