Dyadic shell models are infinite systems of ODEs that retain the quadratic nonlinearity, energy identity, and scaling of the 3D incompressible Navier-Stokes equations. Such models can be realized as averaged Navier-Stokes equations at the PDE level, and some of them blow up in finite time despite satisfying the energy identity, showing that these structural features do not preclude singularity formation. Among dyadic models with these features, the Obukhov model is distinguished by its regulated cascade mechanism: the inviscid version is globally regular. However, the inviscid proof relies on exact energy conservation, which viscosity destroys. We prove that the viscous Obukhov model is globally regular for all initial data in the critical Sobolev space and above. The proof introduces a critical rescaling that shows a viscous activation threshold for each shell, and establishes regularity by showing that two competing scales, viscous damping and nonlinear growth, become incompatible at high frequencies.