Eventually linear partially complete resolutions over a local ring with $m^4=0$

We classify the Hilbert polynomial of a local ring $(R,m)$ satisfying $m^4=0$ which admits an eventually linear doubly-infinite resolution $C$ which is 'partially' complete --- that is, for which the cohomology of $Hom_R(C,R)$ eventually vanishes. As a corollary to our main result, we show that an $m^4=0$ local ring can admit certain classes of asymmetric partially complete resolutions only if its Hilbert polynomial is symmetric. Moreover, we show that the Betti sequence associated to an eventually linear partially complete resolution over an $m^4=0$ local ring cannot be periodic of period two or three.