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REVIEW 4 major objections 5 minor 42 references

Reply to "Is $MgB_2$ a superconductor? Comment on "Evidence Against Superconductivity in Flux Trapping Experiments on Hydrides Under High Pressure" "

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The Reply argues that the Comment's charge of hidden data comes from evaluating the model below the penetration field, where no trapped moment exists.

desk verdict The Reply's physics is right but its 'blatantly false' charge overreaches; the original paper never stated the H<Hp restriction, so claims 6 and 7 are not fully refuted. read the letter →

arxiv 2412.05291 v1 pith:OVUX5LV2 submitted 2024-11-22 cond-mat.supr-con

J. E. Hirsch , F. Marsiglio This is my paper · ORCID
classification cond-mat.supr-con
keywords fluxtrappinghydridesuperconductivityBeancritical-statemodelzero-field-cooledprotocolpenetrationfieldMeissnerstatetrappedmagneticmomentquadraticdependence
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper is a Reply to a Comment that accused the authors of hiding simulated data points and of failing to describe the Meissner state. The Reply argues that the accusation comes from applying the model's equations in the field region below the penetration threshold $H_p$, where the model predicts zero trapped moment and where no valid curve exists. Read correctly, the Reply claims, the Bean critical-state model predicts that zero-field-cooled (ZFC) trapped flux grows quadratically with $H - H_p$, not linearly. It then asserts that hydrides under pressure show linear ZFC trapping, while known superconductors show quadratic trapping, so the linear signal is evidence against superconductivity in the hydrides.

What carries the argument

The machinery is the Bean critical-state model for a disk under the ZFC protocol. When a field is applied and removed, currents occupy an annulus between $r_2$ and $d/2$ and a disk inside $r_1$, yielding $m = m_s[1 - 2(r_1/(d/2))^3 + (r_2/(d/2))^3]$. As $H$ approaches $H_p$ from above, both $r_1$ and $r_2$ approach $d/2$, so the trapped moment goes to zero quadratically in $H - H_p$. The Reply's central move is to insist that the alternative linear formula, Eqs. (2)-(3), describes a field-cooled protocol, not a ZFC one, and that any use of the equations below $H_p$ produces radii outside the sample and is therefore unphysical.

What would settle it

Perform a combined measurement on one hydride sample in a diamond anvil cell, without background subtraction, that records both the onset of magnetic screening (the penetration field $H_p$) and the lowest applied field at which a remnant moment appears after field removal; if a trapped moment appears below $H_p$ or if a known superconductor under identical conditions shows a linear rather than quadratic onset, the Reply's central claim is falsified.

Watch

Extended reading notes

Core claim

The core claim is that the Comment's "hide/delete" charge rests on a misreading of Eqs. (4) and (5) of the original paper, which give the inner and outer current-front radii $r_1$ and $r_2$; these radii must lie inside the sample, and using the equations for $H < H_p$ pushes them outside, which the model never asserts. The model's actual ZFC prediction is $m = m_s[1 - 2(r_1/(d/2))^3 + (r_2/(d/2))^3]$, which vanishes for $H \le H_p$ and rises as $(H - H_p)^2$ just above threshold. From this, the Reply defends the original conclusion: the hydride data's reported linear dependence is inconsistent with the Bean model applied to a superconductor, and the reported trapping threshold of 42 mT conflicts with the independently measured penetration field of roughly 95 to 108 mT.

Load-bearing premise

The argument assumes that the Bean critical-state model with a single uniform penetration field $H_p$ - and the particular value of $H_p$ measured in separate screening experiments on the same materials - applies to the trapped-flux measurements, so that a superconductor cannot trap flux below $H_p$.

