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Your revision is accepted. The panel found that every concern from the previous round was addressed. The circular-mean identity is now credited as standard (Sec. 4.1, Ref. [9]). Sec. 7 quantifies how each correction changes the source's guidance. Appendix A gives line-referenced TeX excerpts with a verifier. Appendix B pinpoints the failing covariance bound. Proposition 3.3 handles the complete-flow and σ∝h regime. A reviewer questioned the C² hypothesis in Proposition 3.3. We judged it sufficient, because the derivative of DF·F involves only D²F. Please attend to the editorial notes before publication, in particular the unsupported Fig. 4 remark in Section 5.
Verdict: accept as a rigorous correction note on arXiv:2609.28999v2. All six concerns from the round-1 major revision are addressed in the revised manuscript. Decisive points. Reviewers 2 and 3 confirm the novelty reframing in Sec. 1 and Sec. 4.1 with Ref. [9]. They also confirm the quantified consequences in Sec. 7, the precise failing and holding bounds in App. B Eqs. (42)–(43), and the complete-flow and σ=κh analysis in Prop. 3.3 and Eq. (16). For Appendix A, round 1 asked for TeX excerpts with line references, and the revision supplies them with archive hashes and a verifier. The reviewers' not_enough_evidence verdicts reflect their evidence window, not a failure to deliver that request. Reviewer 2's Matilda fact-check (creation date 2026/09/25) supports the 25 September date. Reviewer 3's new concern on Prop. 3.3 is overruled. DG[v]=D²F[v,F]+DF·DF·v involves only second derivatives, so K is well defined under F∈C², as stated in Eq. (11). Reviewer 1's persistent reject rests on the source's qualitative conclusions surviving (Secs. 7, 9). That objection is overruled. Sec. 7 shows the printed geometric criterion is unsatisfiable for every step, the ancilla shot prefactor differs by 2,500×, and the width ceiling shifts by 3.24×. These are changes to quantitative guidance, which is the standard a correction note must meet. Reviewer 1's new point on the source Fig. 4 mismatches (Sec. 5) concerns a peripheral, non-load-bearing remark. It is handled editorially.
Published with the paper; the authors were not required to address them before publication.
The note audits arXiv:2609.28999v2 and claims its printed geometric-distortion definition keeps an initial separation that its Taylor step drops, so the normalized directional-map quantity is at least one. It recasts the reference-phase fringe as a standard circular mean, gives a same-fringe different-mean pair realized by a conservative Liouville flow, 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 proposes a KvN algorithm that reconstructs nonlinear ODE trajectories from averages of transported densities. What is new is a set of local repairs to formulas the note transcribes from that preprint; Section 4.1 and Reference [9] now correctly treat the circular-mean identity as standard, and Sections 3.2 and 6 label the Taylor and Gaussian-moment repairs as standard.
Dear Editor,
Thank you for the major-revision decision and the specific requests. We have revised the existing submission, preserving its scope as a correction and reproducibility study. The revised manuscript does not claim to refute Koopman embeddings or the nonlinear cloud-to-trajectory discrepancy. The revisions are summarized below.
Established circular statistics and the source-specific contribution. Section 4.1 now explicitly calls Proposition 4.1 a standard circular-moment identity and Corollary 4.2 a standard symmetry consequence. Reference [9], Mardia and Jupp's Directional Statistics, supplies the established context. Publisher metadata and chapter contents were inspected; the full chapter was unavailable, and the identity is derived in the manuscript. The introduction and abstract now distinguish this standard identity from the source-specific correction to the contrast-to-phase inference. Proposition 4.3 retains the explicit positive four-node pair with the same entire fringe and different arithmetic means. We also added a smooth, compactly supported probability-density realization carried by an explicit conservative Liouville flow in Section 4.3. This realization is non-Gaussian; the text does not assert nonidentifiability within a known exact Gaussian family or when extra prior information is supplied. The restricted minimality statement concerns fixed, known, nonaliased support: at most three distinct support points determine their weights from normalization and one complex moment; four suffice for our counterexample. We make no unrestricted global-minimality claim.
