Remark: Perturbed energy

Recovery permits arbitrary integrable $E_e$. Actual bare-energy conservation additionally requires $[E_e(t),H_0]=0$ and an appropriate initial sector. It is not implied by a small norm error.

For $0\le H\le H_{\max }I$, a trace-distance comparison of size $e_{\mathrm{comp}}$ bounds the expectation discrepancy by $H_{\max }{e}_{\mathrm{comp}}$. The full $KN$-cell radiation Hamiltonian has maximum energy $KN\omega$, not $K\omega$, unless conservation of the initial sector is separately established.