A Pure Mathematical Proof of the 4-Colour Theorem

Abstract This work "A Pure Mathematical Proof of the 4-Colour Theorem" is related to the previous proof assited by computer. "Triangulations of Euler Convex Polygon" provides a fresh beginning point for the proof. The central concept is to discover an extended invariant property of Standard Graph’s boundary, which is described as "3-Colour All Phase States (3CP)" in this work and it is demonstrated that the standard graph’s boundary and sub-bound are 3CP and 4-colorable（4-3CP） via the expanded operation e(+, p i ) and e(-, p i ). It's exciting that this regularity was discovered for the first time and the 4-3CP invariant can naturally derive the 4-Colour Theorem. The majority of the definitions, theorems, and proof strategies are shown in this work.