Beginings of a proof that only 4 regions can come together at a point
It can be shown geometrically that the sum of the angles of all lines meeting at a point must be 360 degrees. I can show that every exposed corner in my application is either 90 degrees, or in the case of edge following regions will when added to the angle of the edge they are following be 180 degrees. This means I can prove an upper limit for regions coming together at a point of 4.