In viscous flow, the circulation enclosed by a deterministic material loop generally changes through vorticity diffusion. It is shown that the source-bearing unsteady Kiknadze–Krasnov vortex possesses a distinguished cylindrical material surface on which this viscous circulation transfer vanishes identically. For a single distributed mode, the scaled squared radius $x=β(t)r^2$ satisfies \[ \frac{\mathrm D x}{\mathrm D t} = 4νβ(t)\big[(s-1)-x\big], \] where $β(t)$ is the inverse squared radial scale, $ν$ is the kinematic viscosity, and $s$ is the incomplete-gamma shape parameter. For $s>1$, the cylinder $x=s-1$ is transported exactly by the radial flow and preserves the inside/outside ordering of fluid trajectories. The same factor $(s-1)-x$ governs the radial vorticity gradient and the material circulation-transfer rate. Consequently, the magnitude of the distributed axial vorticity $|ω_z|$ attains its annular maximum on this material cylinder, $\partial_rω_z=0$ there, and the enclosed circulation remains exactly $ΓP(s,s-1)$ for every admissible prescribed strain history. Neighbouring material cylinders continue to exchange circulation viscously, while the distinguished moving cylinder carries a fixed fraction of the distributed circulation throughout the admissible evolution. In the steady source–strain limit, the surface reduces to the radial stagnation cylinder. The complete velocity and pressure fields, exact radial-scale evolution, admissible parameter ranges, and qualifications for multiple radial scales are also derived. The resulting solution-specific material-circulation invariant provides an exact benchmark for numerical material-surface tracking, vortex-core evolution, and viscous circulation transport, with the principal analytical results additionally supported by machine-checked formal verification.