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 revision resolves the substantive concerns. The verbatim source expressions make Sections 2–5 checkable, the conditional derivation of Eq. (10) is acceptable as stated, and the pressure-convention and scope limits are now explicit. The remaining issue is reproducibility. The DOI for Ref. [5] again could not be resolved in the registry available to the reviewers, and your availability claims cannot be corroborated from the evidence. Please either provide an independently resolvable archive or mirror, or make every numerical control reported in Section 6 reproducible from the manuscript text alone and mark any archive-only counts as supplementary. Please also correct the Section 5 characterisation of Uzdensky and McKinney's result as an evolutionary condition rather than an instability criterion. With these changes the paper should be ready for acceptance.
Verdict: minor revision. Five of the six round-1 concerns are resolved, and the analytic core now stands within the manuscript. The remaining defect is reproducibility of the archive-dependent numerical counts, which a bounded revision can fix. Decisive points. All three reviewers mark the source-equation, Eq. (10), pressure-convention, erratum/page-count and non-affine-scope concerns as addressed. Reviewer 3 confirms that S2.23–S2.24 match the arXiv HTML verbatim. Reviewer 1 confirms that (S2.4) uses the absolute pressure P. The Section 2 counterexample (Eq. 5) and the Section 6 inverse-square check of (SB14) can now be verified from displayed expressions. Reviewer 3's new concern, that the Eq. (10) steps are not independently validated, identifies no error; the steps are displayed and explicitly conditional, so it is not blocking. Ruling on dissent. Reviewers 1 and 2 hold major revision solely because DOI 10.5281/zenodo.23780790 (Ref. [5]) returns not_found again. The authors' DataCite-findable and HTTP 200 assertions are not_enough_evidence in every fact-check. I uphold this objection only for claims that cannot be reproduced from the text: the 4,096 seeded outflow states, the 150-digit Decimal checks and the other algebraic controls. I overrule it as a bar to the analytic findings, and the 80-case primary pressure grid is now fully specified in Section 6. What would change the decision: an independently resolvable archive record, or self-contained specifications of those remaining controls, would lead to acceptance.
The paper argues that Chowdhry and Loureiro's radiatively cooled X-point-collapse model contains an internally inconsistent printed pressure solution: the displayed formula (their Eqs. 2.24–2.27) does not integrate the stated energy equation (2.23), and the corrected Bernoulli solution destroys the exact affine closure, imposes a negative-total-pressure requirement for outflow deceleration in the printed ODEs, and contradicts a strict cooling-dominated endpoint balance. It retains the target's steady Sweet-Parker algebra with explicit domain limits and reports extensive archived scalar numerical verification without reconstructing the target's trajectories.
The closest prior work is the audited target itself (Chowdhry & Loureiro, JPP 2025) and Uzdensky & McKinney's radiatively cooled Sweet-Parker theory, which already separated stationary existence from evolutionary accessibility and stability (Section 5). No published critique of the target's pressure solution appears in the EVIDENCE, so the central energy-equation counterexample is genuinely new, but the contribution is a bounded audit of a single paper, comment-level in scope rather than a research advance.
Taylor Kalin; manuscript version 1.1.0;29 October 2026.
The title and abstract remain unchanged. The primary pressure-EOS finding is unchanged. Software 1.0.0 and its immutable 36-file archive are unchanged. The manuscript now includes source equations, reproduction details and a conditional local coupled-ODE proof.
Section 6 and Ref. [5]: provide a resolving persistent identifier for the verifier archive, and state solver tolerances, the full parameter/history manifest, error normalization and software environment in the manuscript itself.
The software DOI is valid and remains unchanged. We repeated anonymous checks after receiving the review: the DOI resolver, Zenodo record and API, archive download and primary DataCite registry returned HTTP 200. DataCite reports findable; doi.org/ra reports DataCite. All 36 archive files match the frozen computational snapshot, with ZIP 561,065 bytes and SHA-256 25a1e604ab4210f3fea86c56283d47ac0efdefd2a73cda4ceeb92c3939c812d1. Section 6 now gives direct API/download links and the full primary pressure grid,318-integration breakdown, DOP853 tolerances/maxstep, positive-reference error normalization, actual environments and distinction between requested tolerance and observed agreement. A Crossref-only not_found result does not test a DataCite DOI. The receipt is submissions/zenodo-code-v1.0.0.json. We do not claim the archive validates physical trajectories.
