Geometric stability and observable inference in dual Koopman–von Neumann embeddings: Corrections and reproducible counterexamples

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.

• Section 5: either document the claimed source Fig. 4 width-annotation and panel mismatches in the manuscript (quoting the captions and line references) or remove the sentence, since it currently rests only on an external audit ledger.
• Proposition 3.3: add one sentence noting that DG[v]=D²F[v,F]+DF·DF·v requires only F∈C², pre-empting the regularity question raised by Reviewer 3.
• Appendix A: state the Limit.tex line containing the printed value 2.29 in source Eq. (45), since this decimal is the calibration claim of Section 5.
• Appendix A: note that the Matilda entry's creation date (2026/09/25) is consistent with the arXiv v2 revision date, which resolves the aggregator discrepancy.
• Section 7: state how the 2,500× shot-count factor scales with packet width (as 1/(4σ²) for the symmetric case), so readers can transfer it beyond σ=0.01.