We reformulate the stellar structure equations in the language of dynamical systems and show that the maximum mass of stellar sequences arises from the existence of a fixed point in the relativistic regime. In an appropriate representation of the Tolman-Oppenheimer-Volkoff equations, this fixed point becomes manifest and is directly associated with the turnover of the mass-radius curve. The existence of a fixed point implies an effective reduction in dimensionality near the onset of instability, which provides a simple explanation for several equation of state–insensitive relations and predicts new ones. In the weakly relativistic limit, we identify a complementary universal structure shared by stellar sequences at their maximum mass, which we term the “compressible limit,” and derive distinct universal relations governing the maximum mass in the Newtonian and post-Newtonian regimes. Combining these theoretical results with current astrophysical constraints, we show that the $\mathrm{J}0740+6620$ pulsar is unlikely to lie near the Tolman-Oppenheimer-Volkoff maximum mass unless the equation of state exhibits a strong first-order phase transition at densities just above its central density.

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• General relativity
• QCD phase transitions
• Neutron stars & pulsars