Stanley\u27s nonunimodal Gorenstein h-vector is optimal

We classify all possible h-vectors of graded artinian Gorenstein algebras in socle degree 4 and codimension ≤ 17, and in socle degree 5 and codimension ≤ 25. We obtain as a consequence that the least number of variables allowing the existence of a nonunimodal Gorenstein h-vector is 13 for socle degree 4, and 17 for socle degree 5. In particular, the smallest nonunimodal Gorenstein h-vector is (1, 13, 12, 13, 1), which was constructed by Stanley in his 1978 seminal paper on level algebras. This solves a longstanding open question in this area. All of our results are characteristic free