Between 1994 and present I posted perhaps a thousand or 2 thousand posts on 4 color mapping. What I am doing now is condensing that wisdom into one proof to be published in Arxiv.
The essence of 4 Color Mapping is that there is never a 5th mutual ?adjacency, and I proved it using the Moebius theorem.
Actually, 4 Color Mapping is a ridicules take off of the 2 color mapping. Given any Plane with closed objects in the plane that are adjacent, then the Jordan Curve theorem proves that all Mapps are 2 colorable. We see this all the time in picture photographs of black and white or we see it in mapps of countries whose border is black and interior is white.
But then some people feel that black borders with all white interiors should be spruced up a bit with colors other than white and so they throw onto the 2 COLOR MAPPING that of 5 Color Mapping of black borders with 4 Colors for interiors. So instead of just the Jordan Curve theorem we need to apply the Moebius theorem of 4 maximum adjacencies.
In the Plane there is 2 Color Mapping and so 4 Color Mapping is redundant and really that of 5 color mapping.
So, all one really has to do with the 4 Color Mapping is color the ?borderlines black, a 5th Color, so that there are 5 Colors in all and ?then use the Moebius theorem as the proof of 5 Color Mapping. This was the real mathematics proof of Color Mapping and the Appel & Haken's versions was a hijacking of what the true logical statement of Color Mapping had to ?be. Because the definition of a country or region is integral with the ?borderlines of the region.
Appel & Haken threw out the borders of countries. This is like throwing out the sides of a triangle and then expecting to be able to work with a figure that is no longer existing. One cannot separate away a country and its borders, for one does not ?exist without the other. For professors of mathematics who lifelong ?sit away in ivory towers never dirtying their hands over real work and never being practical but mostly always idealistic with their heads and minds in cloud 9, to them, they think that a country can exist that has no borders. They would also think that biology exists without ever having DNA of A,C,G,T, or that physics can exist without there being atoms present. The moment one hands a problem like 4 Color Mapping to some ivory ?towered idealist, of course he is going to mangle that conjecture into being a fakery of ideas.
Here is 4Color Mapping which is truly 5Color Mapping in its essence:
MMMMMMO ? MMMMMMO ? BBJJJO ? BBJJJO ? OOOO
Now, clearly, can you see why 4 mutual adjacency is a maximum? ?Can you see that a 5th is never allowed because the J country ?was covered over by the O country? So that when you include borderlines, you can apply the Moebius ?theorem, and thus dispense with a Computer looking for zillions of stupid and silly case studies. When you do a Appel and Haken by ignoring borderlines, you also do not ?have countries either, because the definition of a country has to have borders. So what Appel and Haken and other 4 Color Mappers are doing is not real math but a shadowy outside world of fake math where borders are ignored. So, now, why did Appel and Haken resort to math fakery? The answer is really quite simple in that their computer aided fake proof was during the 1970s when computer sales were lagging and where a math proof using computers would have a dramatic sales pitch for the industry. So what Appel & Haken did was offer a fake proof and allow the computer sales to explode.