A First-Principles Derivation of the Weak Gauge Coupling from an Eleven-Dimensional Covariant Master Action

We determine a current-normalized weak gauge coupling in the local weak branch of an 11D–4D information-projection theory and establish the normalization through two co-principal calculations. The first is an action–current determination. Seven independently calibrated source tangents are carried, by exact recovery, Riesz representation and one common stationary reduction, into seven unit currents in the irreducible three-dimensional weak adjoint space. Adjoint \\(SU(2)\\) covariance fixes the isotropy of their quadratic response and the calibrated trace fixes its magnitude, \\[\\widehat{K}_W^{\\mathrm{act}}=\\frac{7}{3}\\mathbb{I}_3,\\qquadg_{2,434}=\\sqrt{\\frac{3}{7}},\\qquad\\alpha_{2,434}=\\frac{3}{28\\pi}.\\] A direct derivative of the selected scalar action, with the action unit and physical current fixed in advance, gives the same local kinetic operator. The second principal result is a geometrically independent reconstruction. Rank-one weak directions define the primitive Hopf associated line over \\(SU(2)/U(1)\\simeq S^2\\), with \\[\\frac{1}{2\\pi}\\int_{S^2} f_H=1.\\] The selected seven projective source lines carry a positive occupation measure. A source-owned quadrupole functional removes their spin-two anisotropy and selects the latitude \\(t=1/3\\) before the source number is used; restoring the seven unit responses then gives the isotropic second moment \\(7\\mathbb{I}_3/3\\). A positive line-resolved hopping energy, varied over the full \\(SU(2)\\) link space, selects a unique transporter. Its full Wilson matrix has the half-solid-angle phase, and its continuum limit obeys \\(P\\mathcal{F}P=f_HP\\). Converting each projector to the unit physical current \\(X=P-\\mathbb{I}_2/2\\) yields \\[\\widehat{K}_W^{\\mathrm{geom}}=\\sum_{r=0}^{6}X_r\\otimes_J X_r=\\frac{7}{3}\\mathbb{I}_3.\\] The two routes are not additive contributions to the action. They determine and audit one operator in a common current convention: \\[\\widehat{K}_W^{\\mathrm{joint}}:=\\widehat{K}_W^{\\mathrm{act}}=\\widehat{K}_W^{\\mathrm{geom}}=\\frac{7}{3}\\mathbb{I}_3.\\] Topology fixes the primitive associated-line period, projective geometry fixes the distribution of seven unit responses over three weak directions, and the action together with the physical current fixes the absolute kinetic strength. Their agreement supplies a same-source, independent-route test of the seven-to-three normalization. The native \\(434\\) momentum-subtraction prescription specifies the reported local coefficient. Subsequent sections distinguish this result from finite scheme conversion, threshold evolution and pole observables, and apply the normalization to an electroweak kinetic–mass problem, a fixed-input charged-lepton calculation and finite event dynamics. Version 2.0 update. The central weak-coupling result and its reported values are unchanged from Version 1.0. This version identifies and version-locks the foundational source as The 11D–4D Covariant Master Equation for a Unified Field Theory, Version 5.0, distinguishes the structures inherited by the action–current and geometric routes from the weak-sector results proved in this article, and adds exact chapter-, equation-, and page-level provenance. These additions include an inherited-versus-present proof map and Appendix R's version-locked parent-theorem genealogy and equation crosswalk. The title has also been revised to reflect the explicit derivation from the eleven-dimensional covariant master action.