arXiv:2609.17492v3 Announce Type: replace Abstract: In this work we propose a new existence theorem for numerical-flux functions for supercritical fluids that are pressure-equilibrium-preserving (PEP). In particular, we characterize the existence of consistent numerical-flux functions that satisfy an algebraic PEP property. Our theory links the existence of PEP schemes to geometric properties of the equation of state describing the thermodynamics of the fluid. When these geometric properties fail for a pair of states on the same isobar, no PEP schemes of the considered form can exist on a domain containing those states. In our analysis the equation of state itself can be fully general only needing to satisfy some fundamental thermodynamic principles. Recently, PEP compatibility conditions for general equations of state have been derived [1] that rely on the existence of certain thermodynamic derivatives which are not guaranteed to be defined under fundamental thermodynamic principles. The geometric conditions in our theory do not depend on the existence of these derivatives and recover the recent compatibility conditions in the case that these derivatives are defined. Finally, using numerical experiments we demonstrate for two supercritical fluids that our existence conditions are restrictive and thus that no PEP schemes of the form we consider exist on the domain we specify for these fluids. Using our geometric perspective, we also shed light on mechanisms by which PEP schemes can develop numerical issues, which we also demonstrate using numerical experiments.