Changes to practical numerical prescriptions. New Section 7 separates the quantitative consequences. For the same nonlinear shear with width 0.01 and step 0.001, the printed normalized geometric quantity is about 1.0000098691 while the increment statistic is about 0.0044399602. Thus the printed definition rejects this example regardless of step refinement under its directional-map interpretation; a replacement definition requires a separately justified threshold. For readout, a symmetric packet with variance about 0.0001 has reduced contrast but no shifted phase maximum, whereas the explicit same-fringe pair has an arithmetic-mean gap of about 0.0001966 that cannot be removed by increasing the number of samples of that fringe. For a root-mean-square precision target of 0.001, the coordinate and ancilla experiments require approximately 100 versus 250,000 independent samples under the stated variances: a factor of 2,500 in this example, with the same inverse-square exponent. These are RMS counts, not fixed-confidence guarantees. Finally, applying the same source heuristic R <= 0.1 yields nominal width ceilings of about 0.04367 from the printed 2.29 coefficient and 0.14142 from the displayed definitions, a factor of about 3.24. Other constraints may be stricter; both accept the small width used in the source's plotted example. We do not infer failure of that plotted orbit.
113 of 263 blocks of the round 2 manuscript are new or changed since round 1; 65 of 215 blocks of round 1 no longer appear. A block is a paragraph, heading, list, table or code block; a re-worded paragraph counts whole and a moved one does not count. This compares the two submitted texts and is not the response letter.
| Section | Changed | Removed |
|---|---|---|
| Geometric stability and observable inference in dual Koopman–von Neumann embeddings: Corrections and reproducible counterexamples | 1 | 1 |
| Abstract | 1 | 1 |
| 1. Introduction and scope | 1 | 1 |
| Proof | 24 | 14 |
| 3.2 A width-uniform complete-flow remainder | 2 | — |
| Proposition 3.3: Complete-flow increment bound | 9 | — |
| 4.1 The exact statistic measured by a reference scan | 2 | 2 |
| Proposition 4.1: Standard circular-moment identity for the reference fringe | 2 | — |
| Corollary 4.2: Standard symmetry consequence: positive variance without phase bias | 2 | — |
| 4.2 A fixed-frequency fringe need not identify the arithmetic mean | 1 | 1 |
| 4.2.1 Smooth realization in a conservative Liouville flow | 9 | — |
| 4.3 Direct mean encoding and its resource assumptions | 1 | 1 |
| 4.4 Two sampling experiments have different variances | 4 | 4 |
| 5. Calibration of the Lotka–Volterra example | 7 | 7 |
| 6. Regularity required by the Gaussian remainder | 1 | 1 |
| Proposition 6.2: A sufficient fourth-derivative bound | 1 | 1 |
| 7. Consequences for numerical and measurement choices | 2 | — |
| The geometric acceptance test. | 2 | — |
| The readout target and stopping precision. | 2 | — |
| Shot allocation. | 2 | — |
| Packet-width calibration. | 2 | — |
| Observable error budgets and covariance constants. | 2 | — |
| 8. Reproducible experiments | 3 | — |
| 9. Discussion and limitations | 1 | — |
| 10. Conclusion | 1 | — |
| Funding and competing interests. | 2 | — |
| Prior dissemination and parallel-submission disclosure | 1 | 1 |
| References | 2 | 1 |
| A. Targeted source passages and version scope | 2 | 1 |
| A.1 Geometric definition and criterion | 1 | 1 |
| A.2 Phase fringe and the claimed precision scale | 1 | 1 |
| A.3 Sampling and observable regularity | 1 | 1 |
| A.4 Ehrenfest–Reynolds calibration | 1 | 1 |
| A.5 Disclosed approximations and preserved conclusions | 1 | 1 |
| B. Covariance and width-normalization repairs | 12 | 9 |
| C. Pinned execution evidence | 4 | 4 |
| Proposition 4.1: Exact phase-reference fringe | — | 2 |
| Corollary 4.2: Positive variance without phase bias | — | 2 |
| 7. Reproducible experiments | — | 2 |
| 8. Discussion and limitations | — | 1 |
…and 2 more sections.
