ℓ¹ Native Cohomology of Finite Defect Fields: Quotient Norms, Dual Circulations, Sparse Representatives, and Integral Optima

Abstract

Title. L1-Native Cohomology of Finite Defect Fields: Quotient Norms, Dual Circulations, Sparse Representatives, and Integral Optima Abstract. For a finite directed graph with strictly positive edge weights, we study the weighted ℓ¹ norm induced on the first cochain quotient Q¹ = C¹ / im(d₀). The quotient infimum Φ₁([ω]) = inf_f ||ω + d₀ f||_{1,w} is attained, is a genuine norm, and has an exact dual description as the maximum circulation pairing max_z { zᵀ ω : Bᵀ z = 0, |z_e| ≤ w_e } subject to coordinate capacity bounds. A primal residual r = ω + B f and dual circulation z certify one another by complementary slackness z_e = w_e sgn(r_e). We also prove that every class has an optimal representative supported on at most the cycle rank β = m - n + 1 and that integral input admits an integral optimum under total unimodularity. These are finite convex and total-unimodularity results. They do not constitute an ℓ¹ Hodge theorem, a claim about continuum cohomology, or evidence of terminal stabilization. Why a researcher should care. Least squares spreads error across all edges. Weighted ℓ¹ quotient minimization instead localizes the sparse representative of an equivalence class while retaining an exact dual circulation certificate. This provides a machine-checkable bridge between graph cohomology, basis pursuit, network flow circulation theory, and certified finite witnesses. What this does not claim. This paper does not claim an ℓ¹ Hodge decomposition, continuum limit cohomology, universal physical norm selection, infinite-dimensional closed-range properties, uniqueness of non-sparse optima, or evidence of terminal stabilization. It grants no AuthorityEffect and creates no publication or physical authority. Reproduction. Unpack the deposit and run: python -m pytest code/tests/test_l1_cohomology.py --cov=l1_cohomology --cov-fail-under=95 -p no:debugging One-click notebook launchers. Enter notebook: jupyter_notebooks/000_enter_003_l1_native_cohomology.ipynb Falsification notebook: jupyter_notebooks/001_falsify_003_l1_native_cohomology.ipynb