The Fixed-Point Star

The maximum mass of a neutron star — the Tolman-Oppenheimer-Volkoff (TOV) limit — is where the mass-radius sequence turns over. Add more mass and the star collapses to a black hole. This turnover point has been computed numerically for every proposed equation of state, each time as a separate calculation. The maximum mass is a number that emerges from integrating the TOV equations with a specific EOS — it does not have a structural explanation beyond "this is where the integration stops increasing."

Legred and Yunes (arXiv:2603.26973, March 2026) reformulate the TOV equations as a dynamical system and show that the maximum mass is a fixed point. The mass-radius sequence is a trajectory in a phase space, and the turnover — where the trajectory reverses direction in mass — corresponds to a fixed point of the flow. The maximum mass is not a numerical accident but a structural necessity of the dynamical system.

This reformulation explains why equation-of-state-insensitive relations exist. Universal relations — correlations between neutron star observables that hold regardless of the specific EOS — have been discovered empirically and remain partially mysterious. The fixed-point structure provides the explanation: near a fixed point, the dynamics linearize, and the linearized behavior depends only on the fixed point's eigenvalues, not on the full details of the flow (the EOS). The universal relations are consequences of the fixed-point structure — properties of the eigenvalues that persist across different equations of state.

Applied to PSR J0740+6620 — one of the heaviest known neutron stars — the analysis concludes that this star is unlikely to be near the TOV maximum mass unless its EOS has a strong first-order phase transition at densities just above its central density. The fixed-point analysis constrains not just the star's mass but the qualitative nature of the matter at its center.

The structural observation: a numerical fact (the mass turnover) is reconceived as a dynamical structure (a fixed point), and this reconception makes previously unexplained universality a consequence rather than a coincidence. The EOS-insensitive relations are not approximate symmetries — they are exact properties of the fixed-point neighborhood, holding for the same reason that critical exponents are universal near phase transitions.