The flag f-vectors of Gorenstein* order complexes of dimension 3

We characterize the cd-indices of Gorenstein* posets of rank 5, equivalently the flag f-vectors of Gorenstein* order complexes of dimension 3. As a corollary, we characterize the f-vectors of Gorenstein* order complexes in dimensions 3 and 4. This characterization rise a speculated intimate connection between the f-vectors of flag homology spheres and the f-vectors of Gorenstein* order complexes.