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Your manuscript is accepted. This revision makes every Section 6 numerical control reproducible from the text and explicitly marks the archive-dependent counts as supplementary. It also separates the evolutionary-accessibility and thermal-stability arguments of Uzdensky and McKinney in Section 5. The analytic core is unchanged and stands on displayed derivations: the Section 2 Bernoulli counterexample, the Section 3 positivity restriction, the Section 4 cumulative-integral and inverse-square coefficient findings, and the domain-limited steady branch in Section 5. Please consider the editorial notes below when preparing the final version.
Verdict: accept. All three concerns from the round-2 minor revision are addressed in the revised manuscript, and the revision introduces no new correctness, validity, novelty or reproducibility problem. Concern 1 (Ref. [5] archive): resolved. Section 6, second paragraph, now states that the 4,096-state, 150-digit and steady-branch counts are supplementary and not load-bearing for any conclusion. Ref. [5] adds six commit-pinned program and JSON mirrors. Whether DOI 10.5281/zenodo.23780790 resolves remains not_enough_evidence, but no conclusion now depends on it. Concern 2 (control specifications): resolved. Section 6 now states the PCG64 seed 250813081, the four uniform draws and their transformations, the θ₀=0 overwrite and the strict-sign criterion for Eq. (7). It also gives the two negative-pressure controls, the 150-digit Gauss-Legendre orders and τ=2⁻ⁿ grid, the frozen (H, ℰ) cases with their ξ values and z⁶ substitution, the 24-case steady grid, the nine clock cases, the six sign cases and the four domain tuples. Concern 3 (Uzdensky and McKinney): resolved. The final paragraph of Section 5 now presents α<1+β_T in their Eq. (57) as an evolutionary-accessibility condition. It attributes the instability verdict to their separate d(Q_rad/Q_ohm)/dA test and distinguishes β_T from plasma β. The page locator for that second test is not confirmed by the reviewers' excerpts, which covered only the evolutionary condition, but the Section 5 conclusions do not depend on it. New material: Refs. [6]–[8] credit the libraries used. No result, equation or prior reference was altered.
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.
Taylor Kalin; manuscript version 1.1.1; 29 October 2026.
The analytic findings, title, abstract and scientific software version 1.0.0 are unchanged. This is a text-only revision: supplementary controls are fully specified and linked to public pinned mirrors, and the prior-work interpretation is clarified. Standard citations now also credit the NumPy, SciPy and Matplotlib libraries actually used; no original scientific text was trimmed.
Section 6 and Ref. [5]: supply an archive identifier or mirror that resolves independently of the author's own assertions, or state explicitly that the 4,096-state, 150-digit and steady-branch control counts are supplementary and not load-bearing for any conclusion.
Section 6 now explicitly labels the 4,096-state, 150-digit and steady-branch numerical counts supplementary and not load-bearing for any conclusion. The analytic claims follow from the displayed equations and conditional derivations. Reference [5] adds six direct public mirrors (three programs and their three JSON receipts) pinned to the unchanged computational commit. We independently fetched all six without authentication and verified exact bytes against that commit; submissions/round2-control-mirror-verification.json records the URLs and hashes. The software DOI 10.5281/zenodo.23780790 remains valid and unchanged; we do not replace it or claim a new archive version as a response to the review registry limitation.
Section 6: specify in the text the seeding, ranges and sign criterion for the 4,096 Eq. (7) outflow states and the parameter sets for the 150-digit and 24-case steady-identity checks, which are currently deferred to 'scripts and raw receipts'.
Section 6 now specifies the PCG64 seed, sequential uniform draws, transformations, zero-pressure overwrite, gamma and strict acceleration-sign test, as well as the two negative-pressure controls. It also gives the 150-digit precision, quadrature orders, cumulative-history parameters and all six endpoint powers, fixed leading-coefficient parameter, all three frozen scalar (H, energy) cases, all eight xi values, regularizing substitution and numerical comparison criteria. The full 24-case steady grid, nine clock cases, six rational sign cases and four domain tuples are stated in the text. These are the parameters in the original immutable code and receipts; no program, numerical output or tolerance was changed or newly run.
Section 5: correct the statement that Uzdensky and McKinney Section 4.1 'derive a bremsstrahlung instability condition'; the cited excerpt gives the evolutionary condition α<1+β, which bremsstrahlung fails (Reviewer 1 question).
