A Deterministic Limit for Quantum Key Distribution: Deriving the Critical Fidelity Radius via Entanglement Distribution

The prediction of channel security and maximum transmission distance in Quantum Key Distribution (QKD) networks relies heavily on probabilistic Bell-state fidelity equations and statistical optical attenuation rates. While these stochastic frameworks effectively estimate aggregate signal-to-noise ratios and dark-count probabilities, they fail to deterministically define the exact spatial boundary where localized channel decoherence triggers Entanglement Sudden Death (ESD). This paper introduces a strict continuum framework for quantum optics. By modeling the optical waveguide as a dynamic kinematic balance between the spatial capacity for phase propagation (entanglement distribution) and the localized rate of optical scattering (channel decoherence), we derive a universal critical fidelity radius (Rfidelity). We demonstrate that ESD is not a probabilistic decay over distance, but an exact deterministic limit where localized optical noise strictly overpowers the advective phase-preserving capacity of the surrounding waveguide. We propose a blueprint for Absolute Quantum Repeater Scaling to guarantee 100% unhackable data transmission by engineering node placement around this strict geometric boundary.