Vortex Objecthood, Admissibility Collapse, and Rebirth

This paper introduces an existence-first framework for vortex breakdown in which vortex dynamics are defined only when vortex objecthood is admissible. Rather than treating breakdown as an instability within a fixed state space, the work formalizes breakdown as an admissibility collapse: a regime-level loss of coherent vortex objecthood despite bounded flow quantities. A quantitative Distance-to-Admissibility (DtA) functional is introduced to measure proximity to collapse, together with a second-order coherence criterion governing existence. The framework classifies vortices into five structural classes, explains abrupt collapse, hysteresis, intermittent disappearance, and multiplicity of post-breakdown states, and formalizes rebirth as regime re-entry via geometric selection rather than continuation. Geometry-attached time is introduced as the only notion of time consistent with conditional objecthood, advancing if and only if vortex objecthood exists. The result is a closed, equation-independent regime-geometric account of vortex breakdown consistent with decades of experimental observations.