Finite-Memory Reconstruction in a Coarse-Grained Brownian Benchmark

We study a one-dimensional overdamped Brownian dynamics in a four-well potential, observed through noisy quantization and then reduced to a discrete metastable description. The aim is not to claim universal validation, but to analyze transparently how the reported reconstruction behaves under finite-memory feature embeddings and how the conclusions depend on the chosen memory window. The protocol separates calibration from validation, estimates cluster-based microstate labels in standardized feature space, and compares empirical and theoretical laws of short macro-histories together with support violations, total variation distance, goodness-of-fit tests, conditional mutual information, and absorption-time summaries under right censoring. The supplied outputs report memory-window runs for k in {1,2,3,4,7}, and the manuscript restricts itself to those values only. The main empirical message is nuanced. On the supplied runs, the macro-history support is preserved exactly and the support-violation rate is zero throughout. The residual conditional dependence becomes very small as k increases, reaching approximately 1.733 x 10^-6 for k=7. At the same time, microstate reconstruction accuracy is highest for k=1 among the reported orders, so larger memory does not uniformly improve every metric. The observed history laws are broadly consistent with the fitted macro model, but the chi-square and G-test p-values remain finite-sample sensitive and do not support an absolute claim of validation. Right censoring remains substantial and is therefore handled with Kaplan-Meier estimation and restricted mean survival time rather than naive full-mean estimates.