  In this article, we study the Ricci flow neckpinch in the context of metric
measure spaces. We introduce the notion of a Ricci flow metric measure
spacetime and of a weak (refined) super Ricci flow associated to convex cost
functions (cost functions which are increasing convex functions of the distance
function). Our definition of a weak super Ricci flow is based on the coupled
contraction property for suitably defined diffusions on maximal diffusion
components. In our main theorem, we show that if a non-degenerate spherical
neckpinch can be continued beyond the singular time by a smooth forward
evolution then the corresponding Ricci flow metric measure spacetime through
the singularity is a weak super Ricci flow for a (and therefore for all) convex
cost functions if and only if the single point pinching phenomenon holds at
singular times; i.e., if singularities form on a finite number of totally
geodesic hypersurfaces of the form $\{x\} \times \sphere^n$. We also show the
spacetime is a refined weak super Ricci flow if and only if the flow is a
smooth Ricci flow with possibly singular final time.
