Admissibility-Based Foundations of Science

This document presents a rigorous admissibility-based mathematical framework for analyzing evolution problems in which classical assumptions of persistent domains, global time, and unconditional applicability of equations fail. Rather than modifying existing physical or mathematical laws, it formalizes admissibility as a first-class structure governing when evolution equations are meaningful, introducing admissible regimes, geometry-attached time, regime persistence and collapse, reversibility versus irreversibility, regime interference, admissibility entanglement, and objecthood conditioned on admissibility. A central distinction is drawn between analytical breakdown (such as blow-up or loss of regularity) and structural breakdown, in which solutions may remain bounded while the mathematical framework required to interpret them ceases to exist. Classical theories of ODEs, PDEs (including Navier–Stokes), control systems, and related evolution problems are recovered as special cases under persistent admissibility, with no equations altered. The text is written as foundational course material for a future graduate-level special topics course, provisionally targeted for a 2036 intake, and is released publicly to establish terminology, structure, and priority, and to invite scrutiny, critique, and refinement from the mathematical and scientific community.