P Vs NP as a Physical Problem: From Topological Invariants to Hardware Implementation

Abstract

The relevance of the research is determined by the fundamental nature of the P vs NP problem, which has been considered a central unsolved problem in theoretical computer science for more than half a century. In this article, for the first time, it is proved that the problem of solving the P vs NP problem is not a mathematical problem, but a physical problem, the solution of which depends on the physical implementation of the computing system. The aim of this paper is to prove that the solution of the P vs NP problem depends on the type of physical system used for calculations. The leading approach to research is to synthesize three independent paradigms:

1. Algebraic topology that provides invariants for complexity classification.

2. Quantum physics, which provides parallel information processing.

3. Apparat implementation on the GPU that demonstrates the physical limitations of classical computing. 

Author's results include: A formal proof that P ≠ NP holds in classical physics (Newtonian mechanics, deterministic computations) and a simultaneous proof that P = NP holds in quantum physics (superposition, entanglement). Experimental confirmation of the physical implementation through a series of computational experiments. Development of hybrid system that allows you to switch between modes depending on the available resources. The theoretical significance lies in rethinking the fundamental problem of complexity theory as a physical law. Practical significance is the creation of an adaptive computing system, the efficiency of which is determined by the available physical platform.