{"id":"35a323d9-34ce-4ac6-8148-de6e24723673","arxiv_id":"2507.10199","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A Pearl vortex's stray field can greatly enlarge or invert the chirality of a Neel-type skyrmion, deform it off-center, and help stabilize vortex-antivortex pairs.","lead":"This review explains how superconducting vortices can enlarge, flip, and displace nanoscale magnetic textures called skyrmions in hybrid superconductor-ferromagnet stacks. The effects matter for building topological quantum-computing platforms where skyrmion-vortex pairs host Majorana states.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper's own admission in Sec. IVB2 that the a_add≈0.2ℓw minimum lies below the precision of the second-order expansion shows the truncated variational free energy can manufacture uncertified features; the same functional sets the phase boundaries, so their quantitative positions inherit this…","rationale":"The reader's weakest_assumption correctly identifies the two-parameter variational ansatz as the load-bearing element. My stress test narrows this to the specific truncation error admitted in Sec. IVB2: the second-order energy (58) is precise enough to locate the main stable configurations but demonstrably not precise enough to certify a shallow minimum. Since the same truncated functional is used for the quantitative phase boundaries and critical values, the review's strongest quantitative statements are conditional on an uncontrolled approximation. This is a genuine concern, but it is already reflected in the CONDITIONAL verdict. I do not see a fatal internal inconsistency: the derivation of the ansatz (33) is algebraically coherent, the exact Euler–Lagrange comparison in Fig. 3 and the micromagnetic points in Figs. 5, 6, and 10 provide real independent support, and the paper explicitly disclaims the a_add feature. The main qualitative predictions are therefore credible as predictions, while the specific a_add minimum should not be cited as established. No shipped code or full simulation data are provided, which compounds the reproducibility issue but does not by itself change the verdict. The single most useful additional check is a direct test of whether the unexpanded ansatz reproduces the truncated energy landscape; that would settle whether the admitted artifact is an artifact of truncation or a real metastable state.","tokens_in":36049,"tokens_out":5976,"duration_ms":78640,"concrete_test":"For the parameter set near the a_add minimum (e.g., ϵ=0.45, γ≈0.119 in Fig. 7), evaluate the free energy of the full normalized ansatz (45)–(48) without expanding the denominator or truncating at γ^2, and compare F(a) with Eq. (58). If the a_add minimum changes depth or position by more than its quoted depth, or disappears, the second-order truncation is the cause. Then repeat the comparison for the γcr(ϵ) boundary extraction over ϵ∈[0.25,0.49]; a shift of more than roughly 10% in γcr would mean the quantitative phase boundaries in Fig. 8 are not settled by the present analysis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The quantitative claims—the coexistence boundaries γcr(ϵ), γ−cr(ϵ), γ+cr(ϵ) in Fig. 8, the radius/distance curves near γcrit in Fig. 6, and the F(a) landscapes of Fig. 7—are all derived from the free-energy functional (58), obtained by inserting the two-parameter ansatz (44) and truncating at order γ^2. The review itself flags a concrete failure mode: in Sec. IVB2, the additional minimum at a_add≈0.2ℓw has a depth outside the precision of this second-order expansion and is not confirmed by micromagnetic simulations. That is an explicit, located example in which the truncated functional predicts a metastable feature that the authors do not certify. Because the same functional, with no error estimate, is used to construct the phase diagram and to locate γcr for each ϵ, the quantitative positions of these boundaries are only as secure as that truncation. This does not overturn the qualitative central effects: coaxial radius increase and chirality inversion are supported by exact Euler–Lagrange solutions and OOMMF simulations at selected parameters, and eccentric deformation and vortex–antivortex stabilization are supported by the same machinery plus simulation points. The objection is therefore a robustness/precision concern about the variational machinery, not a disproof of the main physical picture.