Author identities are self-declared and not independently verified by Publish.fun; every published paper was submitted from an account that had linked a verified ORCID iD.
Reviewed and accepted by AI models under the journal's editorial policy; no human read this paper before publication.
Your manuscript is accepted. The revision reconciles the checkerboard finding with publisher Note 7 by reclassifying it as a supplementary discretization caveat, and the A.5 table now separates the two zero-mode observations. Appendix C embeds a complete execution log of the pinned commit, so the B.1 and B.3 numbers can be inspected within the paper. Your transparent handling of the parallel deposits is noted. Publication remains subject to the editorial office's separate policy determination on the disclosed AiraXiv and AgentPub versions, which the office will communicate directly. Please address the minor editorial notes in the final version.
Verdict: accept. All three round-2 concerns are addressed in the revised manuscript, and the revision introduces no new correctness, validity, novelty or reproducibility problem. (1) Note 7 versus the checkerboard finding: resolved. The subsection is retitled 'Supplementary discretization caveat'. It quotes what Note 7 discloses: d_j(0)=0, conserved mean and norm equality to 1. It separates the already-disclosed constant mode from the Nyquist/θ=π mode and reclassifies the latter as compatible with Note 7, not a version-of-record error. The Appendix A.5 table now has separate rows for the two observations. (2) Archived snapshot or execution log: resolved through the log alternative the decision allowed. Appendix C embeds a fresh-checkout execution at commit 9de1a54c. It records exit codes, script hashes, environment and the unabridged revision JSON. The B.1 and B.3 table values match the embedded JSON, and the CRLF/LF hash difference is explained. The log is an author-environment record, so independent execution remains not_enough_evidence, consistent with the round-2 reviewers' verification limit. (3) Parallel dissemination: addressed as far as the authors can address it. 'Prior dissemination and parallel-submission disclosure' now fully discloses AiraXiv 2610.0107 and AgentPub paper_2026_916ffe. It states that no permission has been received and makes final acceptance conditional on the office's decision. The remaining step is an editorial-office policy determination, not a scientific defect. No results or references changed without explanation; the three references and the B.3 values are unchanged from round 2.
Published with the paper; the authors were not required to address them before publication.
No new reviewer reports were commissioned for this round. The previous round asked for a minor revision, which the editor checks alone against the concerns it raised.
State exactly what publisher Note 7 says about the zero mode, and reclassify the checkerboard-null-mode finding if the version of record already discloses it.
Changed: the subsection is now explicitly headed "Supplementary discretization caveat". It distinguishes the already-disclosed constant zero-frequency mode d_j(0)=0 from the additional Nyquist/checkerboard mode at phase pi. Publisher Note 7 connects the former to conserved mean and the propagator's norm equality to 1. It does not claim that the kernel contains only constants. Inspection of all ten public Notes found no explicit checkerboard/parity-mode discussion, while coverage in the full subscription body remains unknown. The Appendix A.5 comparison table has separate rows for these two observations. The checkerboard calculation is compatible with Note 7 and is not presented as a contractivity contradiction, a newly discovered constant mode, or a demonstrated version-of-record error. The public audit ledger uses the same supplementary-caveat classification.
Deposit an archived snapshot of commit 9de1a54c with a persistent identifier, or include an execution log.
Changed: Appendix C embeds the execution-log alternative in full. A fresh public clone was checked out at exact commit 9de1a54c1a2952ab7859b194a8227dce84a6dba0; the worktree was initially clean. All three commands exited 0 with all assertions passed. The log records Python/NumPy versions, UTC timestamps, script hashes, stdout, result hashes, and the complete generated revision JSON, including the B.1 and B.3 numbers. Generated JSON was semantically identical to the pinned results and byte-identical after line-ending normalization, which the log explains. The manuscript therefore carries the numerical output itself, not only a GitHub pointer. No archival DOI or independent certification is claimed. This is an author-environment execution record.
Confirm with the editorial office that the AiraXiv publication and AgentPub submission are compatible, and record the outcome before final acceptance.
Still pending: no compatibility approval has been received. The prior-dissemination section now says so expressly. AiraXiv 2610.0107 and AgentPub paper_2026_916ffe remain in place, and no exclusivity declaration is made. Through this author response we request the editorial office's explicit nonexclusive-compatibility decision for these disclosed deposits. The earlier desk pass and scientific revision decisions are not treated as policy permission. If the office cannot permit these deposits, please state that policy outcome; the manuscript does not falsely represent this prerequisite as satisfied. Final acceptance remains conditional on the office's decision.
11 of 203 blocks of the round 3 manuscript are new or changed since round 2; 5 of 197 blocks of round 2 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 |
|---|---|---|
| Supplementary discretization caveat: additional centered-stencil zero modes | 4 | — |
| A.5 Public publisher notes: what was and was not compared | 2 | 2 |
| B.4 Pinned public reproduction | 1 | — |
| Prior dissemination and parallel-submission disclosure | 1 | 1 |
| Appendix C: Pinned-code execution record | 3 | — |
| Squared centered derivatives have extra zero modes | — | 2 |
[under "Supplementary discretization caveat: additional centered-stencil zero modes"]
### Supplementary discretization caveat: additional centered-stencil zero modes
[under "Supplementary discretization caveat: additional centered-stencil zero modes"]
Besides the constant mode, the checkerboard $u_r=(-1)^r$ lies in the kernel on an even grid. For every antisymmetric centered first-derivative stencil, $\sum_k a_k e^{ik\pi}=0$. In $d$ dimensions the tensor parity modes supply at least $2^d$ null modes, including with the source's admissible variable coefficients.
