Observer-Dependence in P vs NP

Abstract

We present a new perspective on the P vs NP problem by demonstrating that its answer is inherently observer-dependent in curved spacetime, revealing an oversight in the classical formulation of computational complexity theory. By incorporating general relativistic effects into complexity theory through a gravitational correction factor, we prove that problems can transition between complexity classes depending on the observer’s reference frame and local gravitational environment. This insight emerges from recognizing that the definition of polynomial time implicitly assumes a universal time metric, an assumption that breaks down in curved spacetime due to gravitational time dilation. We demonstrate the existence of gravitational phase transitions in problem complexity, where an NP-complete problem in one reference frame becomes polynomially solvable in another frame experiencing extreme gravitational time dilation. Through rigorous mathematical formulation, we establish a gravitationally modified complexity theory that extends classical complexity classes to incorporate observer-dependent effects, leading to a complete framework for understanding how computational complexity transforms across different spacetime reference frames. This finding parallels other self-referential insights in mathematics and physics, such as Gödel’s incompleteness theorems and Einstein’s relativity, suggesting a deeper connection between computation, gravitation, and the nature of mathematical truth.