Probability Leakage Hypothesis: Observational Arrows and Gravitational Redshift

We introduce the Probability Leakage Hypothesis (PLH) as a unifying framework for analyzing the statistical structure of open quantum systems and their role in the emergence of the arrow of time. In this work, we establish three main theoretical results. First, we formulate the no-uncollapse theorem, proving that the cumulative probability of observed events (expressed through the hazard integral) is a monotone quantity that cannot be decreased by any local completely positive trace-preserving (CPTP) dynamics. This result clarifies the irreversible character of quantum measurement and rules out information-retrieval strategies based solely on local dynamics. Second, we derive covariant continuity equations for observational currents in curved spacetime. Under causality constraints, the visible current satisfies a non-positivity condition, ensuring that information propagation remains subluminal and consistent with relativistic structure. This provides a bridge between the stochastic master equations of quantum measurement theory and the covariant conservation laws of relativistic physics. Third, we demonstrate that observational rates undergo gravitational redshift in curved spacetime, generalizing the classical redshift of proper time to stochastic observational processes. The result establishes that quantum measurement cadences, when viewed as physical clocks, inherit the same redshift factor as ordinary proper time intervals. To support the theoretical developments, we propose a set of experimental protocols involving quantum optics platforms, including interferometric tests of the no-uncollapse theorem and atom-photon setups capable of probing gravitational rate shifts at laboratory scales. These protocols highlight the empirical accessibility of the PLH framework and connect the abstract results to realizable experimental contexts. Overall, the Probability Leakage Hypothesis offers a mathematically rigorous and conceptually transparent approach to unifying the statistical mechanics of open quantum systems with the geometric structure of spacetime. By identifying hazard integrals as observer-independent monotone quantities, PLH provides a foundation for a new class of operational time observables, with implications for quantum information, statistical physics, and gravitational theory.