There is however no difference between "positive" and "negative" proofs. There is a very big difference in the scope of some proofs. It is easier to prove the baseball is not in the toolbox than it is to prove the baseball is not in the Empire State Building. The scope of one proof is a toolbox. The scope of the other proof is the Empire State Building. The larger the scope, the more difficult the proof. The "positive" proof that the baseball is in the Empire State Building can be equally as difficult as the "negative" should the baseball be in the last place you look.
Every proof requires that something else could not have happened. In that sense all proofs are negative proofs. So called "positive" proofs start with the assumptions that certain other things could not have happened, but without bothering to prove those.
The proof that the baseball is not in the Empire State Building is just as dependable as the proof that the baseball is not in the toolbox if you realize that the Empire State Building is just a bunch of spaces the size of a toolbox. You have more to check, but once you've checked them all you are as certain as with one toolbox sized space.
It might be fair to say that you cannot prove anything with infinite scope. That however can be the same for "positive" and "negative" proofs. Some might complain here that if you find the baseball then it's "positive" proof, which is "possible" where a negative proof is not. It remains true though that if the scope is truly infinite no proof is possible positive or negative. Once you find the baseball you need to check the autograph to ensure it's authentic. Then you have to check the person who made the autograph to ensure he's authentic. Understand that it never ends. So with truly infinite scope there is no proving things positive or negative.
The purpose of crime scene tape is to limit the scope of the positive proof who commited the crime. The more traffic allowed through the scene the larger the scope of the proof and the more difficult it becomes.
While it might well be true that the universe is "infinite" in time and space and the number of objects, it is apparently not at all infinite in anything else. There does not appear an infinite number of chemical elements or their properties. Actual possibilities in physics are apparently extremely limited. The only predictability we have depends on those limits. The existence of gravity is proved. It is indeed the "law" of gravity because it is proved as far as anything can be proved. Just as it can be proved that nothing ever defies the law of gravity it can be proved that life never assembles itself without an intelligent designer. See that proof here, Intelligent Design Is Scientific Fact.