Mechanism-Resolved Fractional–Volterra Modeling for Thermal-Hazard Screening: Lithium-Ion Battery Intervention Evidence, PDE Transfer, and Stochastic Verification

Thermal-hazard screening in heterogeneous systems may require separate representations of long-tail persistence, delayed heat return, spatial exchange, and stochastic variability. We formulate a causal three-zone fractional–Volterra model family at the Caputo endpoint; a Hilfer extension is given in the appendix. The numerical implementation is assessed with manufactured solutions, nested-Brownian self-convergence, an exact stochastic endpoint, four-model discrimination, and transfer to an out-of-family nonlinear PDE. The application analysis uses 51 published NIST lithium-ion battery heat-interruption experiments. Normalizing intervention time by a format- and state-of-charge-matched heating-to-ignition delay increases the format-stratified AUC for immediate ignition from 0.634 to 0.881. Propagating uncertainty in the published reference-delay summaries gives a median normalized AUC of 0.872 (95% interval 0.715–0.968) and a median AUC gain of 0.233 (0.058–0.419); the gain is positive in 99.5% of resamples. Leave-one-immediate-ignition-out analyses give normalized AUCs of 0.861–0.906, and ordinal and SOC-adjusted analyses yield consistent results. Sensitivity analyses using Newton/radiative heat loss and additive noise retain the same dissipative regime. Repeated excitation improves separation of α and memory strength, whereas the composite low-frequency gain ℓv/κ is better identified than ℓv and κ separately. In an auxiliary NOAA benchmark, locked climatology outperforms the memory models. These results support the model family as a screening method; they do not constitute battery-specific calibration or define a universal hazard threshold.