Off topic but anyway.
RM: Side effects are behavioral illusions when, as in the case of the power law, they are thought to tell you something important about how behavior works.
“Important” here is a subjective statement. Behavior works in an environment and always causes side effects. It is interesting and often important to study why certain side effects follow in certain cases – especially if there are somehow surprising regularities (or irregularities) in these side effects. An interesting research topic is why curved movements often cause regular power law phenomena – that requires explanation. You have been saying that it does not require explanation because a) the phenomenon is not really existing; or b) there is a mathematical explanation; or perhaps for some other reason. But measurements show very convincingly that the phenomenon is real, surprisingly regular, but not at all universal. Because it is not universal, no mathematical explanation can do. Your mathematical explanation is tautological saying that if there is 1/3 power law then there is 1/3 power law. But why there is sometimes other power law and sometimes none?
If something is a side effect it can still be important, interesting and relevant.