Low Earth Orbit (LEO) satellite mega-constellations enable orbital edge computing for latency-critical geocentric applications, but satellite motion compresses line-of-sight communication windows to mere minutes. To maintain logical geographic proximity, a paradigm termed Virtual Stationarity (VS), applications must continuously migrate across passing satellites in the constellation. Traditional live migration mechanisms optimize individually initiated relocations or replication, but not the continuous handover between successive hosts that orbital motion requires. We reformulate live migration as a steady-state execution model and introduce a novel virtual stationarity runtime paradigm. We formalize the viability of continuous migration through a lightweight model capturing latency and availability service-level objectives (SLOs), continuous migration overhead, and the convergence bound of state replay. To satisfy these constraints, we derive a Virtual Stationarity Runtime (VSR) framework whose Migration Membrane combines Capability Virtualization and Continuous State Projection to rebind host resources and reconstruct evolving execution on a successor. Evaluation on physical heterogeneous hosts shows 95th-percentile downtime below 6 milliseconds across 300 recurrent handovers, migration time effectively invariant to memory-write rate, and no cumulative growth in migration time or continuation state. A 1,600-satellite simulation further confirms completion within orbital handover windows.

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