This paper presents a formal proof for the validity of General Relativity and Quantum Mechanics based on the ‘Axiom of Finite Information.’ We demonstrate that if the speed of information transfer were infinite, or if space-time were perfectly continuous, the universe would contain states of infinite information density, leading to logical singularities. Furthermore, we prove that because physical laws must be encoded within the universe itself, a ‘Theory of Everything’ is precluded by Godelian incompleteness, ensuring that a universal theory can never exist.

We propose a conceptual framework to resolve the dichotomy of the Millennium Prize Problems by categorizing mathematical systems based on their capacity for logical simulation. We distinguish between Class I (Structural) problems (e.g., Poincaré, Hodge, Yang-Mills), which rely on symmetries, conservation laws, and coercivity estimates that constrain degrees of freedom effectively, and Class II (Simulational) problems (e.g., P vs NP, Navier-Stokes), which theoretically possess the fidelity to simulate Universal Turing Machines. While not a formal proof of independence, we argue that Class II problems face obstructions isomorphic to the Halting Problem, inhibiting standard analytic techniques. We posit that the ‘intractability’ of these problems arises because they inhabit a complexity class where asymptotic behavior is determined by generalized computation rather than geometric structure.