Complete energy in a declared model

An energy ledger begins with a named Hamiltonian, not with the words “discard” or “recycle.”

In Hawk04-B's regulated source–scalar model, the declared Hamiltonian has the form

with

For its vacuum bath preparation, the supplied energy proposition records, on every finite interval,

These are source-model statements CheungHawk04B (prop:energy); their Hamiltonian construction is not repeated here.

The notation matters. The bath-energy quantity called $E_B$ in that proposition is an expectation of $H_{\mathrm{bath}}$. It is not an energy automatically assigned to every factor of the complete residual $B$ in the channel partition. Selected source, carrier, uncollected-mode, and apparatus ledgers must not be identified by a shared letter.

Em03-E records conservation of a declared free Hamiltonian in its different finite model CheungEm03E (thm:evolution, thm:release). A free Hamiltonian is not the thermodynamic Helmholtz free energy $F=E-TS$. Neither conservation statement determines switching work, memory-reset cost, or the cost of restoring a controller.

An external switching schedule, readout apparatus, or reset mechanism would need its own Hamiltonian and work accounting. Those components are not hidden inside (eq:energy-source).