Given two data feeds, find out if they capture the same information Say, there are two camera feeds, how can I establish if they were filming the same scene? It seems plausible that there are algorithms that somehow calculate mutual information and detect "causality violations", determining that the other feed is most probably not another perspective of the same source, and that in the presence of noise.

If it is unknown what kind of properties (e.g. dimensionality, physics) the environment that the data feeds capture has, are there algorithms that can construct the transition logic from scratch?

In computer vision, this problem is apparently called 'image registration'.
The problem of finding the relationship between two scenes shot from two different angles is known as the correspondence problem and there is a lot of literature on how to approach it; I suggest you start by reviewing the research literature on methods for solving this.  In particular, I'd suggest that if you can consistently find a correspondence between the two images, you might conclude that they are from the same feed; and if you can't, you might conclude that they aren't.