We present F-Rosette, a stable space-ground network structure for low-earth orbit (LEO) satellite mega-constellations at scale. Due to the dynamic many-to-many space-ground mapping in high mobility, existing LEO mega-constellations with IP protocol stack suffer from frequent user IP address changes (every 133-510s per user) and network routing reconvergence (thus ≤20% network usability). To provably stabilize the space-ground network under high dynamics, F-Rosette aligns the network design in the cyberspace with the mega-constellations in the physical world. It devises a recursively stable network structure over the Rosette constellation, decouples the hierarchical network addresses from mobility for stability, and aligns the geographical routing on the ground with the topological routing in space for efficiency and high usability. Our hardware-in-the-loop, trace-driven emulations validate F-Rosette's stability, near-optimal routing (≤1.4% additional delays), and affordable overhead (<1% CPU, <2MB memory) for resource-constrained satellites.