Work statistics of autonomous quantum energy pumps

We develop a terminal-resolved theory of work statistics for autonomous quantum energy pumps. By modeling the systems that supply and receive energy as explicit quantum terminals, the energy exchanged with each terminal is defined directly from its Hamiltonian change. When the terminals are modeled by ideal clocks, these full-space observables admit exact representations on the pump Hilbert space and recover the conventional phase-derivative currents of periodically and quasiperiodically driven systems. The framework resolves transported work from energy accumulated in the pump. It also incorporates arbitrary initial pump--terminal correlations and identifies when correlations can enhance directional energy transfer under uncertain driving phases. For periodic pumps, we derive finite-cycle work statistics in Floquet eigenstates, relate their fluctuations to Floquet quantum geometry, and show that the long-time terminal currents become mutually compatible. An exactly solvable two-terminal qubit exhibits noise matching, in which transport becomes sharp through cancellation of common-mode terminal fluctuations even though the individual terminal energies remain noisy. Finally, for physical terminals beyond the ideal-clock limit, we introduce a positive work-variance gap that quantifies the fluctuations missed by a pump-only description. A coherent-cavity benchmark shows systematic convergence toward the ideal-clock regime with increasing occupation while demonstrating that agreement of the mean current alone does not guarantee accurate work statistics.