[under "Geometric stability and observable inference in dual Koopman–von Neumann embeddings: Corrections and reproducible counterexamples"]
28 October 2026 · Version 1.1.0
[under "Abstract"]
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. Applying the standard circular-mean description to the source’s phase-reference circuit shows that finite variance can reduce contrast without shifting the maximizing reference. An explicit pair of positive grid distributions has identical fringes but different arithmetic means, including in a conservative transported-density realization. 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.
[under "1. Introduction and scope"]
The present note concerns statements that can be checked independently of a large-scale quantum implementation. Its main contributions are: (i) an exact initial-separation obstruction to the geometric criterion printed in source Eqs. (30)–(31); (ii) an explicit nonidentifiability construction for the proposed reference-phase readout and a correction to its contrast-to-phase substitution; and (iii) reproducible corrections to sampling prefactors, a benchmark calibration, and regularity assumptions. Each correction is local to the statement identified. Standard ingredients such as Taylor’s theorem, characteristic functions, and quantum amplitude estimation are credited rather than presented as new general theory.
[under "Proof"]
Thus the approximately $0.1$ condition printed in source Eq. (31) cannot hold under the directional-map interpretation given in its geometric discussion. If the intended map were instead the complete flow, the time-zero contradiction in (3) would still remain: the normalized raw separation tends to $1$, rather than $0$, as $h\to0$. The following nonlinear example also gives an all-$h$ obstruction for the complete-flow interpretation, rather than only a time-zero objection.
[under "Proof"]
Then $\dot q=\partial_pH$, $\dot p=-\partial_qH$, and $\Phi_h(q,p)=(q+h\sin(2\pi p),p)$. Both own-coordinate derivatives vanish. Here the complete flow equals the sole nonidentity directional factor exactly; the other factor is the identity. Thus (5) applies to both interpretations for every admissible $h$. A packet with means $(1/2,1/2)$ and a $p$ marginal symmetric about $1/2$ has $\mathbb E\!\left[\sin(2\pi P)\right]=0$. Its mean therefore remains at the fixed-point trajectory $(1/2,1/2)$, exactly, on any interval without coordinate wrapping. A smooth compactly supported symmetric packet can ensure this condition. The printed geometric condition nevertheless fails. This example concerns the geometric condition; its covariance need not remain constant, and no constant-covariance error theorem is invoked.
[under "3.2 A width-uniform complete-flow remainder"]
### 3.2 A width-uniform complete-flow remainder
[under "3.2 A width-uniform complete-flow remainder"]
The remainder for a complete flow can be bounded uniformly as the packet width shrinks. A bound on two separate trajectory remainders would give a loose $O(h^2/\sigma)$ estimate after normalization. Keeping their difference inside the integral instead gives the following standard auxiliary ODE estimate. It concerns the exact complete flow, rather than a global error theorem for a Trotter approximation.
[under "Proposition 3.3: Complete-flow increment bound"]
#### Proposition 3.3: Complete-flow increment bound
[under "Proposition 3.3: Complete-flow increment bound"]
Let $U\subset\mathbb R^d$ be open and convex, and let $F\in C^2(U;\mathbb R^d)$. Suppose $B(x,\sigma)\subset U$, $\sigma>0$, and all trajectories starting in that ball remain in $U$ for $0\leq t\leq h_0$. Assume the uniform bounds
[under "Proposition 3.3: Complete-flow increment bound"]
$$
\sup_U\left\lVert F\right\rVert_2\leq M,\qquad
\sup_U\left\lVert DF\right\rVert_{\rm op}\leq J,\qquad
\sup_U\left\lVert D^2F\right\rVert_{\rm bil}\leq H_{\rm v},\qquad
K:=H_{\rm v}M+J^2,\qquad\text{(11)}
$$
[under "Proposition 3.3: Complete-flow increment bound"]
where $\left\lVert D^2F\right\rVert_{\rm bil}$ is the operator norm of the vector-valued bilinear map $(u,v)\mapsto D^2F[u,v]$. Put $A_h(z)=\Phi_h(z)-z$ and
[under "Proposition 3.3: Complete-flow increment bound"]
$$
R_h(x,\delta):=A_h(x+\delta)-A_h(x)
-h\bigl(F(x+\delta)-F(x)\bigr).\qquad\text{(12)}
$$
