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.
Using map opens you to claims such as that a map containing the North Sea, the kingdom of the Netherlands, the kingdom of Belgium, the French Republic and the Federal Republic of Germany can’t be four-colored because they all border each other (the kingdom of the Netherlands and the French Republic on Saint Martin in the Caribbean)
Historically, other examples likely existed, as land ownership tended to be patchy in feudal Europe, and, later, colonies added borders between various European countries.
Theories are not laws; only takes one counter-example to disprove them.
A theory is a hypothesis. It's a plausible-but-unproven guess.
A theorem is a fact which is known because it has been proven.
As you're probably aware, this confusing patchwork of similar and overlapping yet distinct meanings is ripe for abuse: it's what allows people to get away with saying things like "evolution is just a theory".
[1] Here it means "the collective body of work of some school of thought".
[2] "A widely-accepted, well-tested scientific explanation for a phenomenon".
[3] "A subfield of mathematics; the set of definitions, theorems, and techniques related to a particular type of abstract object" (in this case, groups).
I agree, apparently it only took a couple of minutes in mspaint to disprove yours :)
On a serious note, what hermitdev did we've probably all done at some point in our past, especially in our youth (disparaging long-standing theories about life, science, philosophy, etc.), no need to be mean.
Few times this leads to groundbreaking revelations that help humanity progress, but most times it leads to the embarrassing conclusion that past generations were not idiots.
He was a good guy. Took me several days to finally understand my mistake, but afterward I had a much better understanding of the problem
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).
Again, as mentioned in my earlier comment, I would suggest thinking of things / describing things in terms of plane graphs rather than "maps". It'll make everything easier.
I might also suggest learning some of the relevant graph theory? It sounds like you're trying and failing to embed a K_5, or perhaps an arbitrary K_n, into the plane. That can't be done, and the proof that this is impossible is much easier than the full 4-color theorem. (There are also the 5-color and 6-color theorems, which again have much easier proofs than the 4-color theorem, and which are another reason you can't embed an arbitrary K_n in the plane.)
The third and fifth rings (counting from the inside) don’t touch each other so, you can give them the same color.
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.