Fix the per-fold scaler in a churn pipeline

An AI wrote a scikit-learn churn pipeline that calls StandardScaler().fit_transform(X) on the whole dataset and then cross-validates. Cross-validation reported 0.94 ROC-AUC, and production was far worse. Your team checks the numbers by standardizing each validation fold the leak-free way, one feature at a time.

An assistant wrote standardize_within_fold(values, fold_ids, fold). It is in the editor. It has a bug. Find it and fix it.

Input and output:

• values: one feature's value for each row.
• fold_ids: each row's fold number, at the same index.
• fold: the fold being validated. Its rows are the held-out rows, and every other row is a training row.
• Return the standardized held-out values, in input order.

Rules the tests check:

• If the lists differ in length, use only the first n rows, where n is the shorter length.
• Compute the mean and the standard deviation from the training rows only. Use the population standard deviation: divide by the number of training rows, not that number minus 1.
• If there are no training rows, return an empty list.
• If the standard deviation is 0, use 1 instead.
• Each held-out value becomes its value minus the mean, divided by the standard deviation.
• Answers within 0.000001 of the expected values pass.