You can only disprove.
The only way to prove a positive if there is a finite number of possibilities and you have disproven all but one. But even then, someone could conceivably come up with an alternate description that preserves the current understanding but makes additional predictions or is a simpler model making the same.
As Feyman said: "We can never know if we are right, we can only be certain if we are wrong".
But otherwise, there is nothing special about positive or negative statements. You can express any positive statement as the negation of a negative statement, so to the extent that science can "disprove negatives", it can equivalently "prove positives".
For example:
Every continuous symmetry of action in a physical system with conservative forces has a corresponding conservation law. (Noether's Theorem)
There must be two antipodal points on Earth with exactly the same temperature and barometric pressure (as a result of the Borsuk-Ulam Theorem)
As far as I know these are absolutely proved positively because they are mathematical consequences of the properties of continuous functions etc. I'm not a scientist, but there are thousands of things like this where we are definitely absolutely certain we are right because of the possibility of a mathematical direct proof.
Likewise, we assume at an operational level that temperature and barometric pressure are continuous functions (as assumed in Borsuk-Ulam), but it's not something you can conclusively prove aobut reality.
There's an argument that you are still doing science if you construct a logical proof showing that "if the world is like X, then it will behave like Y". A lot of theoretical physics is like this, and people call that science. But I think there's truth to what OP is saying in that science does not conclusively positively prove things about reality.