[under "Proposition 3.3: Complete-flow increment bound"]
Then, for $0\leq h\leq h_0$ and $\left\lVert \delta\right\rVert_2\leq\sigma$,
[under "Proposition 3.3: Complete-flow increment bound"]
$$
\left\lVert R_h(x,\delta)\right\rVert_2
\leq \frac{K h^2e^{Jh}}{2}\left\lVert \delta\right\rVert_2.\qquad\text{(13)}
$$
[under "Proposition 3.3: Complete-flow increment bound"]
Consequently, writing $H_i=\sup_{B(x,\sigma)}\left\lVert D^2F_i\right\rVert_{\rm op}$, the complete-flow increment statistic satisfies
…and 98 more blocks not shown.[under "Geometric stability and observable inference in dual Koopman–von Neumann embeddings: Corrections and reproducible counterexamples"] 28 October 2026 [under "Abstract"] [block of 1505 characters not shown; it begins "We examine version 2 of arXiv:2609.28999, which reconstructs conservative nonlin…"] [under "1. Introduction and scope"] The present note concerns statements that can be checked independently of a large-scale quantum implementation. Its main contributions are: (i) an exact initial-separation obstruction to the geometric criterion printed in source Eqs. (30)–(31); (ii) an exact characterization and nonidentifiability example for the proposed reference-phase readout; and (iii) reproducible corrections to sampling prefactors, a benchmark calibration, and regularity assumptions. Each correction is local to the statement identified. Standard ingredients such as Taylor’s theorem, characteristic functions, and quantum amplitude estimation are credited rather than presented as new general theory. [under "Proof"] Thus the approximately $0.1$ condition printed in source Eq. (31) cannot hold under the directional-map interpretation given in its geometric discussion. If the intended map were instead the complete flow, the time-zero contradiction in (3) would still remain: the normalized raw separation tends to $1$, rather than $0$, as $h\to0$. …and 61 more blocks not shown.
This correction note identifies and repairs five specific errors in arXiv:2609.28999v2 (a dual Koopman–von Neumann embedding paper), supplying proofs, explicit counterexamples, a new width-uniform complete-flow increment bound (Proposition 3.3), a smooth conservative realization of the nonidentifiability example, and a new section quantifying the practical consequences of each correction for packet-width selection, shot allocation, and calibration thresholds.
The closest prior work is the source paper itself (https://arxiv.org/abs/2609.28999), whose abstract confirms the Ehrenfest-type covariance estimate and stability criterion being corrected. The genuinely new content is the specific error identification, the four-node nonidentifiability construction now realized in a conservative Liouville flow (Section 4.2.1), and Proposition 3.3's width-uniform complete-flow bound; the mathematical tools remain standard Taylor, Grönwall, and Gaussian-moment arguments applied to one preprint.
The paper is a correction and reproducibility note on arXiv:2609.28999v2 (Fredon et al.), claiming that the source's geometric-distortion criterion retains an initial separation its Taylor expansion drops (normalized quantity at least one), that its phase-reference readout yields a circular mean and cannot identify an arbitrary arithmetic mean (explicit four-node same-fringe pair), and that its sampling prefactor, Lotka-Volterra calibration, covariance-transpose identity, and C² Gaussian-remainder assumption each need repair, all supported by proofs and deterministic checks.
The closest prior work is the audited source itself (EVIDENCE: https://arxiv.org/abs/2609.28999, whose abstract confirms the Ehrenfest-type covariance estimate and Ehrenfest-Reynolds stability criterion) together with the KvN background of Joseph (https://arxiv.org/abs/2003.09980) and the Trotter transport scheme of Zylberman et al. (https://arxiv.org/abs/2508.15691). The revision honestly downgrades the circular-mean identity of Prop. 4.1 to a standard directional-statistics fact (Sec. 4.1, Ref. [9]) and restates contribution (ii) as the nonidentifiability construction (Prop. 4.3 plus the new Sec. 4.2.1 Liouville realization), so the genuine novelty now rests on the source-specific obstruction and counterexamples, which are narrow but real and, per Sec. 7, quantitatively consequential.
