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We present a first example of a flag vector of a polyhedral sphere that is not the flag vector of any polytope. Namely, there is a unique   3  ‐sphere with the parameters(   f  0   ,   f  1   ,   f  2   ,   f  3   ;   f  02   )   =   (  12  ,  40  ,  40  ,  12  ;  120  )    , but this sphere is not realizable by a convex   4  ‐polytope. The   3  ‐sphere, which is   2  ‐simple and   2  ‐simplicial, was found by Werner [Linear constraints on face numbers of polytopes.  PhD Thesis , TU Berlin, Germany, 2009]; we present results of a computer enumeration which imply that the sphere with these parameters is unique. We prove that it is non‐polytopal in two ways: first, we show that it has no oriented matroid, and thus it is not realizable; this proof was found by computer, but can be verified by hand. The second proof is again a computer‐based oriented matroid proof and shows that for exactly one of the facets this sphere does not even have a diagram based on this facet. Using the non‐polytopality, we finally prove that the sphere is not even embeddable as a polytopal complex.

A FLAG VECTOR OF A 3‐SPHERE THAT IS NOT THE FLAG VECTOR OF A 4‐POLYTOPE