An Empirical Relation Among Six Precisely Known Standard Model Masses

We report an empirical relation m_μ · m_p · m_n = (9/8) m_e · m_τ · m_Z among six precisely measured Standard Model masses—three charged leptons, two nucleons, and the Z boson—satisfied to 4.6 ppm. The relation contains no quark masses and no free parameters beyond the rational coefficient 9/8. A high-statistics Monte Carlo null test (10⁷ random spectra) shows that only 0.026% of random mass spectra contain any analogous relation at comparable precision, giving p = 2.6 × 10⁻⁴ (3.5σ). The result is robust under sector-preserving randomization (p = 3.7 × 10⁻⁴). The formula is unique: no other relation of this type exists among the nine most precisely known SM masses. It connects QCD confinement (m_p, m_n), lepton Yukawa couplings (m_e, m_μ, m_τ), and electroweak symmetry breaking (m_Z)—three sectors independent in the SM Lagrangian—and can be rewritten as Λ²_QCD ∝ (y_e y_τ / y_μ) v². The coefficient 9/8 = N²_c/(N²_c − 1) is the unique rational number that simultaneously equals β₀(n_f=3)/(β₀(n_f=6) + 1) for integer N_c; the identity selects N_c = 3 via 3N²_c − 11N_c + 6 = 0. We discuss additional relations involving quark masses, present concrete predictions for m_s, m_c, m_d, and m_b testable by lattice QCD, and show that the four-formula core system is internally consistent via a derived cross-check among precisely known masses alone. We describe decisive tests at Belle II, lattice QCD, and FCC-ee.