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The paper claims that a single dimensionless combination, α²|Q|, decides whether Regularized Maxwell black holes exhibit first-order coexistence or second-order criticality, with the threshold at α²|Q| = 1.

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load-bearing objection The paper's central α²|Q| = 1 classifier is built on a mass formula that fails the horizon condition; substituting Eq. (10) into Eq. (2) gives f(r_h) = −α²|Q|, so the threshold should be 1/2. the 3 major comments →

arxiv 2510.20983 v1 pith:WJGRST4V submitted 2025-10-23 gr-qc hep-th

Topological Signatures and Geometrothermodynamics of Critical Phenomena in Regularized Maxwell Black Holes

Y. Sekhmani , G.G. Luciano , S.K. Maurya , J. Rayimbaev , M.K. Jasim , I. Ibragimov , S. Muminov This is my paper
classification gr-qc hep-th
keywords black hole thermodynamicsthermodynamic topologyDuan topological currentRuppeiner geometryRegularized Maxwell electrodynamicsnonlinear electrodynamicsphase transitionsenergy conditions
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to establish that the thermodynamics of charged AdS black holes in Regularized Maxwell (RegMax) gravity is organised by a single dimensionless number, α²|Q|, where α is the nonlinear coupling and Q the electric charge. The claim is that below α²|Q| = 1 the small-black-hole branch is thermodynamically unstable, no intermediate phase exists, and transitions are first-order small/large coexistence driven by free-energy competition; above the threshold the small branch becomes stable, an intermediate branch appears, and vertical tangencies in the temperature curve produce second-order critical points. Why this matters: if true, one parameter of nonlinear electrodynamics decides whether black holes show Van-der-Waals-like first-order behaviour or richer continuous criticality, and it ties that exotic behaviour to where classical energy conditions break down. The paper supports the topological claim with winding-number analysis and with Ruppeiner geometry, showing the microscopic-interaction curvature changes sign between regimes.

Core claim

The paper claims that the Duan off-shell free-energy vector field φ = (∂F/∂S, −cotΘ cscΘ) has a small-radius expansion whose leading coefficient is φ_S(0+) = ½(1 − α²|Q|). From this single coefficient the authors read the entire thermodynamic topology: for α²|Q| < 1 the small black hole emerges on a rising branch of τ(r_h) = 1/T(r_h), so the heat capacity is negative and the winding number is −1, and the phase structure is first-order SBH/LBH coexistence; for α²|Q| > 1 the small black hole emerges on a falling branch, winding +1, and the defect curve generically develops an intermediate branch plus vertical tangencies dτ/dr_h = 0 where the heat capacity diverges and defect pairs are created

What carries the argument

The load-bearing object is the small-radius expansion of the Duan vector field, Eq. (52): φ_S(0+) = ½(1 − α²|Q|). It converts the nonlinear RegMax coupling α and charge Q into a single dimensionless control parameter. The Duan construction assigns each zero of φ — each equilibrium black hole at τ = 1/T — an integer winding number w = +1 (locally stable, heat capacity C > 0, falling τ(r_h)) or w = −1 (locally unstable, C < 0, rising τ(r_h)); vertical tangencies dτ/dr_h = 0 are where C diverges and second-order criticality occurs. The Ruppeiner metric, the temperature-scaled Hessian of the mass, is the companion object: its scalar curvature R_Rup encodes the sign of the dominant microscopic in

Load-bearing premise

The load-bearing premise is that the sign of the leading small-radius coefficient φ_S(0+) = ½(1 − α²|Q|) — Eq. (52) — determines the global shape of the defect curve τ(r_h), so that the entire three-regime classification follows from this one term; the paper verifies this only at two representative parameter points and does not prove that the intermediate branch and vertical tangencies appear for every α²|Q| > 1 or that a genuine second-order point exists exactly at α²|Q| = 1