Editorial extensions

If this is right

  • The Comment's curves extending below $H_p$ are not predictions of the model; the model gives zero trapped moment for $H \le H_p$.
  • Zero-field-cooled trapped flux in known superconductors rises quadratically in $H - H_p$, so the hydrides' linear rise is anomalous.
  • A sample that excludes fields below roughly 95 to 108 mT cannot trap flux at 42 mT; therefore the trapped moment in the hydrides is not superconducting in origin.
  • The omission of red curves for $H < H_p$ in the original figures was not data-hiding but a direct consequence of the model's boundary condition $m = 0$ for $H \le H_p$.
  • Control measurements after pressure release are irrelevant because cracked diamonds and reduced pressure change both sample and environment.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If this quadratic-onset criterion is adopted as a diagnostic, any future flux-trapping claim for hydrides must exhibit $m \propto (H - H_p)^2$; a linear onset would point instead to ferromagnetic impurities or magnetomechanical artifacts.
  • The dispute could be settled empirically without new theory: measure trapping and screening on the same sample in the same cell, which would make $H_p$ and the trapping threshold directly comparable.
  • The Reply's logic suggests a cheap reanalysis of existing ZFC datasets on other candidate high-pressure superconductors: classify them by the power of the field onset, which may serve as a screening test independent of resistance measurements.
  • A strong test of the Reply's premise would be a trapped-flux measurement on a known type-II superconductor inside a diamond anvil cell; if it shows linear ZFC onset under pressure, the quadratic criterion would need to be relaxed.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. This manuscript is a formal Reply by Hirsch and Marsiglio to a Comment by Talantsev, Minkov, Ksenofontov, Bud'ko, and Eremets, which accused the authors of the original paper (J. Supercond. Nov. Mag. 35, 3141 (2022)) of using a wrong model, hiding/deleting parts of simulated datasets that disagree with the Meissner regime, restricting access to their computer code, and other methodological misconduct. The Reply argues that all ten accusations in the Comment are unfounded. Its central defense is that the Hirsch-Marsiglio model, based on the Bean critical-state model, predicts zero trapped magnetic moment for applied fields below the penetration threshold Hp, and that the Comment's accusation rests on applying Eqs. (4)--(5) of Ref. [3] outside their intended domain H_M >= Hp. The Reply also asserts that the model was subsequently validated against flux-trapping measurements on known superconductors (Bud'ko et al.) in Refs. [11,14], that the hydride ZFC data show a linear rather than quadratic field dependence and are therefore inconsistent with superconducting behavior, and that the Comment's authors were informed of the boundary-condition issue by a letter reproduced in the Appendix. The Reply includes a history of the code-sharing exchange and an acknowledgment that one of their own figures contained a sign error, which it deems irrelevant to the main issue.

Significance. If the Reply's arguments are correct, they would clear the authors of the data-hiding and misconduct charges and reinforce their earlier challenge to the interpretation of trapped-flux experiments on hydrides as evidence for high-temperature superconductivity. The Reply usefully clarifies the Bean-model prediction of quadratic ZFC trapped-moment field dependence and honestly acknowledges a sign error in a previous figure. However, the Reply's central historical defense is weakened by its own concession (Sec. III.A) that the original paper never explicitly stated the domain restriction H_M >= Hp for Eqs. (4)--(5). As a result, the Reply's strong claim that the Comment's reading is 'utterly and blatantly false' is overstated. The Reply also relies on a transferability assumption between separate screening and trapping experiments that is not defended. The paper makes a plausible technical case but does not fully establish that all ten accusations are unfounded.

