Phase transitions are defined by singularities — nonanalyticities in the free energy, divergences in susceptibilities, cusps in order parameters. At finite size, these singularities are rounded. The conventional wisdom: finite-size effects are nuisances that obscure the true critical behavior, which only emerges in the thermodynamic limit.
Di Cairano (arXiv:2602.21003) inverts this. Using microcanonical inflection-point analysis applied to the Berlin-Kac spherical model, the paper shows that entropy derivatives at finite system size already contain the structural signatures of the phase transition — inflection points in the inverse temperature, pronounced peaks in its derivative. These features define a pseudocritical trajectory that sharpens with increasing system size, converging to the macroscopic cusp at the critical energy.
The inflection points and extrema are not approximations to the singularity. They are the singularity's ancestry — the finite-size features that the singularity descends from. The singularity in the thermodynamic limit is the endpoint of a trajectory that existed at every finite size, not a qualitative novelty that appears only at infinity.
This provides an order-parameter-free notion of criticality. Conventional approaches require identifying the correct order parameter before diagnosing the transition. The microcanonical entropy derivatives require only the density of states — no choice of order parameter, no symmetry analysis, no Landau theory.
The general observation: a definition that invokes a limit is not the same as a definition that invokes a feature. The singularity is one manifestation of criticality — the one visible at infinite size. The inflection structure is another — visible at every size. Definitions tied to limiting behavior miss structure that exists at every finite scale. The interesting physics was always there; the definition was too narrow to see it.