Well, I am neither mathematician nor logician. Your quip about the division fallacy gave me pause so I thought I would formalize things.This is why mathematicians and logicians don't like each other.
Even if I grant you all your assumptions, your proof doesn't mean much if P(C) itself is an extremely low number. That's why you have to be careful about drawing logical deductions about specific individuals based on probabilities across a population. Put simply, unless P(C|T) = 1, then at least some of Trump's appointees are not corrupt. Ranger and I both think that evidence strongly suggests Mattis is one of those guys, so we aren't willing to go into attack mode unless genuine contrary evidence arises.
In other words, your math can be both technically correct and practically meaningless at the same time.
Actually, the exercise was kind of interesting. The value of the exercise is that at least you and I know pretty precisely what we disagree about. I wouldn't have gone to the trouble without your prompt and the condition P(T|C)>P(T|~C) is a bit stronger than I had thought. If P(T|C)<P(T|~C) for example then you would argue that working for Trump should IMPROVE someone's reputation for non-corruption! That strikes me as counter-intuitive. Your point about P(C) being low is precisely where prior reputation matters. The argument that P(C) is low in Mattis' case was something we all previously took for granted. The new information matters because it raises P(C). It would be interesting to look at the derivative of P(C|T) wrt P(C)...it is non-linear and might be interesting. My guess is that when P(C) is low and increase in P(C) has bigger effect on P(C|T) than if P(C) were large. Would be interesting to check.
I am not in "attack mode" on Mattis I think the proper response is to be in "alert mode". But there are full 5 alarm fires going on elsewhere...as has been pointed out. People, as a rule, don't learn enough from what is right before their eyes. They should be more Bayesian.