Representation-directedness of the actual finite interval algebra

theorem MagnitudeConjecture.RightModule.FiniteIndecomposableSkeleton.standardIntervalSkeletonDirectedFinite {k A : Type u} [Field k] [IsAlgClosed k] [Ring A] [Algebra k A] [FiniteDimensional k A] [IsNoetherianRing Aᵐᵒᵖ] (S : FiniteIndecomposableSkeleton k A) (m : ℕ) : FiniteDimensional k (S.standardFormIntervalAlgebra m)

theorem MagnitudeConjecture.RightModule.FiniteIndecomposableSkeleton.standardIntervalSkeletonDirectedNoetherian {k A : Type u} [Field k] [IsAlgClosed k] [Ring A] [Algebra k A] [FiniteDimensional k A] [IsNoetherianRing Aᵐᵒᵖ] (S : FiniteIndecomposableSkeleton k A) (m : ℕ) : IsNoetherianRing (S.standardFormIntervalAlgebra m)ᵐᵒᵖ

theorem MagnitudeConjecture.RightModule.FiniteIndecomposableSkeleton.standardFormIntervalSkeleton_acyclic {k A : Type u} [Field k] [IsAlgClosed k] [Ring A] [Algebra k A] [FiniteDimensional k A] [IsNoetherianRing Aᵐᵒᵖ] (S : FiniteIndecomposableSkeleton k A) (m : ℕ) : (S.standardFormIntervalSkeleton m).HasAcyclicNonzeroNonisomorphisms

The actual finite right-module skeleton of every standard-form interval is directed.