Logic 101, bro.
Presumption is negative. Burden of proof falls on the affirmative.
Why? Because if presumption was affirmative, we'd have to prove negatives to rebut it. And you can't prove a negative. QED.
You should have learned that in high school. Consider suing your teachers for doing a really shitty job of educating you.
Uh oh. I think you might want to double-check your own logic teachers. The old yarn about "not proving a negative" isn't in any way a QED it's a popularism which, while related to an actual fact of logic, is actually a fallacy in the way that you've deployed it.
In fact, "proving a negative", as you say, is one of the basic argumentation techniques for proof. You take a statement you want to prove, you negate it, you show that that leads to absurdity, this is QED for the original statement. The crucial point, in fact, relates to the square of opposition (cf. Arisitotle's Περὶ Ἑρμηνείας, Latin: De Interpretatione). Contradictories divide up the space between them with nothing left so that either A or ~A is true, it's impossible for it to be otherwise. When you're dealing with propositional/sentential negation then you are dealing with contradictories so it is quite useful to employ negation in proofs. On the other hand, contrary statements allow for the middle ground to also be true and this is where you have to be careful employing negation.
All cats are black [is contradictory to] Some cats are not black (= not(all cats are black)).
All cats are black [is contrary to] No cat is black.
In the contradictories, either the left or right is true, no other options (in fact, I believe in the real world that the right side is true, some cats are not black). However, in contraries, it's possible that neither the left or right is true (and, btw, I believe this is the case in the real world for this example).
The old yarn about "you can't prove a negative" is a popularism which is often employed without considering how vague it really is. If you really want to understand syllogistic logic and negation, start with the square of opposition.
Best!