Many transition phenomena across science — phase transitions in thermodynamics, regularization paths in statistical learning, and rate–distortion trade-offs in information theory — share a common mathematical structure. In this paper we propose a unified geometric framework for such transitions based on a one-parameter family of variational functionals Phiₗambda (F) = K (F) + lambda · D (F), where K measures complexity and D measures fidelity. The associated value function Psi (lambda) = inf over F of Phiₗambda (F) is concave. Transitions correspond to changes in the exposed face of the lower convex hull of the attainable set (K (F), D (F) ). The function Psi can be expressed as the Legendre–Fenchel transform of the minimal complexity f (D) = inf over F such that D (F) = D of K (F), so that regime changes correspond precisely to singularities of this transform. Within this framework, abrupt (first-order) transitions appear as non-differentiable points of the value function, while continuous (second-order) transitions correspond to higher-order singularities where the minimizer evolves continuously but its stability changes. These transitions can be realized through rupture operators, defined as maps between configurations producing a geometric jump, a strict decrease in complexity, and the preservation of structural invariants. An operational interpretation of complexity is introduced through the Minimum Description Length (MDL) principle, making the framework computationally meaningful. The resulting geometric perspective reveals a common variational mechanism underlying classical constructions such as the Maxwell construction in thermodynamics, regularization paths in sparse learning (e. g. , LASSO), and rate–distortion trade-offs in information theory. A minimal numerical example illustrates the convex-hull geometry governing regime changes. The ideas, conceptual structure, and scientific decisions presented in this work are the author’s own. Artificial intelligence tools were used as assistance for writing, formatting, and editorial refinement.
Luc de Veigy (Sun,) studied this question.