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript reviews and extends the authors' previous theoretical work on the magnetostatic interaction between Néel-type skyrmions in a thin chiral ferromagnet and Pearl vortices in an adjacent thin superconductor. The central technical tool is a variational ansatz for the skyrmion magnetization in a weak inhomogeneous magnetic field, constructed as the free-skyrmion 360-degree domain-wall profile plus a linear-order correction built from the no-skyrmion response to the vortex field (Eqs. 33 and 44). Using this ansatz, the authors analyze coaxial configurations, obtaining radius blow-up, chirality inversion, and a maximum radius Rmax = ζℓw/(2∓επ) at finite Pearl length (Eq. 41); eccentric configurations, where second-order terms in the effective vortex strength determine the equilibrium skyrmion-vortex distance and lead to a phase diagram with coaxial/eccentric coexistence (Fig. 8); and stabilization of a vortex-antivortex pair by a skyrmion (Sec. V). The paper also discusses higher-order skyrmions and compares with micromagnetic simulations (OOMMF, Ubermag) and with three recent experiments. The presentation is structured as a review of Refs. [42-44] with additional context, a table of material parameters, and a discussion of experimental status.","tokens_in":2029,"tokens_out":2310,"duration_ms":74776,"significance":"If the results are robust, the manuscript provides a usable analytical framework for a system of current interest: coupled skyrmion-vortex complexes are a proposed platform for Majorana bound states and topological quantum computing. The paper's strengths are the transparency of the derivation, the explicit cross-checks against direct numerical solution of the Euler-Lagrange equation and against micromagnetic simulation in selected parameter regions, and the parameter-free nature of the maximum-radius expression (41), which is a falsifiable prediction depending only on ferromagnetic material constants. The review also usefully quantifies the regime of validity of the Pearl approximation and provides a table of experimental parameters that allow estimating the dimensionless coupling γ. The principal weakness is that the quantitative phase diagram and the critical values γcr(ϵ), γ±cr(ϵ), and γ*cr(ϵ) are all derived from a free-energy functional truncated at second order in γ, and the paper itself identifies a located example (the additional minimum at a_add ≈ 0.2ℓw in Sec. IVB2) where this truncation produces an uncertified feature that micromagnetic simulations do not confirm.","major_comments":[{"comment":"The paper explicitly states that the additional local minimum at a_add ≈ 0.2ℓw lies \"outside the precision of our second-order expansion in γ\" and that micromagnetic simulations did not confirm its presence. This is a concrete instance in which the truncated free-energy functional (58) predicts a metastable feature that the authors themselves do not certify. The same functional, with the same truncation and no error estimate, is used to construct the phase boundaries γ−cr(ϵ), γcr(ϵ), and γ+cr(ϵ) in Fig. 8 and the radius/distance curves in Fig. 6. While the authors compare some of these curves with simulations (green circles and diamonds in Fig. 8), the comparison is limited to a few parameter points. The quantitative positions of the boundaries therefore inherit an unquantified uncertainty from the γ^2 truncation. I ask the authors to provide an estimate of the truncation error, for example by evaluating the magnitude of the leading omitted O(γ^3) terms or by performing a convergence check at selected (ϵ, γ) points, and to state explicitly how this error affects the reported critical values. Without such an estimate, the claim that the phase diagram is quantitatively reliable is stronger than the evidence supports.","section":"Sec. IVB2 and Eq. (58)"},{"comment":"The variational ansatz (33)/(44) restricts the skyrmion profile to a two-parameter family (R, δ) and a fixed functional form for the γ-correction. The paper validates this ansatz against exact Euler-Lagrange solutions and micromagnetic simulations at a few parameter sets, for example ϵ = 0.325, γ = 0.479 in Fig. 3, and states that the method \"yields reliable results over a wide range of parameters ϵ and γ.\" This claim is not backed by a systematic scan. In particular, the phase diagram in Fig. 8 covers a range of ϵ from about 0.2 to 0.6 and γ from 0 to 0.7, but the comparison points are sparse. Given that the ansatz is exactly the kind of restricted functional that can miss true minima or create spurious ones, as shown by the a_add minimum, the paper should either provide a denser benchmark of the variational results against direct ELE solving across the (ϵ, γ) plane, or explicitly delimit the parameter region in which the two-parameter ansatz is controlled. A short statement of expected error bounds would turn this from a caveat into a quantitative validation.","section":"Sec. IIIB and Sec. IVA"},{"comment":"The prediction that a skyrmion can stabilize a vortex-antivortex pair relies on the same