It's just the epitome of hubris. People have been trying to find counterexamples since at least the 19th century, without success, and now that a formal proof has come around, none of the thousands of professional research mathematicians alive has found any flaw with it.

But no, they must all be wrong, because of your (trivial and very easy to come up with) counterexample! Nope, the counterexample was shown wrong in minutes.

The worst part isn't that you didn't see how your example could be 4-colored. That's fine, everyone makes mistakes in mathematics. The worst part is that you have such a low opinion of everyone else's intellectual abilities that instead of asking "am I understanding the problem statement wrong?" or "what's the four-coloring of this graph that I'm missing?" you simply assert that you're right and the entire field of specialist mathematicians is wrong. Do you really think your counterexample is so shockingly clever that you're the only person capable of coming up with it in 150 years?

You are trying to construct a complete graph on N vertices in the plane, giving an example for N=5. This cannot be done. Indeed, every graph is either planar xor contains either the complete graph on 5 vertices or the bipartite graph of 3 vertices in each group as an implied subgraph.

You can embed a complete graph on N vertices in 3 dimensions easily, with thin rods. But this doesn't work for 2 dimensions because that connecting line divides 2-d space, so you can't cross it with another line. This property imposes some sharp constraints on what planar graphs have to look like, with implications for its colorability. The Four-Color theorem amounts to an exhaustive enumeration of the possible situations arising from these constraints and then showing that all of them can be colored with only 4 colors.

It's a rather easy formula

So what are you waiting for? Draw it up. You'd shake the very foundations of mathematics.

Right... Either you're misunderstanding what the theory says or your crazy concentric countries idea doesn't work (hard to tell from your description).

Then please do, we'd all love to see it.

You are correct that O(N) colours are needed if all the islands of one ring-country must have the same colour. But to encode that constraint graph-theoretically you would need "bridges" (edges) over isthmuses, which would make the thing non-planar.