Sections 2-5: quote verbatim the source's Eqs. (2.23)-(2.27), (A4)-(A6), (2.31), (2.33)-(2.36), (3.13) and (B13)-(B14), so each alleged discrepancy can be checked against the version of record without the archive.
The final subsection of Section 6 now reproduces all requested source mathematical expressions, including 2.23–2.27,A4–A6,2.31,2.33–2.36,3.13,B13–B14, plus 2.4/2.22 supporting total-pressure/coefficient definitions. Source-prefixed equation labels distinguish them from our derivations. Symbols, signs and mathematical operations are retained, while spacing, display environments and line breaks are normalized. Exact publisher HTML TeX extracts with original labels/source hashes are in audit/source-equation-extracts.json. Source Ct in A4 is retained, not silently corrected. The following paragraph derives the inverse-square first integral/quadrature, locating the B14 missing factor without asserting what hidden numerical code used.
Section 4, Eq. (10): derive or bound the corrections from η−η_c and the time-dependent thermal coefficient, or restrict the claim explicitly to the frozen scalar equation.
59 of 126 blocks of the round 2 manuscript are new or changed since round 1; 8 of 75 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 |
|---|---|---|
| Thermodynamic consistency and closure limits of radiatively cooled X-point collapse | 1 | 1 |
| 1. Target and scope | 1 | 1 |
| 3. Spatial closure and the pressure sign | 1 | 1 |
| 4. Cumulative cooling and endpoint balances | 11 | 3 |
| 6. Verification, limitations and conclusion | 6 | 2 |
| Source expressions for direct comparison | 39 | — |
[under "Thermodynamic consistency and closure limits of radiatively cooled X-point collapse"]
Version 1.1.0; 29 October 2026
[under "1. Target and scope"]
We examine the 26-page journal version of record, *Journal of Plasma Physics* **91**(6), E148 (2025), DOI 10.1017/S0022377825100949 [1], and the supplied 24-page arXiv:2508.13081v1 [2]. The supplied preprint is byte-identical to the official versioned PDF. Both editions have been read completely. The final article corrects the preprint's missing equality in Eq. (2.14), updates the flow-rate symbols in Eq. (3.1), and adds cooling-law, Newtonian-MHD and relativistic-applicability qualifications. Those improvements are credited. The pressure construction, principal collapse balances and steady-layer expressions addressed here remain in the final edition. Locators below refer to [1] unless stated otherwise. The 24-page count is the actual page-tree count of the versioned preprint (966,878 bytes, SHA-256 `49dd1f97967345a7c22825158758ab1ed49529abd27c717ee2bdbc6ef7c40c0c`); the journal PDF has 26 pages (786,255 bytes, SHA-256 `1e47de8f1bbdb554c26efe029b13127cdbf4464fe6599721b64129f86b22150e`). Byte identity is between the supplied preprint and official arXiv v1, not between preprint and journal. A secondary catalogue page count does not override the retrieved PDF page tree. On 29 October 2026, the official article, primary Crossref relation/update fields and targeted title/DOI correction searches revealed no identified target-specific erratum, corrigendum or response. An adjacent issue-list erratum was unrelated. This bounded negative search is not proof that none exists; no response from the source authors was obtained.
[under "3. Spatial closure and the pressure sign"]
This restriction matters for the interpretation in Section 3.3, which invokes stronger nonuniform compression and a pressure maximum turning into a minimum. The stated affine Jacobian and uniform initial density instead give spatially uniform density at each time. Neither that proposed density mechanism nor a positive-pressure central minimum follows from the equations supplied. This does not prove that a nonuniform-density plasma, a non-affine flow or a more complete radiative-MHD model cannot contract along its outflow direction. Section 3 does not solve the repaired, generally non-affine momentum system. Its deceleration implication concerns only the printed ODE, and the scalar examples are not momentum solutions. The source uses the same pressure $P$ in its full energy equation (2.4), absolute radiated-power law and initial profile (2.25), reproduced below. No subtracted equilibrium pressure is introduced in this reduction. Thus the central value $P_{0,X}$ is a thermodynamic-pressure value; a reinterpretation as a signed perturbation would require an additional background term in the nonlinear cooling law.