variational machinery and on the neglect of the superconducting interaction energy for β ≪ 1 (Eqs. 74-79). The latter approximation is stated and reasoned, but the former inherits the truncation issue identified above. The critical curve γ*cr(ϵ) in Fig. 8 is compared with only a few micromagnetic points, and the bounded and discontinuous behavior of the antivortex distance a_Vbar in Fig. 10 is a strong qualitative prediction that depends on the existence of a maximum in the energy as a function of a_Vbar. The paper would be strengthened by showing that this maximum and the associated criticality survive when the O(γ^3) terms are included or when the ansatz is relaxed. At minimum, the authors should state the expected sensitivity of the vortex-antivortex phase region to the variational truncation, and distinguish which aspects of Fig. 10 are robust (qualitative) versus sensitive (quantitative).","section":"Sec. V, especially Sec. VB and Fig. 10"}],"minor_comments":[{"comment":"The first term in these Euler-Lagrange equations appears as \"ℓ2w r ∂r(r∂rθ)\", which seems to be missing the division by r; the standard form is (ℓ_w^2/r) ∂_r (r ∂_r θ). Please check and correct the typesetting in these equations.","section":"Eqs. (5), (32), (55)"},{"comment":"The displayed expression for |R| contains a formatting artifact \"1p γγ ±∞ − 1\"; it should read sqrt(γ/γ±∞ − 1). Please correct the equation.","section":"Eq. (40)"},{"comment":"Reference [72] (C. Tanguy, arXiv:cond-mat/0106184) appears unrelated to the field approximation quoted in Eq. (38). If the approximation is from another source, please cite the correct reference; otherwise the citation is misleading.","section":"Reference [72]"},{"comment":"The sentence describing the a_add minimum is honest and important, but it is placed parenthetically in the middle of the results. Since this is a known limitation of the method, it deserves a more prominent discussion, perhaps in a dedicated paragraph on the validity of the variational approach, where the implications for other results are stated clearly.","section":"Sec. IVB2"},{"comment":"The phrase \"with sign corrected by multiplication by the skyrmion chirality\" is ambiguous. Define whether the plotted quantity is χθ(r) or θ(r)/χ, and explain the reason for the correction.","section":"Fig. 3 caption"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is essentially a review of the authors' own published results (Refs. 42-44) with a synthesis of experimental comparisons. That is legitimate for a journal that publishes reviews, but the novelty is limited relative to the underlying papers. The most important concern is the unquantified truncation error in the variational free energy, which the authors themselves acknowledge for one feature; this should be addressed with an error estimate or a more extensive benchmark before the quantitative phase diagram can be accepted. The qualitative physical picture is well supported and likely correct."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a review, not a new result, but it is a good review. The authors consolidate their own earlier papers on Pearl-vortex/skyrmion interactions. The three headline effects—radius blowup, chirality inversion, eccentric deformation—appear in Refs. [41–44]. What this paper adds is a unified derivation, a phase diagram, an explicit bound Rmax, and a comparison with three experiments. The physics is credible: the variational ansatz is a sensible expansion in gamma, and the central results are cross-checked against OOMMF/Ubermag simulations at representative parameters. The experimental comparison is qualitative but supports the picture.\n\nThe soft spots are real but not fatal. The two-parameter ansatz is restricted to a specific functional family, and the paper itself admits in Sec. IVB2 that the additional minimum at a ≈ 0.2 lw falls below the precision of the second-order expansion and is not confirmed by simulations. That is a concrete instance of the truncated functional manufacturing an artifact. Because the same functional sets the gamma_cr(epsilon) boundaries in Fig. 8, those boundaries inherit an unquantified error. I would not rely on the precise positions of the phase boundaries. The qualitative effects—coaxial radius increase, chirality switching, eccentric minima—survive because they are also backed by exact Euler–Lagrange solutions and simulations at selected points.\n\nThe self-citation pattern is expected for a review of the group's own work, and the paper is honest about it. No code or data is shipped, which limits reproducibility, but the analytical expressions are given in enough detail to reimplement.