This paper introduces F-Theory, a hypothetical framework that derives the universe’s physical phenomena and mathematical structures from a single universal extremization principle, δF = 0. Acting as a meta-theory, F-Theory generates the laws of physics, informational structures, and thermodynamic behavior through a universal functional F[ϕᵢ]. Conceptually, F-Theory provides a unified approach to cosmology and fundamental physics, offering derivations of phenomena such as the Big Bang, black holes, multiverse scenarios, and dark matter/dark energy. By connecting physical laws and mathematical structures under a single principle, it proposes a pathway toward deep conceptual unification. F-Theory remains hypothetical, with observational and numerical validation left for future research. The paper emphasizes mathematical rigor and theoretical consistency. The manuscript was prepared with the support of AI tools (ChatGPT) for text generation and structural assistance. "A profound theory, formally verified in Lean 4, CI passing."