Work and resources

For stored density ${\rho }_{\mathrm{st}}$, put $\bar{n}=\mathrm{tr}({\rho }_{\mathrm{st}}{N}_{\mathrm{B}})$. Ideal extraction gives

Mode mixing and initial bath vacuum prove these identities. The instantaneous quench work is

The hopping switch has zero mean work: $G_{\mathrm{hop}}$ is conserved during extraction and has initial expectation zero. These are reference-Hamiltonian energies, not free energies or autonomous black-hole mass loss.

The decoder is a complete mathematical specification, not a physically implemented local receiver. Preparation, whole-bath delivery, code inversion, ancillas, and compilation remain separate resources.