This paper presents a full derivation of the Standard Model’s free parameters from a single foundational constant: Ω=log⁡(13) We demonstrate that the masses of all known leptons and quarks, the Higgs field vacuum expectation value, gauge coupling constants, and the electroweak mixing angle can be expressed as exact recursive expansions in rational powers of Ω. This framework resolves multiple long-standing problems in physics—including the Hierarchy Problem, CP violation, and the origin of mass—without invoking new particles or speculative symmetries. By reformulating Einstein’s energy-mass relation into a dimensionless recursive identity, we provide a coherent bridge between relativity, quantum field theory, and emergent vacuum structure. The Omega expansion offers a reproducible, exact system where physical constants emerge from number-theoretic recursion, not empirical fitting. This paper completes the theoretical framework introduced in ENSO (Emergent None-locality Structural Order) and CUT (Constant Unity Theory), establishing Ω = log(13) as the recursive seed of all physical structure. A full parameter table, predictive corrections, and testable resonance values are included.