major comments (4)
  1. [III.A] The Reply concedes that it 'did not state explicitly that Eqs. (5) did not apply to the region HM < Hp,' yet in Section VI it calls the Comment's statements about nonzero moments in the Meissner regime 'utterly and blatantly false.' As printed, Eqs. (4)--(5) of Ref. [3] evaluated at HM = 0 yield r1, r2 > d/2 and a positive moment, so the Comment's reading is a plausible interpretation of the published text even if it is physically wrong. The Reply should acknowledge the ambiguity in the original paper and argue why the physical context makes the boundary condition m = 0 for HM <= Hp obvious, rather than accusing the Comment authors of knowingly making false statements.
  2. [I, III.A, VI] The Reply's blanket claim that 'all of them are unfounded' (Section I) is not supported even by its own arguments. For claims 6 and 7, the Reply's own concession of the unstated domain restriction shows that the Comment's reading was not baseless, even if the Comment's inference of deliberate hiding is unwarranted. For claims 2 and 9, the Reply merely explains (Section III.B) why it did not consider the post-pressure-release reference measurements relevant, which is a scientific disagreement rather than a demonstration that the claim is unfounded. The Reply should be revised to claim that the accusations are 'not supported' or 'incorrect in our reading,' rather than 'unfounded' in every case.
  3. [V] The Reply argues that it is difficult to understand how a sample that excludes magnetic fields smaller than about 95--108 mT (Refs. [17,19]) could trap flux between 42 mT and 108 mT. This argument assumes that the penetration field Hp measured in separate magnetic screening experiments applies quantitatively to the same samples and experimental conditions as the flux-trapping measurements. The Reply does not justify this transferability, and the Comment's position is that the trapping data themselves imply a lower threshold (about 42 mT) in those runs. Without direct evidence that the Hp values from different experimental setups are comparable, this argument does not by itself refute the Comment's claims.
  4. [Appendix and Section VI] The Reply uses the December 8, 2023 letter reproduced in the Appendix as evidence that the Comment authors knew the boundary condition and nonetheless repeated their claims. However, the Comment was posted on December 7, 2023 (Ref. [8]), one day before the letter was sent. The letter therefore cannot establish that the Comment authors knew of the boundary-condition restriction at the time they wrote their Comment. It can only show that they were informed afterward and did not retract. The Reply's rhetorical claim that the Comment authors 'must know' is not supported by the evidence presented.
minor comments (5)
  1. [II, item 3] The phrase 'the authors of this Comment' in item 3 should read 'the authors of the Comment' to avoid confusion with the authors of the Reply.
  2. [IV] There is a typo in the third paragraph: 'misuderstanding' should be 'misunderstanding.'
  3. [VI] The sentence 'Why do the authors of Ref. [1] choose to knowingly write such falsehoods and submit them for publication in a scientific journal?' is accusatory and would be more appropriately replaced by a factual statement that the Comment's interpretation is inconsistent with the Bean model and the intended domain of Eqs. (4)--(5).
  4. [V] In the discussion of Ref. [19], the phrase 'Revaluation of the lower critical field' should be 'Re-evaluation of the lower critical field' to match standard English usage.
  5. [III.A] The Reply says 'in all the examples we showed (Figs. 3, 4, and 5 of Ref. [3]) we showed calculated ZFC values as dashed lines in red, and in all the figures the dashed red lines were plotted only for HM ≥ Hp.' It would be helpful to explicitly note in the Reply that the original paper did not state this restriction, to avoid the appearance of an after-the-fact justification.

Circularity Check

1 steps flagged · score 4.0 of 10

One load-bearing step is self-definitional: the Reply's zero-trapped-moment 'prediction' for H_M < H_p is exactly the boundary condition it assumed but omitted from Eqs. (4)-(5) in Ref. [3]; other claims rest on external data.

  1. self definitional [Sec. III.A ('Claim that we hid/deleted data'), discussion of Eqs. (4)-(5) and the boundary condition]
    "We did not state explicitly that Eqs. (5) did not apply to the region HM < Hp, which would yield values of r1 and r2 larger than the radius of the sample d/2, because it was completely obvious from the context ... We assumed that the boundary condition mtrap = 0 if HM ≤ Hp, which had been explicitly stated in Ref. [8], would be obvious to readers and didn’t state it again explicitly."

    The Reply's refutation of Comment claims 6 and 7 is that 'our model does not predict a trapped magnetic moment for HM < Hp.' But this is not a consequence of Eqs. (4)-(5) as published: for HM < Hp, Eq. (5) gives r1,r2 > d/2 and Eq. (4) gives a finite positive moment. The zero trapped moment below Hp is exactly the extra boundary condition 'mtrap = 0 if HM ≤ Hp' that the authors say they assumed but did not state in Ref. [3]. The 'prediction' is therefore the input assumption by construction; the Reply's assertion that the Comment's plotted red curves are 'not predictions of our model' rests on a retrospective restriction rather than on a derivation from the model's published equations.