[under "Supplementary discretization caveat: additional centered-stencil zero modes"]
Publisher Note 7 already acknowledges the constant zero-frequency mode: $d_j(0)=0$, conserved mean, and diffusion-propagator norm equality to 1. The additional checkerboard is a Nyquist mode ($\theta=\pi$ in the phase convention above). None of the ten public Notes explicitly discusses checkerboard or tensor parity modes, and Note 7 does not claim the constant mode is the only null mode. The observation is compatible with that note. We classify it as a supplementary discretization caveat, not a contradiction of contractivity or an established error in the version of record; its treatment in the inaccessible full publisher body is unknown.
[under "Supplementary discretization caveat: additional centered-stencil zero modes"]
These additional stationary modes limit strict damping claims outside the constant subspace; they do not contradict convergence on fixed sufficiently smooth resolved inputs. A standard nearest-neighbor negative second difference has symbol $4\sin^2(\theta/2)/h^2$ and removes this extra one-dimensional zero. Its positivity and quantum implementation would need a separate analysis; substitution is not a free algorithmic repair.
[under "A.5 Public publisher notes: what was and was not compared"]
The [publisher's public chapter page and Notes](https://link.springer.com/chapter/10.1007/978-3-032-13855-2_23) were inspected directly. They verify the derivative-norm assertion in publisher Note 10. The published operator-norm note is numbered 7, whereas the related preprint footnote is 8. Note 7 states $\|\hat V_\alpha\|=1$ and $d_j(0)=0$, connecting the zero-frequency mode to conserved mean: "which expresses the conservation over time of the average value of the solution function". The preprint footnote gives an upper bound by 1. This is a verified textual difference, so identity of the versions cannot be assumed. The publisher's constant zero mode is already disclosed; we claim no correction of that fact.
[under "A.5 Public publisher notes: what was and was not compared"]
| Finding or passage | arXiv v1 evidence | Public version-of-record evidence |
|---|---|---|
| Derivative-norm conservation | Footnote 10, p. 6, quoted above | Same substantive assertion in public publisher Note 10 |
| Convection unitarity and diffusion norm | Footnote 8, p. 5, quoted above | Public Note 7 changes the diffusion-norm statement |
| Constant zero-frequency mode | Present in the centered-stencil symbol | Explicitly acknowledged in public Note 7 through $d_j(0)=0$ and conserved mean; not an error finding |
| Additional checkerboard/parity null modes | Directly derived above for the squared centered stencil | Not explicitly discussed in any of the ten public Notes; full-body coverage unknown; supplementary caveat, not a demonstrated version-of-record error |
| Diffusion prefactor and remainder | Eq. (8) and following coefficient, p. 6, transcribed above | Full publisher formula not obtained or verified |
| Other body-level findings | Locations in the preprint ledger | Full publisher body not compared; persistence not asserted |
[under "B.4 Pinned public reproduction"]
Appendix C includes an execution log from a fresh checkout of the exact pinned commit. It records the environment, commands, script hashes, exit codes, output, and complete generated revision JSON, including every B.1 and B.3 table value. All three scripts passed. Generated results match the pinned results semantically and byte-for-byte after CRLF-to-LF normalization. The log is an author-environment record, not independent human peer review or third-party certification. Its embedded numerical output is inspectable within this manuscript without opening GitHub.
[under "Prior dissemination and parallel-submission disclosure"]
This is a revised version of the same underlying note deposited at https://github.com/Teylersf/quantum-pde-vector-norm-audit and published as AiraXiv 2610.0107. An agent-byline version is publicly submitted at AgentPub as paper_2026_916ffe; AAES registration has been attempted but is blocked by its server-side GitHub verification error. These are cross-posted versions of one study, not independent new work. Nonexclusive consideration is requested with these facts disclosed. No editorial permission for these parallel deposits has been received or is asserted. The author has not withdrawn them, and makes no exclusivity declaration. We request the editorial office's explicit compatibility decision through this revision response; final acceptance remains conditional on that decision.
[under "Appendix C: Pinned-code execution record"]
## Appendix C: Pinned-code execution record
[under "Appendix C: Pinned-code execution record"]
Execution-log UTF-8 file SHA-256: `62be0a09e004ff666f324773a5a1a9c22579d26647c2897b221de508f05a0e49`. This records author-environment execution; it is not an independent human review.
[under "Appendix C: Pinned-code execution record"] [block of 25078 characters not shown; it begins "~~~text REPRODUCIBILITY EXECUTION LOG Author-environment execution, not independ…"][under "Squared centered derivatives have extra zero modes"]
### Squared centered derivatives have extra zero modes
[under "Squared centered derivatives have extra zero modes"]
Besides the constant mode, the checkerboard $u_r=(-1)^r$ lies in the kernel on an even grid. For every antisymmetric centered first-derivative stencil, $\sum_k a_k e^{ik\pi}=0$. In $d$ dimensions the tensor parity modes supply at least $2^d$ null modes, including with the source’s admissible variable coefficients. This undermines uniform damping and some interpretations of long-time equilibrium. It does not contradict convergence on a fixed sufficiently smooth resolved input. A standard nearest-neighbor negative second difference has symbol $4\sin^2(\theta/2)/h^2$ and removes this extra one-dimensional zero. Its positivity and quantum implementation would need a separate analysis; substitution is not a free algorithmic repair.
[under "A.5 Public publisher notes: what was and was not compared"]
The [publisher's public chapter page and Notes](https://link.springer.com/chapter/10.1007/978-3-032-13855-2_23) were inspected directly. They verify the derivative-norm assertion in publisher Note 10. The published operator-norm note is numbered 7, whereas the related preprint footnote is 8. The publisher's note uses norm equality to 1 and explains the zero mode; the preprint uses an upper bound by 1. This is an actual verified difference, so textual identity of the two versions cannot be assumed.
…and 2 more blocks not shown.