Closed-Form Expressions for Newton's Constant G and the Electron Mass mₑ in Natural Units: A Three-Factor Gravitational Decomposition with 12-Cyclotomic Generation Sharing

We report a parameter-free two-closed-form construction in natural units (ℏ = c = 1) for the gravitational fine-structure constant αG ≡ G mₑ². The core algebraic invariant is I ≡ 4π³ + π² + π, which defines a topological gravitational coupling Gₜop ≡ 4π/I² ≈ 4πα² (squared QED vertex strength), a hierarchy scale ΛT ≡ I⁹/ (4H) with H ≡ 4π + 1, and a 12-cyclotomic algebraic correction χ₁₂ ≡ 1 − 1/Φ₁₂ (π) with Φ₁₂ (x) = x⁴ − x² + 1. Newton's constant admits the three-factor closed form Gₜopo = Gₜop · ΛT⁻² · χ₁₂ = 64π H² χ₁₂ / I²⁰. The electron mass admits a parallel closed form mₑ, topo = (4 H^ (3/2) / (√π · I^ (5/2) ) ) · χ₁₂^ (1/3), whose main term is systematically identified as the optimal candidate within a low-complexity algebraic family, and whose χ₁₂^ (1/3) factor is characterised as a conditionally rigid correction. The product αGᵗh = 1024 H⁵ χ₁₂^ (5/3) / I²⁵ evaluates at 50-digit precision to 1. 7519 × 10⁻⁴⁵, a relative deviation of +4. 36 × 10⁻⁵ from the CODATA 2022 value αGᵉxp = 1. 75181 × 10⁻⁴⁵, consistent with twice the CODATA G uncertainty. A systematic numerical search over 14 cyclotomic siblings Φₙ (π) demonstrates that n = 12 is a unique local optimum of the precision landscape: replacing Φ₁₂ by any other low-degree cyclotomic polynomial degrades the fit by factors of 10² to 10⁶. A sensitivity analysis confirms that the integer coefficients (4, 1, 1) underlying I reside at a sharp, isolated minimum of the error landscape (unconditional rigidity), the cyclotomic index n = 12 is empirically optimal within the Φₙ (π) family, and the generation-sharing exponent ξ = 2/3 is conditionally selected to bring the product precision to 10⁻⁵ given the prior structural choices. We classify the construction as a conjectural algebraic constraint that any successful theory of induced quantum gravity must reproduce or exclude; explicit open problems on the algebraic origin of χ₁₂, the conditional status of the χ₁₂^ (1/3) exponent, and the empirical falsification roadmap are formulated.