What would settle it

Scan the defect curve τ(r_h) for a fine grid of α at fixed Q = 1, P = 0.1, crossing α²|Q| = 1 (e.g., α = 0.9, 1.0, 1.1). If for any α²|Q| > 1 the curve lacks an intermediate branch or a vertical tangency with dτ/dr_h = 0 and diverging heat capacity — or if at α²|Q| = 1 no tangency/divergence occurs — then the sign of the small-radius coefficient alone does not control the topology and the central claim fails.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • For α²|Q| < 1 the SBH branch is always born unstable (w = −1), so no intermediate branch exists; the only phase competition is first-order SBH/LBH coexistence.
  • For α²|Q| > 1 the SBH branch is stable (w = +1) and the defect curve has an SBH–IBH–LBH structure; the vertical tangencies mean the heat capacity diverges at two points, so the system can undergo continuous second-order critical transitions.
  • At α²|Q| = 1 the small-radius coefficient vanishes and J_0 = T/C → 0, so C → ∞; this is a marginal point where topological defect pairs can be created or annihilated.
  • The Ruppeiner curvature is predominantly positive in the subcritical regime (repulsive microstructure) and becomes negative at small radii in the critical/supercritical regimes (attractive microstructure), connecting regime change to microscopic interaction sign.
  • Because larger α pushes the NEC/WEC threshold r_NEC = (√2−1)√|Q|/α inward, a horizon at r_h > r_NEC sits in an energy-condition-violating region; the regime that admits second-order criticality is the same regime that violates classical energy conditions at the horizon.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the paper checks the claimed topology at only two representative points (α = 2 and α = 0.2, with Q = 1, P = 0.1). A numerical scan through α²|Q| = 1 would test whether the intermediate branch appears exactly when the small-radius coefficient changes sign or at some different value.
  • Editorial inference: the same α²|Q| classifier should generalize to other nonlinear electrodynamics black holes whose off-shell free energy admits the same small-radius expansion structure; if a similar theory failed to show the three-regime pattern, the mechanism would not be generic.
  • Editorial inference: if the second-order critical points exist, they should leave signatures in the temperature-dependence of thermodynamic response functions (specific heat, isothermal compressibility) near the tangencies, and possibly in quasinormal-mode or shadow observables for astrophysical black holes; the dimensionless threshold gives a concrete target for constraining α.
  • Editorial inference: the claimed correlation between EC violation and enriched topology is novel enough that it deserves a direct test — constructing a RegMax solution with α²|Q| > 1 but arranging the outer horizon to lie below r_NEC should still show the intermediate branch if the topology is truly driven by α²|Q| alone.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The manuscript studies charged AdS black holes in Regularized Maxwell (RegMax) gravity, combining Duan's topological current method with Ruppeiner thermodynamic geometry. Starting from the RegMax lapse function and a mass function M(r_h) taken from the literature, the authors construct an off-shell free energy, define the Duan vector field, and compute defect curves τ(r_h). Their central claim is that the dimensionless combination α²|Q| is a sharp analytic classifier: for α²|Q| < 1 the small-black-hole branch is unstable (winding w = −1) and the system exhibits first-order SBH/LBH coexistence; for α²|Q| > 1 the SBH branch is stable (w = +1), an intermediate branch appears, and vertical tangencies in τ(r_h) produce second-order critical behaviour; at α²|Q| = 1 the system is marginal with C → ∞. The paper also derives energy-condition radii (r_NEC, r_WEC) and the Ruppeiner curvature, reporting sign changes that are interpreted as a transition between attractive and repulsive microstructures. The analysis is supplemented by numerical plots for representative parameter values.

Significance. If the central classification were correct, the paper would supply a clean, analytic link between the RegMax coupling, energy-condition violations, and topologically richer phase structure. The closed-form energy-condition radii in Eqs. (25) and (28), the explicit off-shell free energy, and the Ruppeiner curvature are useful and transparent; no fitted parameters hide the results. However, the central classifier rests on a mass function that does not satisfy the horizon condition f(r_h)=0. This is a load-bearing algebraic inconsistency: the off-shell free energy, the Duan vector, the defect curve, and the Ruppeiner curvature are not those of the spacetime described by the lapse function (2). Correcting the mass term shifts the apparent threshold from α²|Q| = 1 to α²|Q| = 1/2. The qualitative framework (subcritical/supercritical regimes) may survive after this correction, but all quantitative statements, figures, and the claimed critical value would need to be reworked. Because the present version's main quantitative claim is not established, the paper in its current form is not publishable without substantial revision.