full rationale

The Reply's core defense against the 'hid/deleted data' accusation is that its model predicts m_trap = 0 for H_M < H_p, making the Comment's plotted red curves non-predictions. But as the Reply admits, Eqs. (4)-(5) of Ref. [3] were published without an explicit H_M >= H_p restriction; evaluated at H_M < H_p they give r1,r2 > d/2 and a positive moment. The zero is supplied by the separate boundary condition 'mtrap = 0 if HM ≤ Hp,' which the authors say they assumed. Thus the 'prediction' reduces by construction to the input assumption; the Reply's further assertion that the Comment's reading is 'utterly and blatantly false' depends on this retrospective restriction. This is a genuine circular step in the argumentation. It is, however, the only clear case. The Reply's quadratic-vs-linear dependence is a Taylor-series consequence of the Bean-model geometry and is checked against Bud'ko et al. data in the authors' Refs. [11,14]; those self-citations are not load-bearing circular because they analyze external, falsifiable datasets. The H_p = 95-108 mT threshold used to argue that trapping at 42 mT is impossible is taken from the same group's independent screening measurements (Refs. [17,19]); it is a consistency argument, not a fitted parameter renamed as a prediction. Score 4 reflects one central self-definitional move while the rest of the Reply has independent content.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The Reply's arguments rest on the Bean critical-state model and on fitted parameters Hp, H*, and ms, plus auxiliary experimental numbers taken from the same experimental group. No new physical entities are introduced. The core technical claim about the H < Hp boundary is internal to the model rather than an independent prediction.

free parameters (3)
  • Hp (penetration threshold field) = 42 mT for H3S in the authors' inferred fit; 95-108 mT from Refs [17,19]
    The Reply relies on different estimates of Hp to argue inconsistency. The model's boundary condition m=0 for H<Hp depends on this fitted parameter, and the comparison between datasets requires assigning a value to Hp.
  • H* (full penetration field) = Not specified in the Reply, used as a fitting parameter in the model curves
    The Reply quotes the model equations (4) and (5) that contain H*, and its later fits to known superconductors vary H* to match data. The value is not predicted from first principles.
  • ms (saturation trapped moment) = Set by the maximum trapped moment in each dataset
    The model curves are normalized to ms, the saturation moment, which is read off the measured data. This is a fitting constant, not a prediction.
assumptions (3)
  • domain assumption The Bean critical-state model accurately describes flux trapping in the hydride samples.
    The entire Reply is predicated on the Bean model with a uniform critical current density. This is a standard model for type-II superconductors, but its applicability to the heterogeneous, high-pressure hydride samples is the very point in dispute.
  • domain assumption The ZFC trapped moment is exactly zero for all applied fields below Hp, and only becomes nonzero above Hp.
    The Reply argues this boundary condition is obvious from the model, but it is a modeling assumption about the threshold behavior. It is not independently measured in the Reply.
  • domain assumption The penetration field Hp measured in magnetic screening experiments (Refs. [17,19]) applies to the same samples and conditions as the flux trapping measurements (Ref. [4]).
    Section V uses Hp values from Refs [17,19] to argue that trapping between 42 mT and 108 mT is impossible. This requires that the samples and experimental conditions in the two sets of measurements are directly comparable, which the Reply assumes without direct evidence.

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Cite this review

Pith. "Pith review of Reply to "Is $MgB_2$ a superconductor? Comment on "Evidence Against Superconductivity in Flux Trapping Experiments on Hydrides Under High Pressure" "." pith.science (2026). https://pith.science/paper/OVUX5LV2

@misc{pith2026241205291,
  author       = {Pith},
  title        = {Pith review of: Reply to "Is $MgB_2$ a superconductor? Comment on "Evidence Against Superconductivity in Flux Trapping Experiments on Hydrides Under High Pressure" "},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OVUX5LV2}},
  note         = {Machine review of arXiv:2412.05291}
}
read the original abstract

The preceding Comment [1], previously posted as arXiv:2312.04495 [2], on our paper J. Supercond. Nov. Mag. 35, 3141 (2022) [3] provides a welcome opportunity to clarify what we understand to be pervading misconceptions by Eremets, Minkov and coauthors in regard to our analysis [3] of their trapped flux experiments in hydrides under pressure [4]. We hope that this Reply [5] will help readers interested in hydride superconductivity sort out between different claims and counterclaims in the literature and inform their views based on verifiable facts.

