Upper bound on 'relatedness'?
Very interesting question, it's always hard to prove a negative.
I have a vague idea but I really don't know if it's worth anything or even applicable to this problem, so please take it with a grain of salt! 

The idea is to use randomness and multiple samples in order to compare the result of predicting from random noise vs. actual data X: if the result from X isn't significantly better than from random noise, then you have proved that X doesn't have any predictive power. Of course this relies on the assumption that the model used to predict is sensible enough.

I have seen methods using this idea but I don't remember the details unfortunately. The only directions I can give are these:


the impostor method for authorship verification, which is based on a somewhat similar idea
I've seen some works using the binomial test to formally test whether the outcome from a method X is significantly different from a random baseline.


Sorry for the lack of details, I hope this helps.