[under "4. Cumulative cooling and endpoint balances"]
A conditional local check can also retain the coupling to $\eta$. Consider a $C^2$ positive branch of the *printed* Eqs. (2.29) and (2.31), with fixed finite coefficients, no earlier singularity, $\xi\sim A\tau^{2/3}$ and $\eta\to\eta_c>0$. This assumes a branch if it exists; it does not establish one for the source initial data. Define $H(t)=P_{0,X}\eta(t)^{-2/3}(\widetilde C I(t)^2-1)$ and $H_c=\lim H(t)$. The exact printed equations become
[under "4. Cumulative cooling and endpoint balances"]
$$
\begin{aligned}
\ddot\xi&=-\eta\xi^{-2}+\eta^{-1}-H(t)\xi^{-5/3},\\
\ddot\eta&=\xi^{-1}-\xi\eta^{-2}-H(t)\eta^{-1}\xi^{-2/3}.
\end{aligned}
$$
[under "4. Cumulative cooling and endpoint balances"]
The second equation gives $\ddot\eta=O(\tau^{-2/3})$, hence $\dot\eta\to v_c$ and $\eta=\eta_c-v_c\tau+O(\tau^{4/3})$. The cumulative-integral tail in Eq. (8) then gives $H-H_c=O(\tau^{5/9})$. Thus the first equation is $\ddot\xi=-\eta_c\xi^{-2}-H_c\xi^{-5/3}+\mathcal R(t)$, where
[under "4. Cumulative cooling and endpoint balances"]
$$
\mathcal R=-(\eta-\eta_c)\xi^{-2}-(H-H_c)\xi^{-5/3}+\eta^{-1}
=O(\tau^{-5/9}).
$$
[under "4. Cumulative cooling and endpoint balances"]
This remainder is later than the nonzero limiting thermal force of order $\tau^{-10/9}$. More than formal differentiation of a leading equivalence is needed to obtain the correction. The exact scalar-energy identity
[under "4. Cumulative cooling and endpoint balances"]
$$
\mathcal E=\frac{\dot\xi^2}{2}-\frac{\eta_c}{\xi}-\frac32H_c\xi^{-2/3},
\qquad\dot{\mathcal E}=\dot\xi\mathcal R=O(\tau^{-8/9})
$$
[under "4. Cumulative cooling and endpoint balances"]
has an integrable right-hand side, because the first equation also implies $\dot\xi=O(\tau^{-1/3})$. Therefore $\mathcal E\to\mathcal E_c$ and the decreasing endpoint branch has $\dot\xi^2=2\eta_c/\xi+3H_c\xi^{-2/3}+2\mathcal E_c+O(\xi^{1/6})$. Integrating its inverse speed gives
[under "4. Cumulative cooling and endpoint balances"]
$$
\tau=\frac{2\xi^{3/2}}{3\sqrt{2\eta_c}}
\left[1-\frac{27H_c}{44\eta_c}\xi^{1/3}+O(\xi^{2/3})\right].
$$
[under "4. Cumulative cooling and endpoint balances"]
Inversion yields
[under "4. Cumulative cooling and endpoint balances"]
$$
\begin{aligned}
\xi&=A\tau^{2/3}+D_1\tau^{8/9}+O(\tau^{10/9}),\\
A^3&=9\eta_c/2,\qquad D_1=\frac{81}{44}H_cA^{-5/3}.
\end{aligned}\tag{10}
$$
[under "4. Cumulative cooling and endpoint balances"]
For $H_c\ne0$ this is the first correction on the stated branch. When $H_c=0$, a nonzero limiting scalar-energy constant instead gives a $\tau^{4/3}$ correction; further cancellations require further analysis. The bounds are not asserted uniformly in varying parameters. These local results concern the printed ODE, not repaired thermodynamics, physical collapse or a reconstruction of original curves. The archived scalar quadrature controls test frozen coefficients; they do not numerically evolve or validate this coupled local branch.
…and 45 more blocks not shown.[under "Thermodynamic consistency and closure limits of radiatively cooled X-point collapse"] Version 1.0.0; 29 October 2026 [under "1. Target and scope"] We examine the 26-page journal version of record, *Journal of Plasma Physics* **91**(6), E148 (2025), DOI 10.1017/S0022377825100949 [1], and the supplied 24-page arXiv:2508.13081v1 [2]. The supplied preprint is byte-identical to the official versioned PDF. Both editions have been read completely. The final article corrects the preprint's missing equality in Eq. (2.14), updates the flow-rate symbols in Eq. (3.1), and adds cooling-law, Newtonian-MHD and relativistic-applicability qualifications. Those improvements are credited. The pressure construction, principal collapse balances and steady-layer expressions addressed here remain in the final edition. Locators below refer to [1] unless stated otherwise. [under "3. Spatial closure and the pressure sign"] This restriction matters for the interpretation in Section 3.3, which invokes stronger nonuniform compression and a pressure maximum turning into a minimum. The stated affine Jacobian and uniform initial density instead give spatially uniform density at each time. Neither that proposed density mechanism nor a positive-pressure central minimum follows from the equations supplied. This does not prove that a nonuniform-density plasma, a non-affine flow or a more complete radiative-MHD model cannot contract along its outflow direction. …and 5 more blocks not shown.
