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

Abstract. For finite graph cochains with coordinate-valued edge defects, any edge-local seminorm that is coordinate-permutation symmetric and additive over disjoint coordinate supports is forced to be a nonnegative scalar multiple of coordinatewise ℓ¹ edgewise; positive weights require faithfulness/normhood. Across edges the sharp conclusion is orbit-weighted ℓ¹; a single global scalar requires edge-transitivity or an explicit edge-uniformity assumption. The additivity assumption is justified only for coordinate-separable, replica-extensive observers. Relaxing those assumptions yields alternative obstruction geometries (ℓ², ℓ∞, mixed norms, weighted variants). Why a researcher should care. Graph total variation, componentwise residuals, sparse defect fields, and coordinate-separable model-evaluation errors all decompose observations into channels. If those channels are treated as independent replicas whose disjoint supports must add exactly, then ℓ¹ is not an arbitrary aesthetic choice — it is forced inside that observer class. The paper also says what is not forced when the assumptions are removed. What this does not claim. No finite-structural universal ℓ¹, no Banach/presheaf rigidity (that is GS-001), no projection-obstruction dynamics (that is PO-000), no unweighted global scalar without edge-uniformity, no physical norm selection, no governance authority. Key closed forms (replayable). Finite CE1–CE5 countermodels; edgewise coordinate ℓ¹ under H1–H5; orbit-weighted aggregation under graph invariance, as packaged in the Python and Lean cores. Reproduction. python -m pytest code/tests -q -p no:debugging Version / DOI chain.Concept DOI: 10.5281/zenodo.18944809Preceding published version (v6.1.1): 10.5281/zenodo.20723753This deposit: v6.2