One simple algebra splits the whole category

Splitting the single object X ⊞ Xᘁ and passing to a simple quotient gives an algebra that splits the tensor generator and its dual, a direct summand being a subquotient. The split objects are closed under sums and tensor products, so they contain every mixed power, and over a simple algebra they are closed under subquotients as well (RS.exists_mix_of_isSubquotient), so finite tensor generation carries them to every object. The scalars of that algebra are the complex numbers, so its Γ-algebra has a complex point.

theorem RS.exists_splitting_simple_algebra {C : Type v} [CategoryTheory.SmallCategory C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.SymmetricCategory C] [CategoryTheory.Abelian C] [CategoryTheory.RigidCategory C] [CategoryTheory.MonoidalPreadditive C] [CategoryTheory.Limits.HasFiniteBiproducts C] (ψ : ℂ ≃+* CategoryTheory.End (CategoryTheory.MonoidalCategoryStruct.tensorUnit C)) (P : SchurPackage) (P₀ : SchurPackage) (L : OddLine (CategoryTheory.Ind C)) (X : C) (hgen : TensorGeneratedBy C X) (hgrow : ModerateLengthGrowth C) (hlen : ∀ (Z : C), ∃ (N : ℕ), LengthLE Z N) : ∃ (𝔹 : CategoryTheory.Ind C) (x : CategoryTheory.MonObj 𝔹) (x_1 : CategoryTheory.IsCommMonObj 𝔹), CategoryTheory.MonObj.one ≠ 0 ∧ (∀ (I : CategoryTheory.Subobject 𝔹), IsIdeal 𝔹 I → I = ⊥ ∨ I = ⊤) ∧ SplitsOn L 𝔹 indOf ∧ Nonempty (SuperPoint (gammaAlgebra (CategoryTheory.Ind C) L 𝔹))

One simple algebra splits everything, and its Γ-algebra has a complex point.