I'm gonna blow your mind here.

There's a difference between not being able to prove something because it's untrue, and being unable to prove something something because it's a negative statement.

Going back to the analogy, you can prove that somebody beats their wife, people do that all the time, but if they don't beat their wife then that's a pretty significant obstacle to that undertaking, and it's not because you're making a statement that would require you have every moment of this dude's life recorded from birth to present day in order to come to a definite conclusion.

You can't prove that unicorns don't exist because the data required for that would be too comprehensive. We haven't observed all of existence. But you can't prove that unicorns do exist, because they don't. You can prove that squirrels exist, by looking out your window, and pointing at the squirrel. Proving the existence of something isn't inherently an impossible achievement.

Proving that something exists and is "one" would obviously need some further work to define what qualities being "one" describes, but that's provable, but only if it's true, as opposed to the reverse, which is fundamentally unprovable without completely unfeasible amounts of evidence.

That's the opposite.

Can't prove a negative means you can't prove that Allah isn't one. So in your example, proving that you don't beat your wife is impossible.

"Allah is one" is a positive. You can prove those. The same way you can prove that you do beat your wife.