Proposition: Microscopic energy ledger

For every logical input, the prepared evolution obeys

where $E_I=⟨V_{\mathrm{int}}⟩$. Moreover,

On the prepared finite-time domain,

Proof. The initial bath and interaction means vanish. Autonomous unitary evolution conserves the total Hamiltonian expectation. The form bound in Theorem thm:model-unitarity gives the bath-energy bound.

The occupation estimates in the next subsection give finite moments of every fixed particle-number order on bounded time intervals. Since the frequency band is bounded, differentiation can be justified by finite-particle approximation. The commutator identity and differentiated conservation law then give the currents.