Prove convergence for arbitrary tabulated black-box Gibbs-energy models

Prove that the bifurcation strata of the geometric objective have zero measure, and establish convergence of the geometric solver for arbitrary tabulated or black-box reduced Gibbs free-energy functions rather than only for the real-analytic model families considered in the paper.

Background

The convergence argument relies on an explicit assumption that parameter values at which the defective-minima topology bifurcates form lower-dimensional, measure-zero strata. The paper explains why this is plausible for real-analytic model families such as NRTL, Wilson, UNIQUAC, and cubic equations of state, but does not establish it for arbitrary black-box or tabulated energy functions.

The unresolved problem is to replace this assumption with a proof broad enough to cover genuinely tabulated or otherwise non-analytic thermodynamic models, for which the topology of the dual manifold may change in ways not controlled by real-analytic arguments.

References

This assumption is not proved here, and it is not provable for a completely arbitrary black-box $g$. It is, however, routine for the model families used in this work.

— A geometric reformulation of the bilevel parameter optimization problem to a single level non-linear programming problem with applications to phase equilibria  (2608.17806 - Endres et al., 18 Aug 2026) in Section Conclusions, paragraph beginning “Proof status,” and Appendix, Section “Convergence of the two-stage outer loop under topology-induced discontinuities”