This paper proves a structural classification theorem for seminorms on graph cochains. Let G = (V, E) be a finite directed graph and consider the cochain space C¹ (G, Rᵏ). We show that any seminorm satisfying four natural structural conditions—edge locality, coordinate symmetry, coordinate additivity across disjoint supports, and graph isomorphism invariance—must be a scalar multiple of the ℓ¹ norm. As a consequence, the defect field on graph cochains is isometrically equivalent to the space L¹ (E × 1, …, k) with respect to the counting measure. The resulting ℓ¹ ⊗ ℓ¹ geometry explains why discrete total variation, graph cuts, and related correlation diagnostics naturally adopt ℓ¹-type aggregation. This work forms the third paper in a trilogy studying projection obstructions on presheaf coboundaries. Earlier papers establish the appearance of ℓ¹ aggregation across edges; the present paper proves that the coordinate-level diagnostic must also assume ℓ¹ structure under independence and symmetry constraints. The result provides a discrete combinatorial analogue of classical characterizations of L¹ spaces and clarifies the structural origin of ℓ¹ geometry in graph-based defect diagnostics.
JEREMY H. CARROLL (Tue,) studied this question.