Switching work for both pulses

The source has zero internal Hamiltonian. It does not contain an internal-energy fuel for the process.

A precise bookkeeping realization is available in the doubled representation. Its Killing generator is

with $K_{\mathrm{U}}\Omega =0$. This signed generator implements physical Killing translations on the Weyl algebra. Its change of expectation equals the baseline-subtracted physical radiation energy. It is not a positive total-energy operator for the stationary thermal background.