A Finite-Geometric Obstruction and a Polytopal Separation of Buchstaber Invariants

We construct a polytopal simplicial sphere that admits a mod 2 characteristic map but no mod 3 characteristic map, and hence no integral one. This shows that the Buchstaber number of a polytopal sphere can be strictly smaller than its real Buchstaber number, and gives a negative answer to the toric lifting problem.