The bounded-object conjecture for symmetric tensor categories

This paper proves the Coulembier–Etingof bounded-object classification conjecture for symmetric tensor categories over algebraically closed fields of arbitrary positive characteristic, without finiteness or projective assumptions. The central construction extends the Frobenius–Perron character of the subcategory generated by the tensor product of a bounded object with its dual to a positive dimension character on the category it generates. In odd characteristic, the proof combines the quadratic Adams operation with a refined cyclotomic argument. In characteristic two, it uses terminating Frobenius descent and dyadic lifting, and determines the Frobenius nilpotence height of every noninvertible bounded object. As a further consequence, categories generated by finitely many bounded objects have uniformly bounded growth dimensions of their simple objects.