A Novel Mathematical Framework for Determining Whether We Are Living in a Simulation: Integrating Information Theory, Quantum Mechanics, and Computational Complexity

This thesis presents a comprehensive mathematical framework for determining the probability that we are living in a computer simulation. Building upon Nick Bostrom's foundational simulation argument, we develop a novel integrated approach that combines information-theoretic bounds, quantum mechanical evidence, computational complexity analysis, and Bayesian inference to provide a quantitative assessment of the simulation hypothesis. Our framework introduces several key innovations: (1) a discreteness metric based on quantum measurement precision, (2) computational feasibility analysis across different scales of reality, (3) observer effect quantification as evidence for simulation-like behavior, and (4) a unified Bayesian model that integrates multiple lines of evidence. Through comprehensive analysis of four distinct scenarios (Conservative, Moderate, Optimistic, and Quantum-Focused), we find simulation probabilities ranging from 0.43 to 0.71, with confidence intervals providing statistical rigor to our estimates. Our empirical validation against known physical constants, quantum mechanical phenomena, and computational systems demonstrates strong support for the simulation hypothesis, with an overall validation score of 11.50. The framework reveals that while simulating systems up to solar system scale appears computationally feasible, simulating the entire observable universe exceeds current theoretical limits by 19 orders of magnitude. Key findings include: (1) quantum discreteness at the Planck scale strongly suggests digital rather than continuous physics, (2) the observer effect in quantum mechanics aligns with computational optimization principles, (3) Bell inequality violations can be elegantly explained through simulation constraints, and (4) the fine-tuning of physical constants is consistent with computational parameter selection. This work provides the first rigorous mathematical framework for simulation detection and establishes a foundation for future empirical testing of the simulation hypothesis.