A declared continuum model and endpoint non-attainability

The finite channel includes $\eta =1$ as an exact mathematical endpoint. This does not imply finite-time attainability in a continuum realization. The following independent argument concerns only the model specified here.

Let

Then $\Vert f\Vert =1$. For $t_0\ge 0$, take an initially vacant bath level and

For finite extraction duration $T$, use bounded, piecewise-continuous real controls $v(u),\kappa (u)$ and the generator

Relative to $h\oplus 1$, its perturbation is bounded, self-adjoint, and finite rank. In the interaction picture, the Dyson integrals converge in operator norm on finite time intervals, giving a unitary propagator.

Define

The conserved norm implies

Variation of constants gives, almost everywhere in $\omega$,