Three ledgers and thermodynamic underdetermination

The three ledgers are:

The inequality signs mean that these are different kinds of data, not that their numerical values can never coincide.

For a specified Hamiltonian $H_{\mathrm{tot}}$, complete energy is evaluated from $\mathrm{Tr}(H_{\mathrm{tot}}{\rho }_{\mathrm{tot}})$. A heat or work ledger requires an implementation, designated reservoirs and work storage, and sign conventions. Residual leakage is the dimensionless norm diagnostic in Definition def:leakage.