Tipping as Fiber Degeneration: A Grothendieck-Theoretic Foundation of Admissibility Loss

This manuscript develops a categorical and geometric reformulation of tipping points based on admissible fibrations over forcing histories. Rather than classifying tipping by mechanism—bifurcation-induced, rate-induced, stochastic, or interaction-driven—the framework defines collapse as boundary contact in admissible state space and shows that classical mechanisms arise as projections of a single underlying geometry. The central structural objects are the admissible fiber over each forcing history, Distance to Admissibility as a canonical boundary metric, Joint Margin as a measure of interaction curvature under multi-source forcing, and a Base-Change Theorem unifying tipping mechanisms through categorical restriction. Using the Atlantic Meridional Overturning Circulation (AMOC) as a worked example, the paper demonstrates how diverse tipping behaviors reduce to fiber degeneration within a unified admissibility structure. The resulting theory provides a structural foundation for tipping analysis applicable beyond climate systems to general forced dynamical systems.