\n\nBottom line: this is a solid review for anyone working on skyrmion–vortex heterostructures or Majorana proposals in those systems. It deserves a serious referee. My recommendation: send it to review. The referee should ask the authors to quantify the truncation error, or at minimum mark the phase boundaries as approximate and present the a_add artifact as a caution, not a prediction. I would be comfortable with conditional acceptance after that.","headline":"Self-review of the authors' own prior work: honest and useful, central effects credible, but the unquantified variational truncation should make you treat the phase-boundary numbers as approximate.","tokens_in":36869,"tokens_out":2574,"would_cite":true,"duration_ms":31612,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A Pearl vortex's stray field can inflate a Néel-type skyrmion up to a hard size ceiling, flip its chirality, deform and shift it, and bind a vortex–antivortex pair — all backed by micromagnetic simulations.","keywords":["Néel-type skyrmion","Pearl vortex","superconductor–ferromagnet heterostructure","skyrmion radius","chirality switching","vortex–antivortex pair","micromagnetic simulation","Majorana bound states"],"falsifier":"Measure or simulate the radius of a Néel-type skyrmion coaxially pinned to a Pearl vortex while sweeping the vortex coupling $\\gamma = \\zeta\\ell_w d_S/\\lambda_L^2$ (tunable experimentally through the superconducting film thickness $d_S$): the theory predicts a non-monotonic $R(\\gamma)$ that peaks near $\\gamma = 4\\gamma^\\pm_\\infty$ and never exceeds $|R^\\pm_{\\max}| = \\zeta\\ell_w/(2\\mp\\epsilon\\pi)$, a ceiling set only by ferromagnet constants; a monotonically growing radius, or any measured radius above the ceiling, would falsify the central bound. Independently, a fine-grained micromagnetic scan of the parameter region near the predicted shallow minimum at $a \\approx 0.2\\ell_w$, which the authors' own simulations did not confirm, would settle whether the ansatz hides a stable eccentric state.","tokens_in":35813,"feed_emoji":"🌀","tokens_out":18003,"duration_ms":165834,"temperature":0.7,"pith_summary":"This review paper argues that the stray magnetic field of a superconducting Pearl vortex is not a weak perturbation to a Néel-type skyrmion but a strong handle on its shape, size, and position. Drawing on the authors' recent work, it establishes three linked predictions: a coaxial vortex can inflate the skyrmion radius by more than an order of magnitude up to a ceiling $|R^\\pm_{\\max}| = \\zeta\\ell_w/(2\\mp\\epsilon\\pi)$ set only by ferromagnetic material parameters, and can stabilize a small-radius skyrmion of reversed chirality $\\chi = -1$; an off-center vortex deforms the circular skyrmion profile and sets a stable equilibrium distance, with parameter regions where coaxial and displaced states coexist; and a skyrmion of positive chirality can bind a vortex–antivortex pair that would otherwise repel and annihilate. All predictions are checked against micromagnetic simulations and argued to be qualitatively consistent with recent magnetic-force-microscopy experiments in which skyrmions enlarge below the superconducting transition temperature. The reason to care is that skyrmion–vortex pairs are a proposed platform for Majorana bound states, and the vortex-induced reshaping changes the magnetization profile that determines the quasiparticle spectrum.","feed_headline":"A Pearl vortex can swell a skyrmion to a fixed max radius","feed_subtitle":"It also flips skyrmion chirality and pulls them off-center — key for Majorana-qubit platforms.","key_machinery":"The load-bearing object is the variational magnetization ansatz, which restricts the skyrmion profile to a small functional family and then minimizes the free energy within it. For coaxial geometry the profile is $\\theta^\\gamma_{R\\delta}(r) = \\theta_{R\\delta}(r) + \\gamma\\theta_b(r)\\cos\\theta_{R\\delta}(r)$, where $\\theta_{R\\delta}(r) = 2\\arctan[\\sinh(R/\\delta)/\\sinh(r/\\delta)]$ is the 360-degree domain-wall profile, $\\gamma = M_s\\phi_0/(8\\pi\\lambda\\sqrt{AK})$ is the effective vortex strength, and $\\theta_b(r)$ is the magnetization tilt of the vortex-perturbed uniform background; minimizing the free energy in the two parameters $R$ and $\\delta$ without anchoring them near the free-skyrmion values is what lets weak fields produce large radius changes and chirality inversion. For eccentric geometry the ansatz is $\\mathbf{m} \\approx \\bar{\\mathbf{m}} + \\gamma\\tilde{\\mathbf{m}}$, with $\\bar{\\mathbf{m}}$ the radially symmetric profile in the angular-averaged vortex field and $\\tilde{\\mathbf{m}}$ a first-order deformation built from the difference between the local and the averaged field; the free energy is expanded to second order in $\\gamma \\ll 1$ and minimized over the