Tests for batch systems with high unit value can be expensive, difficult to
operate, slow, or destructive. Testing every unit is then infeasible. Means and
pass or fail counts may also conceal tail risk and variation within a batch.
This paper formulates limited testing as recovery of a finite population
distribution. Auxiliary process variables are known for every unit, but the
target response is observed only for tested units. The procedure first orders
units by principal components. It then forms strata and nested probability
samples (PCA-OS). Generalized regression (GREG) estimates the cumulative
distribution function. A bootstrap rule stops testing when estimated precision
is adequate. We conduct 500 offline replays on one archival industrial batch.
Under a fixed budget, the procedure reduces mean Kolmogorov-Smirnov error from
0.0626 to 0.0421 relative to simple random sampling with a H\'ajek estimator.
The ablation shows that the sampling design and GREG correction both contribute
to this gain. The exploratory stopping rule lowers the mean sample size from
166 to 139.9. Its mean distribution error remains below that of the fixed
budget baseline. These results support methodological feasibility rather than
qualification for a specific application. Because the evidence is limited to
one batch with proxy variable roles and the replay paths are reused for
threshold tuning and evaluation, the stopping result remains exploratory and
has no sequential coverage guarantee.

auxiliary process variables,finite population sampling,GREG estimation,ordered stratified sampling,sequential testing,batch quality