The measurement times are not evenly spaced, so a fixed-step sum gives the wrong integral. Implement the calculation using the supplied positions and the stated integration convention.
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 [time,value] finite numeric pairs in strictly increasing time order. Return area using trapezoids, duration from first to last time, and time_mean=area/duration, each rounded to six decimals. Zero or one sample returns area and duration zero, time_mean null. Reject repeated/decreasing timestamps or nonfinite values.
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.