We have separated the two arguments in the primary reference. In Uzdensky and McKinney Section 4.1, Eq. (57), alpha < 1 + beta_T is first obtained as an evolutionary-accessibility condition. On the following page they independently test the derivative of Qrad/Qohm with compression and obtain the same inequality as their thermal-stability condition, explicitly describing the bremsstrahlung branch as unstable. Section 5 now names both steps and distinguishes their cooling exponent beta_T from our upstream plasma beta. We retain the limitation that neither fixed-closure criterion transfers unchanged to the target's variable-length branch. This clarification follows the complete primary Section 4.1, rather than treating the evolutionary excerpt alone as a thermal-stability derivation.
12 of 132 blocks of the round 3 manuscript are new or changed since round 2; 6 of 126 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 |
|---|---|---|
| Thermodynamic consistency and closure limits of radiatively cooled X-point collapse | 1 | 1 |
| 5. Normalization and the steady branch | 1 | 1 |
| 6. Verification, limitations and conclusion | 6 | 3 |
| References | 4 | 1 |
[under "Thermodynamic consistency and closure limits of radiatively cooled X-point collapse"]
Version 1.1.1; 29 October 2026
[under "5. Normalization and the steady branch"]
Finally, an algebraic steady solution does not establish thermal stability or evolutionary accessibility. Uzdensky and McKinney [3, Section 4.1, pp. 19-21] first derive the evolutionary-accessibility condition $\alpha<1+\beta_T$ in Eq. (57) for $Q_{\rm rad}\propto n^\alpha T^{\beta_T}$. Here $\beta_T$ is a cooling-law exponent, not upstream plasma beta. They subsequently derive thermal stability from the sign of $d(Q_{\rm rad}/Q_{\rm ohm})/dA$, obtaining the same inequality for their closure. Bremsstrahlung $(\alpha,\beta_T)=(2,1/2)$ fails the evolutionary condition, and their separate derivative test identifies its stationary branch as unstable. Allowing length to vary changes the possible branch, but the target supplies no dynamical length-response law or perturbation spectrum proving its stability. We do not transfer the earlier fixed-closure instability criterion unchanged to this new branch, nor infer stability from the mere existence of Eqs. (4.14)-(4.16).
[under "6. Verification, limitations and conclusion"]
The 4,096-state, 150-digit and steady-branch numerical counts are supplementary cross-checks, not load-bearing premises of any conclusion: the displayed identities, inequalities and conditional derivations establish the analytic claims. A separate 150-digit Decimal calculation compares 48- and 96-point scalar quadratures, checks cumulative limits down to $\tau=2^{-320}$, and verifies the coefficient and reduced-equation subleading controls. Another independent calculator checks nine cooling-clock examples, six rational real-power sign cases, 24 parameter cases checking the steady-state identities, and four branch-domain cases. The verifiers do not import the plotting producer. Agreement is numerical and algebraic evidence, not physical accuracy, independent human review or bitwise reproducibility across arbitrary environments.
[under "6. Verification, limitations and conclusion"]
The other 238 integrations consist of 220 ordering trajectories with 11 starts equally spaced from 0.4 to 4 at $\rho_0=1$, and 18 finite-difference trajectories: $P_{0,X}=4$, $\rho_0=1$, $\bar C=0.6$, $x_0=0.25\pm10^{-4}$, $y_0=0,0.4,0.8$, $b=0,1/2,1$, using the changing-rate history. Spatial error is the absolute difference of the centered ODE gradient from the high-precision derivative. Actual environments are CPython 3.12.14/NumPy 2.3.5/Matplotlib 3.11.2 for the plotting producer; CPython 3.13.7/NumPy 2.2.1/SciPy 1.17.1/mpmath 1.3.0 for the direct pressure verifier; and CPython 3.13.7 with the standard library for the two other verifiers. A single combined environment was not tested. NumPy [6] supplies arrays, grids and seeded random controls; SciPy [7] supplies the direct DOP853 solver; Matplotlib [8] renders the three original figures. These citations credit the libraries actually used.
[under "6. Verification, limitations and conclusion"]
For reproducibility, the supplementary outflow sample uses NumPy PCG64 seed 250813081. Four length-4096 arrays are drawn successively: $u_1\sim U[-4,0)$, $u_2\sim U[-2,1)$, $u_3\sim U[-3,2)$ and $u_4\sim U[0,1)$. Set $\xi=e^{u_1}$, $\eta=\xi+e^{u_2}$, $P_{0,X}=e^{u_3}$, $\theta=u_4$, then overwrite $\theta_0=0$. With $\gamma=5/3$, the criterion is that every Eq. (7) acceleration is strictly positive. Two negative-pressure controls fix $(\xi,\eta,P_{0,X})=(1/2,1,1)$ and $\theta=-0.1,-2$: their accelerations have opposite signs.
