Yet you are given their time index values, meaning you have more than the unordered simultaneous pairs you presented in the example.

The example data (here they are again) is a time series, not a set of unordered pairs. (Time is proportional to line number.)

No means whatsoever?

None whatsoever (assuming the random noise that drives the process is truly random, or at least unknowable -- guessing the pseudo-RNG algorithm and its seed doesn't count).

Consider this challenge open to anyone who thinks that there is mutual information between the two columns: calculate it. Prove the validity of the computation by calculating information about the second column given only the first, for a new file generated by the same method.