A Lifting of Classical Fixed State-Space PDE Frameworks to Hazard-Governed Dynamics Geometries

This paper presents a formulation of partial differential equations in which the state space is not assumed to be fixed. The geometry defining admissible states is allowed to depend on a hazard field, so that the state space itself evolves and may degenerate in time. Differential operators retain their standard form and are evaluated with respect to a hazard-dependent metric defined on a fixed reference manifold. Classical fixed state-space PDE theory is recovered as a special invariant case. The framework provides a mathematical setting for representing admissibility collapse and state-space singularity in hazard-driven systems.