A Deterministic Limit for Superconducting Quench: Deriving the Critical Normal-Zone Radius via Cryogenic Diffusivity

The prevention of catastrophic quench events in superconducting magnets relies heavily on statistical thermal stability models, most notably the Minimum Quenching Energy (MQE) criteria. While these lumped-capacitance frameworks effectively estimate macro-scale energy thresholds required to initiate a normal-zone, they fail to deterministically define the exact spatial boundary where a localized resistive hot-spot triggers a cascading transition into irreversible Joule heating. This paper introduces a strict continuum framework for macroscopic electro-thermal scaling. By modeling the superconducting matrix as a dynamic thermodynamic balance between the spatial capacity for advective cryogenic cooling (helium diffusivity) and the localized rate of resistive heat generation, we derive a universal critical quench radius (Rquench). We demonstrate that magnet quenching is not a random statistical anomaly of thermal fluctuation, but an exact deterministic limit where localized Joule heating strictly overpowers the advective thermal quenching capacity of the cryogenic infrastructure. This method provides an exact spatial threshold for quenching, offering superior accuracy over traditional models. We propose a blueprint for Active Quenching Suppression (AQS) using fiber-optic spatial telemetry, enabling real-time quench prediction and mitigation before catastrophic coil failure.