The distribution of relaxation times (DRT) resolves electrochemical impedance spectroscopy (EIS) into mechanistically interpretable battery health features, but the deconvolution is severely ill-posed: at the sparse frequencies of rapid or onboard diagnostics, single-spectrum solutions—and the health descriptors built on them—become unstable. Here we present P³-DRT, which constrains this inverse problem with the battery population itself: a score-based prior, learned from 215,250 quality-screened DRT candidates spanning six public datasets, encodes the DRT morphologies real batteries actually exhibit, while the target spectrum's impedance likelihood anchors every posterior solution to the measurement. With only 12 and 8 frequency points, P³-DRT cuts full-spectrum reconstruction RMSE from 1.92% to 0.64% and from 3.37% to 0.70%, and band-integral descriptor errors by up to 86.6%. Posterior descriptors achieve a leave-one-battery-out state-of-health (SOH) RMSE of 4.43%—one third below single-spectrum Bayesian DRT—and transfer to unseen NCA cells without target-domain adaptation, reducing SOH RMSE by 21.5% and 20.6% at 0% and 100% SOC while delineating an SOC-dependent transfer boundary. Population-level empirical priors thus turn sparse impedance measurements into stable, mechanistically interpretable battery diagnostics.