Seven-Source Information Geometry and Exact-Branch Rigidity of a Current-Normalized $SU(2)$ Gauge Coupling

A gauge algebra fixes transformation laws but not the absolute coefficientof a physical kinetic term. This paper gives a standalone weak-sectorcompression of the author's $11D$--$4D$ information-projectionmaster-action framework and proves a rigidity theorem for its strengthenedexact seven-source branch. The parent framework supplies the rank-sevenevent-channel provenance, the three-dimensional weak workspace and theseven-block action architecture; the theorem data and implications arerestated locally. On the exact branch, a calibrated one-count SLD metric is preserved byexact recovery, the current and action source maps are reciprocal, andthe reduced common quadratic generating functional realizes the samesource metric. A pre-registered full-functional action-unit mapidentifies the natural-log stationary envelope with the dimensionlessphysical-action coordinate, while physical-source completeness identifiesthe independently computed direct weak two-jet with its source pullback.Equivariance with the real irreducible adjoint action of $SU(2)$ then gives\[\widehat K_W^{\mathrm{dir}}=\widehat K_W^{\mathrm{pull}}=\A_{\mathrm{cur}}^{\dagg}\A_{\mathrm{cur}}=\frac73\id_3.\]Under these exact-branch certificates, a renormalized background-field$1$PI realization that is anomaly-free and BRST consistent defines\[g_{2,\CNMOM_{\xi_\star}}^{-2}(\mu_\star)=\frac73,\qquadg_{2,\CNMOM_{\xi_\star}}(\mu_\star)=\sqrt{\frac37}\]in the stated current-normalized momentum-subtraction scheme. Aclassical or bare realization instead fixes the corresponding nativestiffness and boundary datum. Explicit evaluation of a specifiedmicroscopic parent branch and external-scheme matching are not claimed,and no observed weak-sector quantity is used in the derivation.