A Solution to the P Versus NP Problem

People create facts rather than describe them; they formulate mathematical concepts rather than discover them. However, why can people design different mathematical concepts and use different tools to change reality? People establish the correspondence between theory and reality to use theory to explain reality. However, this implies that theory cannot change reality—if theory was capable of altering reality, the correspondence between theory and reality would no longer hold. Similarly, people have introduced the correspondence between mathematical concepts and sets. This implies that people cannot construct different mathematical concepts. If different mathematical concepts could be constructed, the correspondence between mathematical concepts and sets would no longer hold. Unlike traditional approaches that base mathematical concepts on equivalent transformations—and, by extension, on the principle that correspondence remains unchanged—this theory is founded on nonequivalent transformations. By constructing a special nonequivalent transformation, I demonstrate that for a problem P(a) in the complexity class P and its corresponding problem P(b) in the complexity class NP, P(a) is a P nonequivalent transformation of P(b), and P(b) is an NP nonequivalent transformation of P(a). That is, the relationship between P(a) and P(b) is neither P=NP nor P≠NP.