Joint State-Parameter Estimation for the Reduced Fracture Model via the United Filter

The paper presents a clear algorithmic framework (United Filter) and correctly states the score update with a damping h(t) and the reverse-time SDE mechanism, but treats them as approximations and provides no formal convergence/probability theorems. The candidate solution supplies sound high-level arguments (Bayes gradient identity, reverse-SDE exactness under exact scores, importance-sampling consistency, linear–Gaussian reduction to Kalman), yet relies on unspoken regularity/limit assumptions (exact scores, full-support proposals, sufficiency of summaries, R→∞) and stops short of full proofs. Net: the paper is methodologically correct but theoretically incomplete, and the model’s solution is conceptually correct but not fully rigorous. Key points align with the paper’s setup (e.g., the score update (25) and reverse SDE (21) explanation), but the paper itself acknowledges open theory (e.g., the effect of increasing R), while the model asserts more than it proves. See the paper’s EnSF score update with h(t) and reverse-time SDE claims, and the Direct Filter weighting/resampling steps and open-theory remark about R iterations for grounding.