The association of persistent radio sources (PRSs) with repeating fast radio bursts (FRBs) offers unique insights into their circum-burst environments. Building upon the physical link between PRS luminosity ($L_\nu$) and FRB rotation measure (RM), we introduce a novel diagnostic framework utilizing the intrinsic scatter of the $L_\nu - |{ \rm RM}|$ relation as a physical probe of nebula dynamics. We demonstrate that this scatter encodes critical information regarding the temporal evolution of the nebula radius ($R \propto t^\alpha$). By deriving a generic scaling $L_\nu \propto R^\epsilon |\mathrm{ RM}|$ and analyzing the residuals of the five confirmed FRB-PRS systems, we constrain the nebula's evolutionary index to $\alpha|\epsilon| = 1.5 \pm 0.7$ (1$\sigma$ uncertainty). This measurement provides a powerful diagnostic tool for distinguishing among different astrophysical scenarios. Its value deviates from the expectations for supernova remnants (SNRs) in the Sedov-Taylor phase ($\alpha|\epsilon| \sim 0.2-0.4$), reverse shocks during the free-expansion phase of SNR/interstellar medium (ISM) interactions ($\alpha|\epsilon| \gtrsim 3.5$), and pulsar wind nebulae (PWNe) powered by a decaying wind ($\alpha|\epsilon| \sim 0-0.15$). Instead, it is more consistent with forward shocks in the free-expansion phase of both SNR/ISM and PWN/SNR systems ($\alpha|\epsilon| \sim 2.0-2.8$), and young PWNe driven by a nearly constant wind ($\alpha|\epsilon| \sim 1$). These findings suggest that active repeaters are powered by dynamically young, rapidly expanding nebulae. While currently limited by the small sample size, this framework establishes a robust methodology for future population studies to constrain the physical origin of PRSs.
Categories: astro-ph.HE