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Before asking what a genuine coarse-graining exposes, the construction has to survive the case where nothing at all is discarded. Take $\mathcal H_c$ = $\mathcal H_f$ — no reduction in the space of states — and let the “coarse” description be nothing more than the same physics seen through a fixed, time-independent relabeling: a rotated spin basis, a renamed pair of levels, a change of which linear combination of states a lab calls “up.” Model this as $\Lambda(\rho)$ := W$\rho$ $W^\dagger$ for a fixed unitary W , and, because nothing has been thrown away, define the coarse Hamiltonian self-consistently as the same physical generator carried through that same relabeling, $H_c$ := W $H_f$ $W^\dagger$ .

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