The Prime-Factored Model

ABSTRACT We present the Prime-Factored Model (PFM), a unified theoretical framework deriving fundamental constants, particle masses, elemental stability, and dark matter from first principles using only the speed of light (c) as primary input. The framework rests on prime synchronization of bare energy, mediated by the Fine Structure Constant (α ≈ 1/137. 036) reinterpreted as the Refractive Index between Prime-2 (Identity) and Prime-3 (Structural) manifolds. Core Results: The gravitational constant G is derived exactly as G = (2 × 13³ × 137) / (17 × 19² × (√6) ³) × 10⁻¹¹ m³ kg⁻¹ s⁻², landing within CODATA 2018 uncertainty (±0. 00015). All fundamental particle masses occupy discrete prime-snap-points on the √6 manifold, with Prime-17 serving as universal Mass Gate numerator. Elemental stability emerges as a consequence of prime synchronization with the 137 Synchrony Ratio: Iron-56's nuclear binding energy peak corresponds to 12 × 13 = 156 (Carbon-12 baseline × Skeletal prime), a zero phase-deficit state requiring no empirical fitting. Dark Matter Identification: The CMB, with thermal peak at 160. 4 GHz, is structurally determined by the Gate-19² dark matter field at 160. 247 GHz (derived from prime operators, not fitted). Dark matter is gravitationally coupled (via √6¹⁷ anchor) but electromagnetically and nuclearly invisible (absent Prime-3 and Prime-7). When accumulated in massive dark matter halos above a critical density threshold, the Gate-19² field decays to Lyman-alpha photons at 121. 6 nm, with frequency ratio Lyman-α / fDM ≈ 15, 397 (predicted: 15, 397; observed: 15, 389; <0. 1% deviation). This decay mechanism explains: (1) giant Lyman-alpha nebulae in massive halos (e. g. , RO-1001 at z=2. 91 with 1200 M☉/yr star formation) ; (2) starless clouds like Cloud-9 at sub-threshold density; (3) the absence of dark matter decay signatures in low-mass systems.