= Homogeneous stability of a static cosmic-string universe
At fixed string normalization $C=\rho_sa^2$, scale-factor shifts move between static solutions and are neutral, with no restoring force. The acceleration equation gives $\ddot a=0$, but the Friedmann constraint also imposes $\dot a^2=8\pi GC/3-K$. At the static normalization it excludes a nonzero velocity. A slightly larger $C$ admits linearly expanding or contracting branches; a smaller $C$ admits no real closed solution. This distinguishes permitted perturbations from constraint violations.
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