Rank-normalized folded split R-hat

Inputs: $M\ge 2$ equal-length scalar chains with at least four observations each. Let $S$ be the total number of pooled draws.

1. Apply Blom rank-normal scores for pooled draws to all pooled draws, split the transformed chains as in Split R-hat from multiple chains, and calculate the split statistic $R_z$ on the transformed values.
2. Compute the pooled median $m$ by sorting all $S$ draws and averaging the two central order statistics when $S$ is even (using the central order statistic when $S$ is odd). Fold each original draw to $\mathrm{\mid }{x}_{m,n}-m\mathrm{\mid }$, apply the same pooled rank-normal transformation to these folded values, split each folded chain, and calculate the split statistic $R_f$.
3. If either split calculation is not estimable, return not estimable. Otherwise return $\widehat{R}=\max \{R_z,R_f\}$.

The rank-normalized component addresses location and trend differences in a finite output array; folding the absolute deviations makes the reported maximum sensitive to scale differences as well. It is a diagnostic procedure, not a proof of stationarity or convergence.