The selected coordinate of the branch complement

theorem MagnitudeConjecture.BoundQuiver.StringPresentation.arrowBranchRadicalProjection_complement {k A Q : Type u} [Field k] [Ring A] [Algebra k A] [Fintype Q] [Quiver Q] [(x y : Q) → Fintype (x ⟶ y)] (P : StringPresentation k A Q) {x y : Q} (a : x ⟶ y) (z : ↥(Module.jacobson P.quotientCategoryAlgebraᵐᵒᵖ ↑(P.representedVertexModule y))) : (P.arrowBranchRadicalProjectionLinearMap ⟨x, a⟩) ((P.arrowRadicalComplementProjection a) z) = 0

Projection away from a branch has zero coordinate on that branch.