Is there a proof that is true for all cases except for exactly one case? I was curious if there were any such proofs which state that a thing is true always EXCEPT for exactly one instance. As in, for some reason, there is only one instance where the proof is false, but it is true for all other objects. I understand that if it is not true in that one case that it is not necessarily a proof, I was just wondering if there were any "proof-like things" of this form.
Here's a famous one: $\mathbb{R}^n$ has a single differentiable structure (up to diffeo) except for $n=4$, in which case it has uncountably many.

These posts may be of interest:

https://mathoverflow.net/questions/16035/a-reference-for-smooth-structures-on-rn

https://mathoverflow.net/questions/24930/differentiable-structures-on-r3
Each object you can touch in your body is not your tongue, but it fails when the object you touch is your tongue.
Each person you look in this planet is not your mother, but it fails when you look at your mother.
Each person in this question is trying to showoff with advanced math concepts to answer a trivial question like this and is not called Voyska, but it fails for me.