  The uniqueness of Leray-Hopf solutions to the incompressible Navier-Stokes
equations remains a significant open question in fluid mechanics. This paper
proposes a potential mechanism for non-uniqueness, illustrated in a natural
dyadic shell model. We show that, for the Obukhov model with $d>2$, there exist
initial data at the critical regularity that give rise to two distinct
Leray-Hopf solutions. These solutions exhibit an approximately discretely
self-similar structure, with non-uniqueness resulting from a partial breaking
of the scaling symmetry. The fundamental observation is that, in a certain
scenario, the dynamics reduce to a sequence of weakly coupled
finite-dimensional systems. Moreover, the predominant nonlinear interactions
are identical to those arising in convex integration, suggesting the
possibility of a similar construction in the full PDE setting.
