What does this research mean for the field?

For real scalar 1-cochains on finite connected graphs, the induced ℓ¹ quotient norm can be characterized by a finite primal-dual certificate formula via bounded divergence-free circulations, and Hodge inflation in this context reflects ℓ² bookkeeping costs rather than a general failure of harmonic representatives to minimize the ℓ¹ norm. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

The aim is to explore the relationships between different cohomology norms and Hodge decompositions in the context of finite graphs.

May 16, 2026Open Access

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

Puntos clave

The aim is to explore the relationships between different cohomology norms and Hodge decompositions in the context of finite graphs.
Analyzed real scalar 1-cochains on finite connected graphs without 2-cells under an ℓ¹ cochain norm.
Defined the induced quotient norm on C¹/im(d₀) and established the finite primal-dual certificate formula.
Investigated the implications of ℓ² Hodge decomposition and quotient minimization, while refining claims about Hodge inflation.
Demonstrated the relationship between ℓ² Hodge decomposition and ℓ¹ quotient norms in the context of graph cochains.
Clarified that the Cₙ example specifically relates to the cost of ℓ² Hodge component inflation.
Provided bounds on the divergence-free circulations influencing the certificate formulas for these norms.

Resumen

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.

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