Sharp Tensor Nilpotence Bounds for Finite Crossed Products

Let a finite group G act on an additive braided rigid monoidal category by additive strong monoidal autoequivalences. Every tensor-nilpotent object of the crossed product has nilpotency index at most the order of the subgroup generated by its nonzero homogeneous degrees. The bound applies to arbitrary sums of homogeneous objects, without a Noetherianity or support-theory hypothesis. In characteristic p, explicit permutation modules over (C_p)^G give stable matrix units and realize every directed graph on G. A Hamilton path attains the bound |G| for every finite group; a four-group example has index four despite group exponent two. This yields a bound for the first projective tensor power in the corresponding finite tensor category. The result is a scoped contribution to AIM-OTHER-0067, not a solution of the broad AIM transfer problem. Known cyclic examples and categorical constructions are credited. This preprint is unrefereed and makes no certified absolute-priority claim.