This study proposes that the mass hierarchy of elementary particles is governed by a geometric principle of two-dimensional quantization. We show that a single, theoretically-derived parameter γ organizes the masses of leptons and heavy bosons (μ, τ, W, Z, H) on a quadratic lattice m/me = n1² + γ n2² with high precision and no fine-tuning. The theoretical value of γ is derived from a vacuum energy minimization model, where a spontaneous symmetry breaking locks the vacuum's vibrational modes into a narrow angular wedge. This predicted γ incorporates both a geometric term derived from the wedge angle and a small, physically-motivated tension ratio (Ay/Ax). Statistical validation against a null hypothesis confirms the significance of the fit (p ≈ 0.012), and the uniqueness of the integer-pair assignments is ensured by a robust stability margin. The model's applicability is further explored in the hadron sector, where its predictions, while less precise, remain qualitatively successful, suggesting a universal geometric foundation for mass.