As much as I hate it, I don't think we'll ever get a proof of the four color theorem that isn't enumerating cases.
When you have an integral or the sum of an infinite series that comes out to pi, you know there must be some satisfying explanation involving a circle.
Contrast with "Examples of patterns that eventually fail" on math stackexchange[^1]. When a pattern ends at 906150257, you don't really expect the proof for that to be something beautiful. The reason for the exact value of an upper bound is that it isn't smaller and it isn't bigger.
There's a relationship between e, i, pi, and -1 comes from a deeper relationship between complex numbers and rotation.
The relationship between planar graphs, vertex coloring, and 4 might just be because we put planar graphs and vertex coloring in the same room and 4 popped out, instead of 3 or 5.