Sharp Regularity Thresholds for Viscous Katz--Pavlović Dyadic Models

We study the unforced viscous Katz–Pavlović dyadic model with non-negative smooth initial data for two shell settings. For super-exponential scales $N_n=N_0^{b^n}$, with $N_0>1$ and $1<b<2$, we prove that $\alpha =1\mathrm{/}(b+2)$ is the sharp dissipation threshold: every such datum produces a globally smooth solution at and above this value, whereas suitable compactly supported data lose regularity in finite time below it. For geometric scales $N_n=N_0{\mathrm{\Lambda }}^n$, global regularity holds when $\alpha \ge 1\mathrm{/}3$ and $4\alpha -1>(\log 2)\mathrm{/}(2\log \mathrm{\Lambda })$. In particular, for $\mathrm{\Lambda }>2^{3\mathrm{/}2}$, the result reaches $\alpha =1\mathrm{/}3$ and, together with the known classical blow-up theorem, identifies the sharp threshold in classical shell ratios.