I haven't worked through your post in detail, but I have a concern.
The four-colour map theorem fundamentally depends on the fact that the planet is flat (or sphere-shaped, as it turns out). On a planet shaped like a torus, for example, four colours are *not* enough (you can need up to 7). So any proof of the four-colour map theorem has to use the fact that the planet is flat. Otherwise, it would just as well be a proof of the four-colour Torusworld map theorem, which is false, so the proof would be invalid (even for the flat planet case).
But I don't see where the flatness of the planet is used in your proof. Can you clarify?
Well, the only case I can see being a problem is if one other country and your country surround one of the border counters. Think Lichtenstein where Austria and Switzerland have a non-contiguous border broken when they both surround Lichtenstein. Each country would require a unique color, but would break the pattern established by wall segments.
Also, I guess, you could say that ya know, you were using the earth as your prototype to satisfy the overly retentive audience.
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I haven't worked through your post in detail, but I have a concern.
The four-colour map theorem fundamentally depends on the fact that the planet is flat (or sphere-shaped, as it turns out). On a planet shaped like a torus, for example, four colours are *not* enough (you can need up to 7). So any proof of the four-colour map theorem has to use the fact that the planet is flat. Otherwise, it would just as well be a proof of the four-colour Torusworld map theorem, which is false, so the proof would be invalid (even for the flat planet case).
But I don't see where the flatness of the planet is used in your proof. Can you clarify?
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Also, I guess, you could say that ya know, you were using the earth as your prototype to satisfy the overly retentive audience.
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