Steinberg extraction and maximal nilpotence in higher Verlinde categories

This paper proves maximal nilpotence of the higher Verlinde categories and their union in every odd prime characteristic. Together with the known characteristic-two case, this establishes Conjecture 3.2.3 of Coulembier’s *Commutative algebra in tensor categories*. The proof extends the Steinberg splitting method to polynomial representations with bounded coordinate weights. A semisimple coefficient construction transports finite symmetric and exterior algebras, while a Frobenius truncation completes the argument by induction through the Verlinde tower. The paper also determines exact symmetric and exterior orders for broad families of simple objects and gives explicit symmetric-power vanishing bounds for every nonunit simple. In characteristics at least five, it constructs simple objects with fixed orders that generate positive higher Verlinde categories of unbounded rank. These objects disprove the height conjecture and answer negatively the almost-Koszul question posed by Coulembier, Etingof and Newton. A separate bounded-weight argument establishes the fixed-order counterexamples independently of the general Steinberg extraction theorem. Finally, combining maximal nilpotence with existing geometric-reductivity and fibre-functor theorems yields the Hilbert basis property and finite generation of invariants for all finite symmetric tensor categories over algebraically closed fields of positive characteristic.