Proofs are best left to mathmeticians.
In courts of law, things are "proven" by a preponderance of the evidence (more likely than not), clear and convincing evidence, and evidence beyond a reasonable doubt.
That Q and POTUS synch their posts/tweets provides evidence beyond a reasonable doubt that Q and POTUS work in coordination with each other, or are in fact, one and the same.
Launching criminal investigations with no evidence of a crime; launching a counter-intelligence investigation with no intel; these are things done by desperate men out of sheer instinct to survive. They are not the actions of reasonable men acting according to the mandates of their oaths or job descriptions.