The statistics report combines partition summaries rather than raw observations. Implement the merge so the combined mean and population variance include the differences between partition means.
Build the implementation in Python under /app, with /app/run.sh as its entry point. Read one JSON request from standard input and write one JSON result to standard output. Input is a list of summaries with nonnegative integer count, finite mean and nonnegative finite m2 (sum of squared deviations from that partition mean). Zero-count partitions must have m2 zero and are ignored. Merge with the between-mean correction. Return count, mean and population variance m2/count, rounded to six decimals; all-empty input returns zeros. Reject invalid counts or statistics.
Reject inputs that violate an explicit validation rule with exit status 2, no standard output, and {"error":"invalid_input"} on standard error. For valid input, exit with status 0. Keep the implementation reusable for new inputs; it must work offline and must not depend on the example data being the only data it receives.
Repair or replace the unfinished /app/run.sh and add source files under /app. Malformed JSON must use the same invalid-input response. Do not read /tests or /oracle from the application.