Deterministic Limit for Superconducting Quench: Deriving the Critical Hot-Spot Radius via Cryogenic Dissipation

The prediction of catastrophic quench events in high-field superconducting magnets—essential for particle accelerators, MRI systems, and commercial fusion Tokamaks—relies heavily on the classical Minimum Propagating Zone (MPZ) theory and bulk enthalpy margins. While these thermodynamic frameworks effectively estimate stability based on aggregate cooling capacities, they fail to deterministically define the exact spatial boundary where a localized resistive transition triggers irreversible thermal runaway. This paper introduces a strict continuum framework for macroscopic cryogenic stability. By modeling the superconducting matrix as a dynamic thermodynamic balance between the spatial capacity for heat removal (cryogenic dissipation) and the localized rate of resistive heating (Joule heating), we derive a universal critical quench radius (Rquench). We demonstrate that magnet quench is not a probabilistic thermodynamic fluctuation, but an exact deterministic limit where localized Joule heating strictly overpowers the advective thermal-distribution capacity of the surrounding cryogenic lattice. We propose a blueprint for Active Quench Telemetry (AQT) using distributed fiber optic temperature sensors to provide real-time spatial prediction and automated safety triggering prior to catastrophic energy release.