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

Gauge symmetry fixes the form of a weak gauge interaction but does not, by itself, determine its absolute strength. We derive a current-normalized weak gauge coupling in the local weak branch of an eleven-dimensional covariant master-action framework and establish its normalization through two co-principal calculations. The underlying 11D–4D information-projection theory supplies the source and readout architecture, the admissible weak sector, and the common variational framework. The weak-sector calculations and geometric reconstruction are developed in this paper, with the inherited framework fixed to Version 5.0 of the parent work, The 11D–4D Covariant Master Equation for a Unified Field Theory. The action–current calculation begins by assigning definite statistical units to seven independent source variations through the symmetric-logarithmic-derivative quantum Fisher metric. Fixed exact-recovery operations preserve this calibration. The specified source probes and matter-generator metric separately determine seven unit weak-current responses. One common constrained stationary reduction then evaluates the gauge kinetic form, including the response of its shared internal variables. Adjoint \\(SU(2)\\) covariance makes the resulting operator isotropic, and its calibrated trace fixes its magnitude: \\[\\widehat{K}_W=\\frac{7}{3}\\mathbb{I}_3,\\qquadg_{2,434}=\\sqrt{\\frac{3}{7}},\\qquad\\alpha_{2,434}=\\frac{g_{2,434}^{2}}{4\\pi}=\\frac{3}{28\\pi}.\\] Here \\(\\widehat{K}_W\\) is the dimensionless current-normalized weak kinetic operator, \\(\\mathbb{I}_3\\) acts on the real three-dimensional weak-current space, and \\(434\\) labels the native momentum-subtraction prescription. Direct differentiation of the explicit local scalar action, with its action unit and current convention fixed, gives the same kinetic operator. The second calculation reconstructs this normalization geometrically. Projective weak directions define a primitive Hopf associated line with unit first-Chern pairing. A source-defined anisotropy functional selects the isotropic probability second moment before the number of source lines is restored. A positive line-resolved hopping energy, varied over the full \\(SU(2)\\) link space, selects the unique spatial transporter under the stated source conditions. Its full Wilson matrix has the half-solid-angle phase, and its continuum curvature realizes the associated-line flux. Expressing the seven line responses in the same physical-current unit reconstructs the operator \\(7\\mathbb{I}_3/3\\). The two routes are compared after their separate constructions and are not added as distinct action terms. Topology fixes the primitive line period, projective geometry fixes the distribution of unit responses, and the action together with the matter current fixes the absolute kinetic strength. No measured weak-coupling value is used to select the normalization. The reported result belongs to the native current-normalized \\(434\\) momentum-subtraction prescription. Finite scheme conversion, scale evolution, thresholds, and pole observables are treated separately, followed by electroweak, charged-lepton, and finite-event applications of the derived coefficient. Version 3.0 Update Version 3.0 is a major explanatory, mathematical, and reproducibility revision of this work. Its primary purpose is to make the derivation accessible to readers trained in conventional theoretical physics without requiring prior familiarity with information-projection cosmology or specialist quantum-information terminology. The information-projection concepts that provide the generative basis of the theory have not been removed or replaced by conventional field-theory language. Instead, they are now introduced together with their physical meaning, mathematical domain, normalization, and relation to standard gauge theory, effective-action methods, quantum measurement, and differential geometry. A new Reader’s Guide, expanded first-use definitions, and substantially revised explanatory text throughout the main chapters and Appendices A–R clarify the distinct roles of the statistical source metric, the physical matter-current metric, the stationary action response, and the independent geometric reconstruction. The revision also strengthens the mathematical and numerical presentation. It extends and sharpens the uniqueness analysis of the source-selected SU(2) transporter, including its boundary behavior; separates the rounding of the displayed Wilson matrix from the residual of the underlying numerical calculation; formulates the projector connection and curvature covariantly relative to a reference connection; and clarifies the equality and non-additivity of the action–current and geometric routes. The parent-theory provenance registry and equation-level crosswalks have also been restored and synchronized with the present manuscript. The Version 3.0 Supplement has been rebuilt as a standalone reproducibility package. It includes the verified scientific code and inputs inherited from Version 2.0, the new Version 3.0 transporter and covariant-curvature checks, tested dependency specifications, release-verification and reproduction scripts, equation-to-code and chapter-to-execution maps, and renewed numerical and finite-event validation. The central physical result is unchanged. The current-normalized weak kinetic operator remains 7/3 times the identity on the three-dimensional weak-current space, giving g_2,434 = √(3/7) and α_2,434 = 3/(28π). No measured weak-coupling value is used to select this normalization. Version 3.0 supersedes Version 2.0 as the recommended reference and reproducibility release. 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.