We propose a geometric framework for phase transitions based on the concave value function Ψ (λ) = infF K (F) + λD (F) and on rupture operators encoding discrete transitions between competing configurations. The concavity of Ψ naturally leads to regime changes at non-differentiable points of the value function. We analyze the relationship between these singularities, the coexistence of multiple minimizers, and the associated rupture operators. This framework provides a unified geometric interpretation of transitions across different domains. We illustrate the approach on Landau theory and on the LASSO regularization path, highlighting a common envelope structure underlying phase transitions and regime changes.
Luc de Veigy (Thu,) studied this question.