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 ± 0.1 kJy ms and a dispersion measure of 161.3 ± 0.4 pc cm−3, ranks among the brightest extragalactic FRBs at ∼40 Mpc. We obtain 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 μJy beam−1, yielding a 3σ upper limit of <8.4 μJy at the FRB position, corresponding to νLν < 2.4 × 1035 erg s−1. These results represent among the most stringent constraints for a nonrepeating FRB, lying ≳3 orders of magnitude below the νLν of compact PRSs 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 ∣RM∣ ≲ 30 rad m−2, consistent with a clean magnetoionic sight line and progenitor channels such as neutron-star mergers or giant flares from older magnetars.