Figures

Figures reproduced from arXiv: 2412.05291 by the authors.

Figure 2
Figure 2. FIG. 2: Figure 2 of our paper Ref. [ [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Image of email sent to authors of the Comment on December 8, 2023. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

42 extracted references · 41 canonical work pages

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    In this Comment

    On December 18, 2023 and January 15, 2024, one of us (JEH) posted responses to some of the authors’ claims in [2], which are identical to claims in this Comment [1], at OSF [9] and arxiv [10] respectively. In this Comment

    work page 2023

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    the authors have restricted the publication of their simulation code, which we view as a breach of sci- entific integrity and open science principles

    “the authors have restricted the publication of their simulation code, which we view as a breach of sci- entific integrity and open science principles.”

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    the authors 1 engaged in methodological malprac- tice by creating a FORTRAN 77 simulation code rather than performing a genuine data fit as claimed

    “the authors 1 engaged in methodological malprac- tice by creating a FORTRAN 77 simulation code rather than performing a genuine data fit as claimed”

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    relies on the wrong model coupled with selective manipulations (hide/delete) of calculated datasets

    “relies on the wrong model coupled with selective manipulations (hide/delete) of calculated datasets”

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    ignores the reference measurements after the re- lease of pressure

    “ignores the reference measurements after the re- lease of pressure”

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    the authors 1 did not verify their model using any ZFC or FC data measured in well-studied supercon- ductors

    “the authors 1 did not verify their model using any ZFC or FC data measured in well-studied supercon- ductors.”

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    Upon examination of the provided code, we sur- prisingly discovered that the code 1 is a simulation tool

    “Upon examination of the provided code, we sur- prisingly discovered that the code 1 is a simulation tool”

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    This feature of the proposed model was not discussed by the authors1 and obviously contradicts the physics for the trapped magnetic flux in superconductors.”

    “Significantly, the Hirsch-Marsiglio model 1 does not describe the Meissner regime ...Instead, the model predicts a significant positive magnetic mo- ment for ZFC m(Bappl = 0 , T) ... This feature of the proposed model was not discussed by the authors1 and obviously contradicts the physics for the trapped magnetic flux in superconductors.”

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    “the authors 1 in their Figures 3-5 hid/deleted (without reporting this) parts of their simulated ∗Corresponding author: jhirsch@ucsd.edu †email: fm3@ualberta.ca datasets that disagree with the Meissner state. This potentially allowed them to conceal the issue that their model...

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    “We argue that the authors overlooked refer- ence measurements conducted on evidently non- superconducting samples, which demonstrate the absence of such artefacts stemming from the sam- ple environment (diamond anvil cells, rhenium gas- kets, etc.). The reference measurements...

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    On April 12, 2024, we published Ref. [11], where we presented an analysis of flux trapping experiments on known superconductors reported by Bud’ko, Xu and Canfield on September 13, 2023 [12] using the same “Hirsch-Marsiglio model” used in our paper Ref. [3], pro- vided further...

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    On May 26, 2024, we posted on arxiv Ref. [14], as well as submitted for publication, our analysis of a recent paper of Bud’ko et al. on trapped flux in tiny samples of known superconductors under pressure [13] and its rela- tion with the trapped flux experiments on hydrides [4...

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    (Eq. (1) above) assumed the trapped magnetic mo- ment was given by m = Z d/2 r πr′2jchdr′ = ms[1 − ( r d/2 )3] (2) r = r(HM ) = d 2 (1 − HM − Hp 2H ∗ ) (3) which are of course identical to Eq. (1) above. We ex- plained that Hp in the ZFC protocol is the “threshold value of the...

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    (see Appendix) that our model, just as Bean’s model, predicts no trapped moment for HM < Hp. In [8] the authors also posted our computer code. They did this without our permission, which is in violation of intellectual property principles. The authors of [1] accuse us of restr...

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.