The Standard Model relies on approximately 19 free parameters—such as the fine-structure constant (α) and the Planck scale (M_Pl)—which are treated as empirical inputs. We propose that these constants are, in fact, derived geometric outputs of M-theory on a rigid G2 orbifold (S7/Z7). By treating the vacuum as a discrete geometric system, we derive the vacuum quantization error γ and calculate the fine-structure constant (α⁻¹ ≈ 137.036) as the renormalized volume of the 14-dimensional gauge manifold. We validate this hypothesis via a CUDA-accelerated Monte Carlo simulation on a 16⁴ lattice, which identifies a stable Infrared Fixed Point matching the physical coupling to within 0.05%. Furthermore, we show that mass hierarchies emerge naturally from the geometry: the Higgs mass (125 GeV) corresponds to the static projection from the 11-dimensional bulk, while the Planck scale (1.22 × 10¹⁹ GeV) arises from the tunneling probability through the 14-dimensional phase space, corrected by the vacuum impedance of the gauge algebra.