An Empirical Relation Among Six Precisely Known Standard Model Masses

We report an empirical relation m_μ · mₚ · mₙ = (9/8) mₑ · m_τ · mZ 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 (mₚ, mₙ), lepton Yukawa couplings (mₑ, m_μ, m_τ), and electroweak symmetry breaking (mZ) —three sectors independent in the SM Lagrangian—and can be rewritten as Λ²QCD ∝ (yₑ y_τ / y_μ) v². The coefficient 9/8 = N²c/ (N²c − 1) is the unique rational number that simultaneously equals β₀ (nf=3) / (β₀ (nf=6) + 1) for integer Nc; the identity selects Nc = 3 via 3N²c − 11Nc + 6 = 0. We discuss additional relations involving quark masses, present concrete predictions for mₛ, mc, md, and mb 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.