Overflow Regularization and Geometric Structures in Extreme Regimes

This work investigates the role of infinities that arise in extreme regimes of fundamental physics, such as those associated with infinite density, vanishing volume, extreme acceleration, or Planck-scale limits. Rather than interpreting these divergences as physically realizable states or as direct evidence of a breakdown of fundamental laws, it is proposed that they be understood as overflows of the adopted parametrization, that is, as descriptive failures resulting from the extrapolation of variables beyond their domains of validity. Based on this hypothesis, the Principle of Regularization by Overflow is introduced. According to this principle, the appropriate regularization of divergences should not occur through ad hoc modifications of the dynamics, but through a geometric reparametrization of the state space. In this framework, quantities that diverge in conventional descriptions are mapped onto compact domains via simple geometric transformations, in which extreme regimes appear as finite boundaries of the description rather than as physical singularities. The approach is illustrated through an explicit reparametrization of volume and density, in which limits traditionally associated with infinities are reinterpreted as asymptotic boundaries of a compact interval. Compatibility with thermodynamic principles, particularly the irreversibility associated with entropy, is discussed, as well as conceptual implications for the distinction between local dynamics, global geometry, and the topology of space. This work does not propose a new fundamental theory nor does it present direct observational predictions. Its objective is to provide a conceptual framework in which infinities cease to be treated as physical paradoxes and are instead understood as indicators of the need for a change of coordinates. Within this perspective, extreme regimes are interpreted as transitions in geometric description, preserving the continuity of physical evolution without the introduction of new entities or fundamental dynamics.