radius, the domain-wall width, and the center distance $a$. The Pearl vortex itself — a vortex in a superconducting film much thinner than the London penetration depth, with stray-field scale $\\lambda = \\lambda_L^2/d_S$ — supplies the inhomogeneous field through Eqs. (14)–(17).","core_discovery":"The central claim is that the interaction between a Néel-type skyrmion and a Pearl vortex's inhomogeneous magnetic field produces a set of controllable effects rather than a small correction. In a coaxial configuration the skyrmion radius can grow strongly — for the parameters shown in the paper from $R_0 \\approx 0.41\\ell_w$ in the free case to $R \\approx 5.7\\ell_w$, a factor near fourteen — and, for fixed positive DMI, a small-radius skyrmion with opposite chirality $\\chi = -1$ can be stabilized. The radius as a function of vortex strength is non-monotonic and strictly bounded by $|R^\\pm_{\\max}| = \\zeta\\ell_w/(2\\mp\\epsilon\\pi)$, a ceiling that does not depend on the superconductor. In eccentric configurations the vortex deforms the skyrmion away from cylindrical symmetry, and the equilibrium center-to-center distance follows from terms of second order in the small parameter $\\gamma$, yielding a phase diagram with coaxial-only, eccentric-only, and coexistence regimes. Finally, a positive-chirality skyrmion stabilizes a repelling vortex–antivortex pair within a triangular region of that phase diagram. The paper presents these results as predictions confirmed by micromagnetic simulations and qualitatively consistent with recent experiments.","pith_inferences":["Editorial inference: the radius ceiling depending only on ferromagnet constants suggests a materials-characterization application — measure the maximal skyrmion inflation while sweeping vortex strength (via superconducting film thickness) and extract $\\epsilon$ and $\\zeta$ from the observed ceiling.","Editorial inference: because the variational framework is stated to apply to any field profile, the same inflation, chirality-flip, and displacement effects should appear for a vortex in a thick superconducting film or for other stray-field sources such as a magnetic-force-microscope tip; the paper itself calls this extrapolation an open question.","Editorial inference: the shallow free-energy minimum at $a \\approx 0.2\\ell_w$ that the authors report to fall below the precision of their second-order expansion, and that micromagnetic simulations did not reproduce, is the natural stress test for the ansatz — a dedicated simulation sweep near that spot would reveal whether a stable eccentric state was missed.","Editorial inference: if chirality reversal is as robust as predicted, the vortex becomes a local 'write' operation that flips a skyrmion's handedness, a degree of freedom of direct interest for skyrmion-based information storage."],"forward_implications":["There is a hard upper bound on vortex-inflated skyrmion size, $|R^\\pm_{\\max}| = \\zeta\\ell_w/(2\\mp\\epsilon\\pi)$, fixed only by ferromagnet material constants; no amount of vortex strength inflates a given film's skyrmion beyond this ceiling.","Skyrmion radius versus vortex strength is non-monotonic: it grows, peaks near $\\gamma \\approx 4\\gamma^\\pm_\\infty$, and then shrinks, so a single heterostructure can tune a skyrmion through a maximum size by varying the superconducting film thickness.","Chirality is not locked to the DMI sign in the vortex field: a coaxial vortex can stabilize a small-radius skyrmion with reversed chirality $\\chi = -1$, and for $\\epsilon \\lesssim 0.49$ with weak vortices the natural-chirality skyrmion is repelled from the vortex core to a finite off-center distance.","Eccentric and coaxial states can coexist: in the parameter wedge between $\\gamma^-_{cr}(\\epsilon)$ and $\\gamma^+_{cr}(\\epsilon)$ the free energy has two minima in the separation $a$, so either configuration can be realized in the same sample.","A skyrmion can act as a binder for a vortex–antivortex pair: in a triangular region of the $(\\epsilon, \\gamma)$ phase diagram the three-object complex is stable even though vortex and antivortex repel each other, and beyond $\\gamma^*_{cr}(\\epsilon)$ one member is expelled to a separation set by the superconducting energy balance."],"supporting_citations":[{"why":"Supplies the coaxial variational ansatz and the central results of Sec. III: radius blowup and chirality inversion of a Néel-type skyrmion in a Pearl vortex field.","marker":"[43]"},{"why":"Carries the eccentric-configuration magnetization ansatz and the second-order-in-γ free energy that Secs. IV and V build on.","marker":"[44]"},{"why":"First predicted, in the γ→0 limit, that a Néel-type skyrmion sits at a finite distance from a superconducting vortex — the baseline