Quantized Dissipation from the Inverse-Square Anomaly in a Non-Hermitian Klein-Gordon Field

We construct an exactly solvable relativistic model that embeds the anomalous inverse-square interaction into a non-Hermitian Klein-Gordon field theory through a purely imaginary, scale-invariant scalar potential. The stationary field equation reduces to an inverse-square Schrodinger-type problem with a quadratic spectral parameter. Imposing a strictly outgoing boundary condition at the singularity-interpreted as irreversible absorption-selects a unique physical realization and converts the fall-to-the-center instability into a discrete, log-periodic spectrum of complex energies. The resulting decay rates exhibit universal geometric spacing, determined solely by the anomalous scaling exponent and insensitive to microscopic short-distance regularization. This structure defines an emergent kinematic energy scale that controls dissipative dynamics and provides a minimal analytic framework for studying scale anomaly, boundary-condition-induced non-Hermiticity, and quantized dissipation in relativistic open quantum systems.