Gravitation and Relative Complexity: Observer-Dependent Resolution of P vs NP

This paper investigates the hypothesis that gravity imposes universal constraints on information processing across all scale, resulting in a framework that integrates gravitational effects into computational complexity theory. By synthesizing concepts from general relativity, quantum mechanics, and complexity theory, we introduce gravitationally modified complexity classes and derive explicit expressions for how spacetime curvature constrains computational capacity. Our analysis reveals that gravitational effects can induce phase transitions in problem complexity, potentially leading to observer-dependent resolutions of long-standing problems like P vs NP in extreme gravitational environments. We demonstrate that the maximum computational capacity of a system depends not only on its energy but also on the local gravitational field and quantum gravitational effects. Our framework predicts novel phenomena, including gravitationally induced decoherence and potential enhancements to quantum computation in certain gravitational regimes. While this work emphasizes theoretical and mathematical rigor, we propose a series of experimental setups to test our predictions, ranging from Earth-based atomic clock experiments to satellite-based quantum computing tests and astronomical observations. This work has far-reaching implications for quantum computing, fundamental physics, and cosmology, suggesting that the universe may be inherently computational in nature, with gravity playing a crucial role in shaping the informational landscape of reality. It not only offers new perspectives on long-standing problems like the black hole information paradox but also opens up new technological possibilities, such as gravity-assisted quantum algorithms and holographic quantum computation. By demonstrating how gravity fundamentally shapes the nature of computation, this research provides a unified view of information processing in the universe and paves the way for a deeper understanding of the connections between spacetime structure, quantum mechanics, and computational complexity.