A Mathematical Proof of the Four-Color Conjecture (1): Transformation Step

The four-color conjecture has puzzled mathematicians for over 170 years and has yet to be proven by purely mathematical methods. This series of articles provides a purely mathematical proof of the four-color conjecture, consisting of two parts: the transformation step and the decycle step. The transformation step uses two innovative tools, contracting and extending operations and unchanged bichromatic cycles, to transform the proof of the four-color conjecture into the decycle problem of 4-base modules. Moreover, the decycle step solves the decycle problem of 4-base modules using two other innovative tools: the color-connected potential and the pocket operations. This article presents the proof of the transformation step.