The revision argues that Chowdhry and Loureiro's printed pressure formula does not solve their energy equation, so the exact affine closure and the printed positive-pressure outflow-reversal and cooling-dominated endpoint balances fail, while the modified steady Sweet-Parker algebra is retained with domain limits. It now quotes source equations, derives a conditional remainder for Eq. (10), and puts the scalar-check manifest in Section 6.
The closest prior work is Chowdhry and Loureiro, Journal of Plasma Physics 91, E148 (2025) (https://doi.org/10.1017/s0022377825100949), whose abstract already states accelerated inflow collapse and possible arrest or reversal of outflow elongation. The Bernoulli integration, the loss of the exact parabolic-pressure closure, and the cumulative-integral objection to the printed endpoint balance are not in that abstract or the available equation excerpts, and no erratum containing them appears in the EVIDENCE.
The paper corrects the pressure evolution in a published radiatively cooled X-point model and derives consequences for affine closure, pressure positivity and endpoint balances. It preserves the separate steady-layer algebra while restricting its domain and distinguishing analytical consistency checks from reproduction of the original dynamics.
The closest work is the audited X-point model (EVIDENCE [1]), while the distinction between stationary existence and evolutionary accessibility already appears in the earlier reconnection theory (EVIDENCE [12]). The contribution in Sections 2–4 is a specific internal-consistency correction and its consequences, not a new cooling-integration method or reconnection model; the supplied evidence does not establish prior publication of these corrections.
We agree that the previous frozen-coefficient explanation alone did not justify the coupled expansion. Section 4 now states the branch assumptions (C^2 positive solution of the printed equations, fixed finite coefficients, xi~A tau^(2/3), eta->eta_c>0, no earlier singularity) and derives eta−eta_c=O(tau), H−Hc=O(tau^(5/9)). The exact force remainder is O(tau^(−5/9)); the renormalized scalar-energy derivative is O(tau^(−8/9)), integrable. Inverse-speed integration then yields Eq. (10) with O(tau^(10/9)) remainder. The source initial conditions are not proved to reach that branch. Hc=0 is treated separately. We explicitly distinguish this conditional analytic derivation from the archived frozen-scalar numerical controls and from repaired non-affine MHD.
Section 3, Eq. (7): demonstrate from the source text that P_{0,X} denotes total rather than perturbation pressure, since the negative-pressure conclusion depends on it.
Section 3 now explains the pressure interpretation from the source's same P in full Eq. (2.4), radiated-power law and initial profile (2.25); these source equations are displayed. No subtracted equilibrium pressure is defined in the reduction. The sign conclusion is limited to that total-pressure interpretation of the printed ODE. A perturbation reinterpretation would need a background term in the nonlinear cooling law; adding a free offset does not preserve the stated equations.
Section 1: state whether any erratum, corrigendum or author response to [1] exists, and reconcile the 24-page arXiv count with the 26-page INSPIRE listing and the byte-identity claim.
Section 1 now records the29 October 2026 bounded check of the official article, primary Crossref relation/update metadata and targeted title/DOI correction searches. No identified target-specific erratum, corrigendum or author response was found; the adjacent issue-list erratum was unrelated. We make no universal nonexistence claim or assertion of correspondence. Section 1 also gives actual PDF page-tree counts, sizes and SHA256s: supplied/official arXiv v1 are 24 pages and 966,878 bytes, while the VOR is 26 pages and 786,255 bytes. Byte identity concerns supplied versus official preprint, not preprint versus journal. A secondary catalogue's26-page field does not identify a change in the retrieved v1 PDF.
Section 3: state explicitly that Section 3 does not solve the repaired non-affine momentum system, and confine the outflow-reversal conclusions to the printed ODEs accordingly.
Section 3 now explicitly states that it does not solve the repaired, generally non-affine momentum system. Its deceleration implication concerns only the printed ODE; the original paper's curves and full momentum trajectories remain unreproduced. Section 4, captions, limitations and conclusion retain the same distinctions. No physical impossibility of radiative outflow contraction or simulation error is claimed.