These occupation records do not certify inversion

This equality is asserted on the declared source family. It is not an equality of the two channels on all coherent, reference-entangled input operators. A protocol that validates unitary inversion must add suitable coherent probes and its own comparison contract.

The construction is not empty of quantum coherence: before relaxation, for example, the amplitude matrix of $U_{\varphi }|00⟩$ is

whose determinant is $2{\cos }^2\varphi {\sin }^2\varphi \ne 0$. That branch is entangled. Nevertheless the selected occupation likelihood has an exact classical sampling description: sample the hidden source component and then sample the two conditional population transitions derived in the proof. No computational or metrological quantum advantage for this source-weight task is claimed.