This paper studies the cohomological structure of defect fields on finite graphs under the ℓ¹ geometry established in preceding work. Classical discrete Hodge theory provides an ℓ²-orthogonal decomposition of 1-cochains into exact and harmonic components. However, this decomposition is not compatible with ℓ¹ geometry: the ℓ² projection onto harmonic representatives does not, in general, minimize ℓ¹ cost within a cohomology class. We therefore consider the ℓ¹ quotient functional Φ(δ) = min over f in C⁰ of ||δ − d₀f||₁, and show that it defines a norm on H¹(G) and serves as the canonical gauge-invariant measure of defect magnitude induced by the ℓ¹ geometry. In particular, among norms on cohomology arising from the ℓ¹ structure on cochains, Φ is uniquely determined by the quotient construction. We quantify the mismatch between ℓ¹ and ℓ² structures by proving that ℓ² Hodge projections generically inflate ℓ¹ cost, with an explicit ratio of 3 − 2/n on cycle graphs. We further characterize Φ via linear programming duality as the dual norm of bounded divergence-free circulations, connecting cohomology, network flows, and total variation minimization. The results provide a structural interpretation of ℓ¹ optimization on graphs as a cohomological invariant, completing a sequence of works showing that ℓ¹ geometry governs the measurement, aggregation, and decomposition of defect fields across combinatorial, functional, and dynamical settings.
JEREMY H. CARROLL (Sun,) studied this question.