Almost sure convergence of differentially positive systems on a globally orderable Riemannian manifold

Lin Niu, Yi Wang, Yufeng Zhang

correctmedium confidence

Category: math.DS
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The candidate solution reproduces the paper’s structure: (i) SDP implies a strongly order-preserving flow (paper’s Prop. 4.1), (ii) the key anti-chain/dichotomy for ω-limits (paper’s Lemma 2.1), (iii) countability of non-convergent points on simple ordered 1D slices (paper’s Prop. 3.1), and (iv) a Fubini slicing plus a countable chart cover to deduce that D = M\C has measure zero (paper’s Thm. 3.2). The only minor difference is that the model states the dichotomy assuming x ≪ y, while the paper proves it for x ≤ y; since ≪ implies ≤, this is strictly weaker and fully consistent. No substantive gaps or contradictions were found.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The paper establishes almost-sure convergence for differentially positive flows on globally orderable manifolds, a natural measure-theoretic refinement of prior generic results. The proof is concise and logically sound, centered on a clean anti-chain/dichotomy lemma and a Fubini slicing argument. Minor clarifications on where quasi-closedness and Γ-invariance intervene would improve readability. Overall, it is a solid, meaningful contribution.