Finite Operator--Localization Weak Sources and Gauge-Covariant Quasilocal Charged-Current Kernels

At low energy, charged-current weak interactions are represented by a point-contact $V-A$ four-fermion term, but the contact distribution need not be identified with a fundamental normalized source state. We formulate a conservative Euclidean weak branch in which an admissible source has finite operator dimension and a finite state-localization budget. The latter gives a strictly positive second-moment width; on a stronger parent-capacity branch it is recovered from a support-energy ceiling rather than inserted independently. After pulling the conservative second variation to the active-constraint tangent space, we establish a closed-range Moore--Penrose Schur reduction. A Banach-space implicit-function criterion connects the quadratic formula to an actual local nonlinear stationary section, and explicit KLMN bounds yield a lower-semibounded Friedrichs realization. In the centered, current-free, single-scale, $O(4)$-invariant exact quadratic scalar sector, the unique positive normalized minimizer is Gaussian for every $a_W>0$, with width\[\ell_{W,\mathrm{osc}}=a_W^{-1/4}\mu_W^{-1}.\]This is class-universality within the exact quadratic sector; bounded nonquadratic perturbations admit quantitative near-Gaussian bounds. Distributional charged currents are heat-regularized on a dynamically selected charged line bundle and coupled through the flat Euclidean response\[\widetilde G_W^{\mathrm{ql}}(q_E)=\frac{\exp(-\ell_{W,\mathrm{sm}}^2q_E^2/4)}{q_E^2+\mu_W^2}.\]This kernel is an effective response between smeared current insertions, not a standalone physical $W$ propagator. Expanding the bilocal current action gives a kernel-transfer matrix for every fixed Wick contraction. We derive explicitly the one-loop self-energy and a second-order nested two-loop self-energy; in the latter the two independent kernel transfers produce the positive defect matrix $A=\id_8$. More generally, a fixed Euclidean graph is ultraviolet absolutely convergent whenever its action-derived transfer/defect form is coercive on the full cycle space. Hereditary coercivity is a stronger graph-class property ensuring the same conclusion for every ultraviolet-relevant subgraph, while an orderwise result additionally requires graph-family summability. The low-energy $V-A$ Fermi coefficient and the Gaussian-smeared Yukawa profile are recovered as consistency checks. We do not claim automatic finiteness of all electroweak gauge, Goldstone, or ghost graphs, convergence of the full perturbation series, reflection positivity, or a BRST-complete Lorentzian theory.