  We study the gaps $\Delta p_n$ between consecutive rank sizes in
$r$-differential posets by introducing a projection operator whose matrix
entries can be expressed in terms of the number of certain paths in the Hasse
diagram. We strengthen Miller's result that $\Delta p_n \geq 1$, which resolved
a longstanding conjecture of Stanley, by showing that $\Delta p_n \geq 2r$. We
also obtain stronger bounds in the case that the poset has many substructures
called threads.
