Quantum Limits on Relational Time inside a Schwarzschild Black Hole

We ask whether a single quantum state can support a semiclassically sharp notion of relational time throughout a quantized Schwarzschild interior. Building on the relational Kantowski-Sachs framework of Fragolino and Rastgoo [1], and on covariant and Page-Wootters arrival-time constructions [2,5-16], we construct a directional covariant arrival-time positive-operator-valued measure (POVM) for the reduced system Hamiltonian on the black-hole momentum sector . We prove positivity, resolution of the identity, time-translation covariance, and recovery of the classical crossing time by stationary phase. For the real Gaussian family used in the underlying interior model, a linear semiclassical estimate places the loss-of-sharpness surface outside the expectation-value area bounce; direct POVM calculations reproduce the estimate at the few-to-ten-percent level. General squeezed Gaussian states can sharpen the arrival time at one selected interior surface, so this local ordering is not universal. We therefore introduce a pathwise timing cost. For every prescribed finite interval, fixed mean momentum, and positive path weight, minimization over the Gaussian covariance cone yields a strictly positive lower bound. The classical proper-time measure of an Schwarzschild infaller is treated as a physically motivated corollary. Squeezing can redistribute timing uncertainty but cannot remove it from the complete interior history. The result concerns relational time. It is not yet an independent infaller proper-time operator, a detector-based first-arrival law, or a statement about higher curvature moments.