 Disc diagram shell

A shell is an exposed two-cell in a disc diagram, with perimeter split as an exterior boundary arc $Q$ and an interior path $P$. An $i$-shell has $P$ consisting of $i$ interior pieces. Under $C^{\prime }(1/6)$, an $i$-shell with $i\le 3$ has $|P|<|\partial R|/2<|Q|$, so removal replaces a long exterior path by a strictly shorter path.