Coordinate-Wise Additivity and the ℓ¹ Norm on Finite Graph Cochains

This preprint proves a finite combinatorial classification theorem for obstruction diagnostics on graph cochains. For a finite directed graph G and cochain space C¹ (G, Rᵏ), any seminorm satisfying edge locality, coordinate permutation symmetry, and additivity over disjoint coordinate supports is coordinatewise ℓ¹ on each edge. Across edges, the general conclusion is orbit-weighted ℓ¹; a single global scalar requires edge-transitivity or an explicit edge-uniformity assumption. The result isolates the minimal coordinate-separable, replica-extensive observer assumptions under which ℓ¹ geometry is selected. It does not claim that ℓ¹ is universal. Relaxing coordinate additivity, edge locality, symmetry, or uniformity admits alternative geometries such as ℓ², ℓ∞, mixed norms, or weighted variants. This paper supplies the finite combinatorial base case for later obstruction-calculus papers on Banach presheaf coboundaries, M-compatible gauge classification, and projection-induced defects in factorized-state dynamics.