M-Compatible Gauge Classification for Banach Presheaf Coboundaries: The ℓ¹ Maximal-Symmetry Endpoint

This preprint classifies edge-additive obstruction diagnostics on Banach-presheaf 1-cochains under declared local transformation monoids. Given a finite poset P and a Banach presheaf F, the 1-cochain space C¹(P,F) is treated as the finite ℓ¹ direct sum of edge stalks, equivalently the coproduct in the category of Banach spaces and contractive linear maps. The main result shows that any edge-additive, faithful diagnostic that is functorial under all bounded linear endomorphisms of each stalk must be an edge-weighted ℓ¹ sum of the native stalk norms. A single global scalar, or unweighted ℓ¹ form, follows only after adding edge-uniformity assumptions such as edge-transitive or isomorphic edge data, or an explicit normalization across edge types. The paper also develops an M-compatible gauge taxonomy. For a declared admissible transformation monoid M, compatible local gauges correspond to M-contractive symmetric convex bodies. Increasing the allowed transformation symmetry reduces the admissible gauge class. The maximal bounded-linear monoid forces proportionality to the native Banach norm, while weaker monoids admit larger families: rotational symmetry yields Euclidean local gauges, signed-permutation and diagonal-contraction symmetries admit unconditional norms, unitary conjugation yields spectral gauges, and CPTP monotonicity admits quantum distinguishability monotones without uniquely forcing the trace norm. The result is conditional and observer-relative. It does not claim that ℓ¹ is a universal geometry. Rather, it identifies ℓ¹ as the maximal-local-symmetry, edge-additive endpoint of a broader hierarchy of admissible obstruction geometries. The classification supplies the diagnostic-geometry layer used by later finite obstruction-calculus papers, including projection-obstruction and Hodge-style obstruction analyses.