What does this equation mean?

Passive relabeling is one degenerate case; the other is simpler still. Let $\Lambda$ be the identity channel, $\mathcal H_c$ = $\mathcal H_f$ , $H_c$ = $H_f$ : no relabeling, no discarding, the coarse description simply is the fine one. Then trivially $D_\Lambda$ = 0 . Nothing in the paragraphs above required that; it follows on inspection. But the triviality is the point, because this is the operator-language shadow of the oldest and most secure fact the subject has. A system evolving under its own Hamiltonian, with nothing coarse-grained away and nothing relabeled, conserves that Hamiltonian’s own expectation value exactly, because the Hamiltonian generates its own time evolution and commutes with itself. Emmy Noether’s 1918 theorem is the general statement standing behind that fact: whenever an action is invariant under a continuous symmetry — time translation among them — a corresponding quantity is conserved along the resulting dynamics, energy for time translation, momentum for spatial translation [ 2 ] . The ledger built here does not derive that theorem, extend it, or need to; it assumes it, at this one degenerate limit, as the floor every other case in this article is measured against.