ℓ¹-Native Cohomology of Defect Fields - Quotient Norms, Dual Circulations, and Hodge Inflation

Corrected republication of a finite-graph specialization of the General Theory normed-cokernel construction. The paper studies real scalar 1-cochains on finite connected graphs without 2-cells, assuming a declared edge-additive ℓ¹ cochain norm. It defines the induced quotient norm Φ on C¹/im(d₀), proves the finite primal-dual certificate formula via bounded divergence-free circulations, and clarifies the relationship between ℓ² Hodge decomposition and ℓ¹ quotient minimization. The corrected version narrows the Hodge-inflation claim: the Cₙ example concerns inflation of ℓ² Hodge component bookkeeping cost, not a general failure of harmonic representatives to minimize the ℓ¹ quotient norm. It also demotes vector/operator-valued extensions to scoped future-work bridges and clarifies that coarea arguments provide intuition for graph gradients, not a direct cut formula for arbitrary quotient residuals.