Operational recoverability bounds for proper-time interferometry with open quantum clocks

Clock interferometry probes general-relativistic proper-time differences through the entanglement between a particle's center-of-mass path and its internal degrees of freedom. Existing analyses treat the internal clock as either a closed system or study how environmental noise modifies the interferometric visibility. Here we address a distinct question: what fraction of the proper-time-induced visibility loss remains recoverable when the experimentalist is restricted to local unitary control of the internal clock? We model the clock as a two-level system undergoing amplitude damping with branch-dependent proper times in a Mach-Zehnder interferometer and derive the visibility both without and with a spin-echo refocusing pulse. In the closed-system limit ($\gamma\to0$), the echo restores full visibility; for $\gamma>0$, the no-jump survival probability imposes a strict upper bound on echo-recovered visibility that no internal-only unitary can exceed. We establish the conditions under which a proper-time-parametrized Gorini-Kossakowski-Sudarshan-Lindblad equation is valid by appealing to the local comoving frame and the equivalence principle, distinguishing comoving-bath and laboratory-bath parametrizations. Numerical simulations with QuTiP confirm all analytic results. We provide experimental parameter estimates for cold-atom and cavity-QED platforms, showing that the irrecoverable visibility deficit is within reach of near-future clock-interferometry experiments. The echo-recovery deficit constitutes an operational witness that separates irreversible environmental decoherence from the formally reversible path-clock entanglement of the Pikovski mechanism.