Admissibility Lifting and Variational Convergence in Dynamical Systems with Collapsing State Spaces

This paper develops a variational and operator-theoretic framework for dynamical systems in which admissibility itself may degenerate or collapse. Feasibility is encoded through hazard-weighted energies on a fixed ambient Hilbert space, allowing regime change to be analyzed without time-dependent state spaces. Using Γ-convergence and Mosco convergence, the paper characterizes operator and spectral limits under admissibility degeneration and introduces null limits, regimes in which the correct mathematical outcome is absence of a well-defined problem rather than a limiting equation. A recovery result shows that classical fixed-state-space PDE theory is obtained under persistent admissibility.