Quadratic Dispersion From Discrete-Time CPTP Dynamics: A Lattice-Scale Test with High-Energy Neutrinos We derive a concrete, falsifiable propagation law from a discrete-time quantum dynamics in which each temporal layer implements a completely positive trace-preserving (CPTP) map with stochastic, unselective pinching in a fixed pointer algebra. Under mild locality and symmetry assumptions, the continuum limit predicts a universal quadratic correction to the dispersion relation with fixed negative sign, implying strictly subluminal group velocities and time-of-flight delays that scale with the square of the observed energy. We build a hierarchical Bayesian framework that marginalizes over source-intrinsic emission lags and measurement noise, and we recast existing gamma-ray and neutrino observations to obtain a conservative bound on the underlying lattice scale a at the level of order 10^(-32) m for a stacked sample of gamma-ray bursts. We further provide forecasts for next-generation instruments such as IceCube-Gen2 and KM3NeT, showing potential sensitivity to scales near 10^(-33) m under reasonable assumptions. The model thus links operational open-system dynamics to astrophysical propagation tests and provides clear falsification channels through the fixed subluminal sign, E^2 scaling, and absence of superluminal tails.