Uniform arrow-count error for the actual finite interval algebras #
def
MagnitudeConjecture.RightModule.FiniteIndecomposableSkeleton.standardFormInterval_arrow_error_bound
{k A : Type u}
[Field k]
[IsAlgClosed k]
[Ring A]
[Algebra k A]
[FiniteDimensional k A]
[IsNoetherianRing Aᵐᵒᵖ]
(S : FiniteIndecomposableSkeleton k A)
(m : ℕ)
(hm : 2 * S.standardFormIntervalControlHeight ≤ m)
:
|ARCount.arrowCount
(FiniteTauMatrix.arrowMultiplicity
(ofFGFamily
(fun (a : S.standardFormSupportedLabel m) =>
(S.standardFormIntervalAlgebraEquivalence m).inverse.obj (S.standardFormSupportedFamily m a))
⋯ ⋯ ⋯).finiteTauCategoryData.toFiniteRightTauCategoryData) - (↑m + 1) * ARCount.arrowCount (FiniteTauMatrix.arrowMultiplicity S.finiteTauCategoryData.toFiniteRightTauCategoryData)| ≤ ↑S.standardFormBoundaryArrowBound + 2 * ↑S.standardFormIntervalControlHeight * ARCount.arrowCount (FiniteTauMatrix.arrowMultiplicity S.finiteTauCategoryData.toFiniteRightTauCategoryData)
The actual interval arrow count differs from (m+1) times the ordinary arrow count by a constant independent of m.