Circularity of Thermodynamical Material Networks: Indicators, Examples and Algorithms

The paper’s definitions of the cycle-based circularity indicators match the model’s setup, and Propositions 1–3 are correct as stated (and trivial from the definitions) — see the definitions of λ_GA, λ_HA, λ_AA and the set Q, and the proposition statements for the equal-flow cases . However, Proposition 4 contains a dimensional/normalization error: it reports λ_GA(Γ) = m̄ n_φ / (n_φ + |Q|) (its Eq. (22)), which cannot be in [0,1] unless m̄=1 and contradicts the earlier property that λ_GA ∈ [0,1] by definition . The correct specialization, obtained directly from the general formula (Proposition 3, Eq. (20)) by substituting ∑_{γ∈Q} γ = |Q| m̄ and canceling m̄, is λ_GA(Γ) = n_φ/(n_φ + |Q|) . The model explicitly delivers this corrected expression and also notes the standard positivity requirement for the harmonic mean and the nondegeneracy of denominators, which the paper leaves implicit.