Independently verifiable source passages. Appendix A and SOURCE.md now identify the exact versioned PDF and separately retrieved arXiv source archive by byte count and SHA-256. Six short, attributed files in source-excerpts/ preserve 165 selected original TeX lines with original filename and line numbers. The manifest and a standalone verifier fetch the versioned archive, verify its checksum, and compare the excerpt bytes to the original Limit.tex. They cover source Eqs. (30), (31), (35), (45), Appendix C's coordinate definition, Eqs. (D8)-(D10) and the following precision claim, plus the relevant definitions and explicitly disclosed closures. Direct official HTML anchors are supplied, with the warning that HTML appendix equation numbers differ from the PDF. Limit.tex line 1380 explicitly defines x_i as the value of the coordinate after the translation starting from the initial position, rather than as a displacement. Neither the original full PDF nor the full source archive is redistributed in our repository. The selective-excerpt verifier passes against the recorded archive and source hashes.
Version date, aggregation metadata, and author contact. The official versioned arXiv history and the supplied PDF stamp identify v2 as revised on 25 September 2026, at 21:21:38 UTC in the official history. A fresh versioned download is byte-identical to the supplied 17-page PDF. The official record checked on 28 October 2026 lists no later version or journal reference. The review reports a 29 September date on Matilda; our ordinary request to that page returned only a JavaScript application shell. We therefore cannot independently establish either that reported date or the meaning of its field. We retain the primary-record date and explicitly state this limitation. No original author has been contacted, and no author confirmation is claimed. Our conclusions concern the identified v2, not an unexamined later revision or version of record. The optional arXiv DOI was removed from Reference [1]; the versioned arXiv identifier and official links remain. We do not interpret a Crossref lookup miss as proof that a DataCite/arXiv DOI is invalid.
Precise covariance comparison. Appendix B now specifies J, C_0, the exact covariance solution and its Frobenius norm. At t = 0.1, the smaller appendix constant gives an exponential bound of 1.3403624549, below the exact 1.5068242214. The larger main-text definition gives 1.7785221712, which does bound this same example. Thus the text explicitly distinguishes the failing appendix statement from the main-text constant that repairs this example. The rotation demonstration was condensed to its diagnostic matrix entries. The example can be rescaled to bounded localized data for a short interval, and we do not use it to reject all main-text covariance estimates.
Directional versus complete flows, including width proportional to step. The nonlinear conservative shear is both a directional transport map and its own complete flow, so the raw-position obstruction is an exact complete-flow counterexample as well as a directional-map example. Separately, new Proposition 3.3 gives a width-uniform complete-flow increment estimate. Subtracting the two flow-integral equations before bounding yields a temporal remainder at most (K/2) h^2 exp(Jh) times the initial separation, with K = H_v M + J^2 under the stated derivative and field bounds. After normalization by width this remains controlled as width tends to zero. When width = kappa times h, both the spatial-curvature and temporal terms are of order h^2 and both are retained. The proposition is a local exact-flow statement; it is not a new global trajectory-accuracy or splitting-error theorem. A seventh deterministic check evaluates a nonlinear two-component Hamiltonian flow at three steps and three proportional-width constants, with 1,024 versus 2,048 internal RK4 steps. All checked inequalities pass. Refinement differences are reported as numerical checks, not certified continuum error bounds.
Section 4's sampling discussion and Appendix B were shortened, while the practical consequences and complete-flow analysis were expanded. All seven verification groups pass, with the executable script, results and execution log pinned to immutable code commit 1f006c4d0acae13b2ccee9e485cb31288c2cdc0d. The revised PDF and manuscript are version 1.1.0. The earlier GitHub and AiraXiv versions are disclosed preprints of this same work. No other venue has this manuscript under review while this Publish.fun review is active.
The funding disclosure now incorporates the author's confirmation: this study was self-funded by Taylor Kalin, including computing resources and AI tools; no external funding was received. No competing interests were reported for this study. Taylor Kalin remains the author, and AI assistance remains disclosed in the methods.
Taylor Kalin