Is it wise to mention any personal struggle in a statement of purpose (for doctrate program)? If there is a gap in my CV, or perhaps an anomaly due to some personal struggle, is it worth explaining the situation? Or just not saying anything in the SOP?

If one was perhaps wrongfully accused or terrorism/murder or suffered depression? which forced them to take a year out.

On one hand it can demonstrate resilience on the other hand it might be negative.
I think a quick examination of human psychology suggests that being specific about this stuff is a terrible idea.  Even people who want to be neutral would associate you with murder (unfairly, of course).  Better to never put the idea in their head.