Structural Origin of Electromagnetic Coupling and Mass Hierarchy

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 $$\bigl(P(1),P(q),P(q^2)\bigr)=(6,28,496),\qquad\bigl(R(q),S(q^2)\bigr)=\left(\frac{13}{6},\frac{31}{24}\right).$$ Version 1.1 derives the common precursor for the electron and proton corrections directly inside the electromagnetic/mass-hierarchy construction: $$\Psi_{Gp}=\frac{C_e-Y(0)}{X(q)}=\frac{67}{2}.$$ This quantity is fixed before the electron and proton starred corrections are evaluated, giving the forward structural ordering $$C_e\longrightarrow\Psi_{Gp}\longrightarrow\left(\Psi_e^{*},\Psi_p^{*}\right).$$ After the electron and neutron backbones have been generated, the same quantity is recovered through the distinct identity $$\Psi_{Gp}=\frac{1}{q}\left(\Psi_{n0}-\frac{Y(q)}{P(1)}\Psi_{e0}\right)=\frac{1}{2}\left(\Psi_{n0}-\frac{7}{6}\Psi_{e0}\right)=\frac{67}{2}.$$ The second route is an internal reconstruction check rather than the defining route. For the electron, proton, neutron, and muon sectors, the effective structural indices are represented as sparse weighted contractions with a common discrete matrix grammar. The 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 generates the electron mass-energy normalization through two exact structural quantities, $$\Psi_{me}=175882,\qquad\Psi_{me}^{*}=14484.$$ The Version 1.0 product-map representation of \(\Psi_{me}\) is retained, $$\Psi_{me}=qP(1)P(q)P(q^2)+q^{-1}P(q)\Psi_{p0}-P(q),$$ and Version 1.1 makes explicit the equivalent neutron-backbone representation $$\Psi_{me}=qX(q)\Psi_{n0}+qP(q)+q=175882.$$ The secondary quantity admits multiple exact representations, $$\Psi_{me}^{*}=\frac{\Psi_{me}-\Psi_{e0}/q_{\#}}{X(q)}$$ $$=\frac{Y(q)}{q_{\#}}\Psi_{e0}-P(q)-P(1)$$ $$=Y(q)\left(q_{\#}\Psi_{p0}+P(q)\right)+\frac{\Psi_{n0}}{K(1)X(q)}$$ $$=q\Psi_{n0}-P(1)P(q)=14484.$$ These exact equalities provide an internal regression network linking the electron, proton, and neutron backbones. The electron mass-energy is then $$E_e=\frac{c_{\rm km}^{2}}{\Psi_{me}\left[1+\left(\Psi_{me}^{*}\right)^{-2}\right]}\,{\rm eV},\qquadc_{\rm km}=299792.458,$$ giving $$E_e=0.510998950690048\ {\rm MeV}.$$ The adopted proton outer closure remains a discrete structural hypothesis. Historical and base alternatives are retained as falsification comparators, while no gravity-side observable is used to select the construction in this paper. Gravity-sector comparisons are reserved for an independent cross-check in separate work. 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.