Factor-free cd-index formula for mapping cylinders

Derive a version of Proposition~\ref{prop:cdsubdivision} for the cd-index of the non-Hausdorff mapping cylinder of a strong formal subdivision that does not involve the factor of one-half.

Background

Proposition~\ref{prop:cdsubdivision} expresses the cd-index of an Eulerian mapping-cylinder poset in terms of local cd-indices and cd-indices of associated intervals and subposets, but the formula contains a factor of 1/2.

Although the left-hand side has integer coefficients, the averaging argument used in the proof introduces this factor. The paper asks whether an alternative formulation can avoid it, with the derivation formula for the pyramid cited as a special case.