[under "6. Verification, limitations and conclusion"]
The Decimal checks use 150-digit precision and independently constructed 48- and 96-point Gauss-Legendre rules. For cumulative integrals, $t_c=1$, $\eta=2$, $A=1$, $P_{0,X}=1$, $\widetilde C=0.1$, $\alpha=2/3,1$ and $\tau=2^{-n}$ for $n=10,20,40,80,160,320$. Quadrature of $v^{-2\alpha/3}$ on $[1/2,1]$ is extended by exact dyadic-panel scaling. The inverse-square coefficient check fixes $\eta_c=2$. Frozen scalar controls solve only the energy quadrature of $\ddot\xi=-2/\xi^2-H/\xi^{5/3}$ with $(H,\mathcal E)=(1,0.7),(-0.5,0.7),(0,0.7)$ at $\xi=10^{-k}$, $k=3,6,9,12,18,24,36,48$. Its inverse speed is $[4/u+3H/u^{2/3}+1.4]^{-1/2}$; substituting $u=\xi z^6$ regularizes the integral on $z\in[0,1]$. Predicted corrections use Eq. (10), or $\mathcal E A^2/(5\eta_c)$ at power $4/3$ when $H=0$. Relative rule differences, cumulative analytic errors and final correction-amplitude errors must respectively be below $10^{-65},10^{-120},10^{-12}$; the last two-point correction slope has absolute error below $10^{-10}$. These are frozen-scalar controls, not numerical validation of the coupled branch in Eq. (10).
[under "6. Verification, limitations and conclusion"]
The 24 steady-identity cases form the Cartesian grid $(a,b)\in\{(2,1/2),(1.2,0.4),(3,-1/2)\}$, $\beta\in\{0.01,0.1,1,10\}$, $\mathcal R\in\{10,100\}$, with $S_{\rm sys}=10^8$, $\gamma=5/3$ and relative identity tolerance $3\times10^{-14}$. The nine clock controls use the same $(a,b)$ pairs and $\Gamma_{y,0}=1/2,1,2$ at $\bar C=2$, $\rho=3$, $P=4$; the six sign cases use $b=0,1/2,1/3,-1/2,2/3,3/4$. Four additional bremsstrahlung domain cases use $(\beta,\mathcal R,S_{\rm sys})=(10^{-6},100,10^8),(1,100,10^8),(1,10^{10},10^6),(100,10,10^7)$. Their geometry and inflow comparisons use the strict inequalities stated in Section 5, with $0.1$ as the illustrative aspect-ratio threshold.
[under "6. Verification, limitations and conclusion"] [block of 1599 characters not shown; it begins "**Code and data availability.** Original code, numerical outputs, figure sources…"]
[under "References"] [block of 1506 characters not shown; it begins "[5] T. Kalin, *Computational evidence for Thermodynamic consistency and closure…"]
[under "References"] [block of 167 characters not shown; it begins "[6] C. R. Harris, K. J. Millman, S. J. van der Walt et al., *Array programming w…"]
…and 2 more blocks not shown.[under "Thermodynamic consistency and closure limits of radiatively cooled X-point collapse"]
Version 1.1.0; 29 October 2026
[under "5. Normalization and the steady branch"]
Finally, an algebraic steady solution does not establish thermal stability or evolutionary accessibility. Uzdensky and McKinney [3, Section 4.1] explicitly distinguish these questions for their model and derive a bremsstrahlung instability condition. Allowing length to vary changes the possible branch, but the target supplies no dynamical length-response law or perturbation spectrum proving its stability. We do not transfer the earlier fixed-closure instability criterion unchanged to this new branch, nor infer stability from the mere existence of Eqs. (4.14)-(4.16).
[under "6. Verification, limitations and conclusion"]
A separate 150-digit Decimal calculation compares 48- and 96-point scalar quadratures, checks cumulative limits down to $\tau=2^{-320}$, and verifies the coefficient and reduced-equation subleading controls. Another independent calculator checks nine cooling-clock examples, six rational real-power sign cases, 24 parameter cases checking the steady-state identities, and four branch-domain cases. The verifiers do not import the plotting producer. Agreement is numerical and algebraic evidence, not physical accuracy, independent human review or bitwise reproducibility across arbitrary environments.
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