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We examine version 2 of arXiv:2609.28999, which reconstructs conservative nonlinear ordinary differential equations from means of transported probability densities. The printed geometric-distortion definition retains the initial separation between neighboring states while its Taylor expansion omits that separation. For the stated directional transport maps, the resulting normalized quantity is at least one, so it cannot satisfy the proposed small-distortion criterion as written. We give a corrected displacement definition and a local derivative estimate, without asserting a universal replacement stability threshold. We also derive the exact reference-phase measurement fringe: its maximizer is a circular mean, and finite variance can reduce contrast without shifting that maximizer. An explicit pair of positive grid distributions has identical fringes but different arithmetic means. Further corrections distinguish Bernoulli-ancilla sampling from direct coordinate sampling, reconcile the Lotka–Volterra benchmark parameter with the source definitions, and exhibit a twice continuously differentiable observable for which a stated fourth-order Gaussian remainder fails. All numerical examples are deterministic classical computations with analytic checks and a recorded execution log. These results repair specific geometric, inference, and regularity statements; they do not refute Koopman embeddings, the genuine nonlinear mean-trajectory discrepancy, or all covariance-based error estimates.
Thank you for a carefully scoped and reproducible correction note. The panel found no error in your propositions, and the geometric obstruction (Proposition 3.1) and the four-node nonidentifiability example (Proposition 4.3) are clear and checkable. The decision is major revision. The reviewers judged parts of the claimed contribution, notably the circular-mean characterization in Section 4.1, to be standard directional statistics. They also questioned whether the corrections change any quantitative guidance in the source, given your own concessions in Sections 5 and 8. To proceed: - Reframe the novelty claims. - Demonstrate the practical consequences of each correction. - Make the Appendix A transcriptions independently verifiable. - Reconcile the source version date with the Matilda record. - Clarify the Table 1 covariance row against Appendix B. - Address the composed-flow and σ ~ h regime of the repaired criterion. Please also confirm the DOI of Ref. [1] and condense the elementary material in Section 4.4 and Appendix B.
Verdict: major revision. No reviewer found a mathematical error in the propositions. The submission does not yet show that this correction note clears the journal's significance bar, and several evidential points remain open. Decisive points. All three reviewers judge the contribution as local repairs to one preprint using standard Taylor, trigonometric and Gaussian-moment arguments. Reviewer 1 cites directional-statistics sources establishing the circular mean as the standard location statistic, so Proposition 4.1 and Corollary 4.2 are prior art, not contribution (ii). Reviewer 2's reject rests on Sections 5 and 8 conceding that the source's qualitative conclusions stand, which I find persuasive as a significance concern rather than a validity one. All reviewers mark the transcriptions of source Eqs. (30), (31), (35) and (45) as not_enough_evidence. Reviewer 3 did confirm the R definition (source Eq. 25), which supports Section 5. Reviewer 2 cites a Matilda record dating the last visible version 29 September 2026, conflicting with the Section 1 and Appendix A date of 25 September. Reviewer 1 notes that the Table 1 covariance row compares a value against a bound it exceeds, with an inadequate explanation. The DOI flag on Ref. [1] is a registry gap, per Reviewer 2, and is minor. What would change the decision: the following revisions would make the note acceptable as a rigorous comment. - Reframe the novelty claims honestly. - Make the source transcriptions independently checkable. - Show which corrections alter quantitative guidance the source offers. - Resolve the version-date and Table 1/Appendix B inconsistencies.
Verdict: advance to full review. The submission is a structurally complete correction study. It states its contributions (Sec. 1) and gives explicit proofs (Props. 3.1, 3.2, 4.1, 4.3, 6.1, 6.2). It reports deterministic numerical checks with analytic comparisons (Sec. 7, Table 1, App. C), a limitations discussion (Sec. 8), a conclusion (Sec. 9), and references to prior work (Sec. 2). Decisive points: the EVIDENCE items (https://arxiv.org/abs/2609.28999, https://arxiv.org/html/2609.28999) are the audited source paper itself. They confirm the target exists and that it derives an Ehrenfest-type estimate relating reconstruction error to covariance. No EVIDENCE item shows these specific corrections already published. The claimed errors are concrete and checkable against cited source equations (App. A): the dropped initial separation in source Eq. (30), the contrast-to-phase substitution after source Eq. (D10), the R/σ calibration in source Eq. (45), the C² remainder in source Eq. (35), and the missing transpose in source App. B.2. The reference list resolves except one DOI. Novelty and significance are open questions for reviewers rather than desk-reject grounds. The paper itself states its repair tools are standard Taylor and Gaussian-moment estimates (Secs. 3.1, 6), so its value rests on whether the identified source errors are real. Reviewers should verify the transcriptions in App. A against the source PDF.
The paper is a correction/comment note on arXiv:2609.28999v2 (Fredon et al., dual KvN embeddings), claiming that the source's geometric-distortion criterion retains an initial separation its Taylor expansion omits, that its phase-reference readout measures a circular mean rather than an arithmetic mean, and that its sampling prefactor, Lotka–Volterra calibration, and Gaussian-remainder regularity assumptions need repair. It supplies proofs, explicit counterexamples, and a deterministic reproducible code log.
The target paper and its Koopman-von-Neumann context are real prior work (arXiv:2609.28999; Joseph 2020; Zylberman et al.), and the circular-mean result in Proposition 4.1 is textbook directional statistics, already published (e.g., the von Mises MLE characterization in the Wikipedia/EVIDENCE sources). The genuinely new content is limited to the specific error audits of one arXiv preprint (the initial-separation obstruction, the four-node nonidentifiability example, the LV ratio correction, the C^2 counterexample), which are correct in kind but narrow and comment-level rather than a primary research contribution.
The note audits arXiv:2609.28999v2 and claims its printed geometric-distortion definition keeps the initial separation that its Taylor expansion drops, so the normalized directional-map quantity is at least one. It also identifies the reference-phase fringe with a circular mean, gives a same-fringe different-mean grid pair, and corrects sampling prefactors, a Lotka–Volterra ratio, and a C^2 Gaussian remainder.
The closest prior work is Fredon, Ram, Debbasch, Zylberman, and Loureiro, arXiv:2609.28999, whose abstract already gives an Ehrenfest-type covariance estimate for reconstructing nonlinear trajectories from transported densities. What is new is a set of local repairs to formulas transcribed from that preprint; the Taylor, characteristic-function, and Gaussian-moment arguments are standard, and Sections 5 and 8 say the source's qualitative reconstruction claim stands.
This paper identifies and corrects several errors in arXiv:2609.28999v2 (Fredon et al.), including an initial-separation omission in the geometric distortion criterion, a conflation of circular and arithmetic means in phase-fringe inference, a sampling-variance mismatch, a Lotka–Volterra calibration inconsistency, and an insufficient regularity assumption for a claimed remainder order.
The closest prior work is the source paper itself (arXiv:2609.28999, EVIDENCE [1]); the contribution is a set of corrections to that paper's specific statements, using standard Taylor expansion, trigonometric identities, and Gaussian moment calculations. The mathematical content is not novel as general theory, but the identification of the initial-separation obstruction and the constructive nonidentifiability example are specific technical observations not previously published.