A Deep VLA Search for a Persistent Radio Counterpart to the One-off FRB 20250316A

Fast Radio Burst (FRB) 20250316A, detected by CHIME on 2025 March 16 with a fluence of $1.7\pm0.1~\mathrm{Jy\,ms}$ and a dispersion measure of $161.3\pm0.4~\mathrm{pc\,cm^{-3}}$, ranks among the brightest extragalactic FRBs at $\sim 40$ Mpc. We obtained deep Karl G. Jansky Very Large Array follow-up at 15~GHz on 2025 April 5 and 9 and find no persistent radio source (PRS). Our best image reaches an rms of $2.8~μ\mathrm{Jy\,beam^{-1}}$, yielding a $3σ$ upper limit of $<8.4~μ\mathrm{Jy}$ at the FRB position, corresponding to $νL_ν< 2.4\times10^{35}~\mathrm{erg\,s^{-1}}$. These results represent among the most stringent constraints for a non-repeating FRB, lying $\gtrsim 3$ orders of magnitude below the $νL_ν$ of compact persistent radio sources around well-studied repeaters, thereby disfavoring bright magnetar-nebula scenarios and pointing to low-density, weakly magnetized environments. Interpreting our limit through pulsar-/magnetar-wind synchrotron frameworks places joint constraints on ambient density and engine power. If the empirical PRS--rotation-measure trend reported for repeaters extends to one-off sources, our limit implies $\vert \mathrm{RM} \vert \lesssim 30~\mathrm{rad\,m^{-2}}$, consistent with a clean magneto-ionic sight line and progenitor channels such as neutron-star mergers or giant flares from older magnetars.