the refined theory corrects.","marker":"[41]"},{"why":"Shows numerically that the coaxial skyrmion–vortex state can be unstable to displacement and that the skyrmion radius can grow, motivating the analytical framework.","marker":"[42]"},{"why":"Defines the Pearl vortex in thin superconducting films and supplies the vortex–antivortex interaction energy used in Sec. V.","marker":"[59]"},{"why":"Provides the integral expressions (14)–(15) for the stray magnetic field of a Pearl vortex above the film.","marker":"[60]"},{"why":"Derives the three-spin interaction kernel that stabilizes higher-order skyrmions in the vortex field, used in Sec. VI.","marker":"[64]"},{"why":"Establishes bound states of high-order skyrmions with a Pearl vortex, the basis of the Sec. VI analysis.","marker":"[65]"},{"why":"First experimental observation of skyrmion-(anti)vortex coupling in a chiral-magnet/superconductor heterostructure, cited as qualitative support.","marker":"[18]"},{"why":"Recent magnetic-force-microscopy experiment reporting roughly doubled skyrmion radii below the superconducting transition, cited as full qualitative agreement with the predictions.","marker":"[20]"}],"fun_headline_variants":["Pearl vortex swells skyrmions 14-fold and flips chirality","Vortex field expands skyrmion radius up to 14x and flips chirality","Pearl vortex stabilizes opposite-chirality skyrmions and enlarges them","Skyrmion radius grows 14x under Pearl vortex field"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Every prediction in the paper rests on the assumption that the true skyrmion profile is always well described by the two-parameter ansatz $\\theta(r) \\approx \\theta_{R\\delta}(r) + \\gamma\\theta_b(r)\\cos\\theta_{R\\delta}(r)$, so that no free-energy minimum is missed by searching only within that family; the paper validates the ansatz for several parameter sets with micromagnetic simulations but provides no general error bound, its shallow predicted minimum at $a \\approx 0.2\\ell_w$ was not confirmed by simulations, and the higher-order-skyrmion analysis of Sec. VI relies on an even simpler domain-wall profile that the text itself warns may be quantitatively inaccurate.","fun_headline_variants_meta":{"raw":{"variants":["Pearl vortex swells skyrmions 14-fold and flips chirality","Vortex field expands skyrmion radius up to 14x and flips chirality","Pearl vortex stabilizes opposite-chirality skyrmions and enlarges them","Skyrmion radius grows 14x under Pearl vortex field"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00064,"raw_usage":{"total_tokens":2947,"prompt_tokens":943,"completion_tokens":2004,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":559,"completion_tokens_details":{"reasoning_tokens":1928}},"tokens_in":559,"tokens_out":2004,"duration_ms":16367,"temperature":1.0,"reasoning_tokens":1928,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:37:32.884008+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure or simulate the radius of a Néel-type skyrmion coaxially pinned to a Pearl vortex while sweeping the vortex coupling $\\gamma = \\zeta\\ell_w d_S/\\lambda_L^2$ (tunable experimentally through the superconducting film thickness $d_S$): the theory predicts a non-monotonic $R(\\gamma)$ that peaks near $\\gamma = 4\\gamma^\\pm_\\infty$ and never exceeds $|R^\\pm_{\\max}| = \\zeta\\ell_w/(2\\mp\\epsilon\\pi)$, a ceiling set only by ferromagnet constants; a monotonically growing radius, or any measured radius above the ceiling, would falsify the central bound. Independently, a fine-grained micromagnetic scan of the parameter region near the predicted shallow minimum at $a \\approx 0.2\\ell_w$, which the authors' own simulations did not confirm, would settle whether the ansatz hides a stable eccentric state.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the coaxial variational ansatz and the central results of Sec. III: radius blowup and chirality inversion of a Néel-type skyrmion in a Pearl vortex field."},{"cited_title":"Nothhelfer, S","cited_arxiv_id":null,"evidence_quote":"First predicted, in the γ→0 limit, that a Néel-type skyrmion sits at a finite distance from a superconducting vortex — the baseline the refined theory corrects."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the integral expressions (14)–(15) for the stray magnetic field of a Pearl vortex above the film."},{"cited_title":"Everschor-Sitte, J","cited_arxiv_id":null,"evidence_quote":"Derives the three-spin interaction kernel that stabilizes higher-order skyrmions in the vortex field, used in Sec. VI."},{"cited_title":"Hassan, S","cited_arxiv_id":null,"evidence_quote":"Establishes bound states of high-order skyrmions with a Pearl vortex, the basis of the Sec. VI analysis."}],"review_version":1}