Structural Origin of Electromagnetic Coupling and Mass Hierarchy

Abstract

This work presents a discrete structural formulation of the low-energy electromagnetic coupling, the electron mass-energy normalization, and selected charged-particle mass ratios using fixed quantities inherited from the Minimal Axioms of Causal Inheritance. The upstream framework fixes $$(q,q_{\#})=(2,3).$$ and generates the structural hierarchy $$q\longrightarrow(X,Y)\longrightarrow P\longrightarrow(R,S).$$ with the selected values $$\bigl(P(1),P(q),P(q^2)\bigr)=(6,28,496),\qquad\bigl(R(q),S(q^2)\bigr)=\left(\frac{13}{6},\frac{31}{24}\right).$$ For the electron, proton, neutron, and muon sectors, the effective structural indices are written in the common quadratic form $$\Psi_i=\kappa_i\mathbf{u}_i^{\mathsf T}M_i\mathbf{u}_i,\qquadi\in\{e,p,n,\mu\}.$$ The four sectors share the same upper bilinear block while being distinguished by discrete lower-block signatures. At the observable level, the corrections reduce to the two common forms $$1+\Psi_i^{-1},\qquad1-\Psi_i^{-1}.$$ The principal relations are $$\alpha^{-1}(0)=4\pi B_{\alpha}\left(1+\frac{1}{\Psi_e}\right),$$ $$\frac{m_p}{m_e}=\alpha^{-1}(0)A_d\left(1-\frac{1}{\Psi_p}\right),\qquad\frac{m_n}{m_e}=\alpha^{-1}(0)A_d\left(1-\frac{1}{\Psi_n}\right),$$ $$\frac{m_{\mu}}{m_e}=K_{\mu}\,4\pi B_{\alpha}\left(1+\frac{1}{\Psi_{\mu}}\right),\qquad\frac{m_{\tau}}{m_{\mu}}=K_{\tau}A_{\tau}\left(1+\frac{1}{\Psi_{\tau}}\right).$$ The resulting values are $$\alpha^{-1}(0)=137.035999177055,$$ $$\frac{m_p}{m_e}=1836.152673425975,\qquad\frac{m_n}{m_e}=1838.683662002614,$$ $$\frac{m_{\mu}}{m_e}=206.768282701257,\qquad\frac{m_{\tau}}{m_{\mu}}=16.817031722054,$$ and $$\frac{m_{\tau}}{m_e}=3477.228769301753.$$ The same framework also generates the electron mass-energy normalization. Defining $$$$ $$E_e=\frac{c_{\rm km}^{2}}{\Psi_{me}\left[1+\left(\Psi_{me}^{*}\right)^{-2}\right]}\,{\rm eV},\qquadc_{\rm km}:=\frac{c}{1\,{\rm km\,s^{-1}}}=299792.458.$$ giving $$E_e=0.510998950690048\ {\rm MeV}.$$ The proton sector adopts the G1 proton--gravity shared closure through explicit comparison with the historical correction and the shorter \(347\)-base alternative. These alternatives are retained as falsification comparators, and a secondary SI-closure calculation is used only as an auxiliary consistency check. No continuously fitted observable-specific parameter is introduced. Using the quoted reference uncertainties and assigning no additional theory uncertainty, the largest normalized residual among the principal comparisons is approximately $$0.029\sigma.$$ The paper is restricted to electromagnetic coupling, the electron mass-energy normalization, and the charged-particle mass hierarchy. The Higgs, electroweak vacuum expectation value, quark, neutrino, and running-coupling sectors are reserved for separate work.