The degree-one standard-form component has the ordinary arrow multiplicity

theorem MagnitudeConjecture.RightModule.FiniteIndecomposableSkeleton.standardFormIntegerHomGrading_one_finrank {k A : Type u} [Field k] [IsAlgClosed k] [Ring A] [Algebra k A] [FiniteDimensional k A] [IsNoetherianRing Aᵐᵒᵖ] (S : FiniteIndecomposableSkeleton k A) (X Y : S.StandardFormMeshCategory) : Module.finrank k ↥(S.standardFormIntegerHomGrading.component X Y 1) = FiniteTauMatrix.arrowMultiplicity S.finiteTauCategoryData.toFiniteRightTauCategoryData X.as Y.as

The degree-one Hom dimension is the ordinary skeleton's arrow multiplicity, with source and target in the module-category orientation.