The Prime-Factored Model

Abstract The gravitational constant G is measured with lower precision than any other fundamental constant. The muon and Higgs masses have no theoretical derivation in the Standard Model. The fine structure constant α ≈ 1/137 remains unexplained after a century of physics. This foundational paper derives all three — along with the hydrogen ground state, eight planetary orbits, and nine moon systems — from a single closed prime operator set {2, 3, 5, 7, 13, 17, 19, 41, 61, 137} with zero free parameters. The derivation proceeds from the speed of light c alone: the exponent 30/1729 = (2×3×5)/(7×13×19) extracts a base frequency f₀ = c^(30/1729) = 1.4030783 Hz. This same frequency is independently recovered from the Higgs mass and fine structure constant by a completely separate route, agreeing to six significant figures (0.006%). Two independent paths from opposite ends of the scale hierarchy, spacetime propagation and mass generation, converge on the same ground-state frequency. Every particle mass and physical constant in this paper derives from f₀ through the stability operator √6 = √(2×3). Verification: muon mass error 0.000016%, Higgs mass error 0.000001%, G error 0.00000016%. The same operator set simultaneously describes particle physics and orbital mechanics across 27 orders of magnitude without modification or additional parameters. This paper establishes the mathematical constraints, the structural why, not how, any mechanistic theory of fundamental physics must satisfy.