major comments (3)
  1. [§II.A, Eq. (10) and §III, Eqs. (42), (45), (52)–(54)] The mass formula Eq. (10) is algebraically inconsistent with the lapse function Eq. (2). Substituting Eq. (10) into f(r_h)=0 leaves f(r_h) = −α²|Q| identically; all r-dependent terms cancel. Thus the off-shell free energy (42), the Duan vector (45), the defect curve (53), and the Ruppeiner curvature (61) are not those of the RegMax-AdS spacetime (1)–(2). The small-r coefficient in Eq. (52), (1−α²|Q|)/2, is an artifact of the incorrect −3α²r_h|Q| term. Solving f(r_h)=0 gives a mass whose small-r expansion has linear coefficient (1−2α²|Q|)/2, shifting the claimed critical threshold from α²|Q|=1 to α²|Q|=1/2. This invalidates the central classifier and all regime assignments built on it.
  2. [§III, Eqs. (35), (45)–(48)] The vector field is defined in Eq. (35) as φ = (∂F/∂S, −cotΘcscΘ), but Eq. (45) is not ∂F/∂S. Since S = πr_h², ∂F/∂S = (1/(2πr_h)) ∂F/∂r_h, whereas the right-hand side of Eq. (45) equals ∂F/∂r_h as verified by direct differentiation of Eq. (42). Consequently the small-r limit in Eq. (52) is ∂F/∂r_h at 0+, not φ_S(0+), and the identities J0 = ∂_S φ_S = ∂_S T = T/C in Eqs. (47)–(48) do not follow from the displayed definitions. An r_h-based Duan field is legitimate and preserves winding numbers, but the coordinate choice must be made consistently and the Jacobian/winding-number dictionary must be derived for that coordinate.
  3. [§III, after Eq. (52); Figs. 6–7] The global phase classification is inferred from the sign of the leading small-r coefficient in Eq. (52), but this is verified only for two representative parameter points (α=2 and α=0.2 with Q=1, P=0.1). The existence of an intermediate branch and vertical tangencies for all α²|Q|>1 (or α²|Q|>1/2 after correcting Eq. (10)) is an extrapolation. An exact analysis of the number of extrema of τ(r_h) or a systematic parameter scan across the critical threshold is required to establish that the leading coefficient alone determines the full defect-curve topology.
minor comments (6)
  1. [Notation (throughout)] The symbol Q is used both for the charge and its absolute value; in Sec. IV the statement “without loss of generality Q > 0” conflicts with the |Q| notation used elsewhere. Please define conventions once and use them consistently.
  2. [Eq. (2)] The second equality with the infinite series does not reproduce the explicit −2α²|Q| term; clarify the relation between the closed form and the series expansion.
  3. [Typos] There are numerous typographical errors, e.g., “RegMAx” (Sec. II.E), “geothermodynamic” (Sec. II intro), and “Reissner-Nordstrm” (missing umlaut). A careful proofread is needed.
  4. [Figs. 6(c), 7(c)] The winding-number contours are difficult to read; please enlarge the plots and label the contours C1, C2, C3 more clearly.
  5. [Ref. [26]] Reference [26] is cited only as an arXiv preprint; update to the published version if available.
  6. [Eq. (60)] The scalar-curvature formula for a 2D metric with g22 = 0 is non-standard; provide a brief derivation or reference for the expression used.

Circularity Check

0 steps flagged

No significant circularity: the α²|Q| classifier is an explicit analytic consequence of the stated mass function, not a fitted or self-referential input.

full rationale

The central derivation is self-contained: the mass formula M(rh) (Eq. 10, inherited from the external RegMax solution [49]) is substituted into the off-shell free energy F = M − S/τ (Eqs. 34 and 42), giving the vector component φS (Eq. 45) and its small-radius expansion φS(0+) = ½(1 − α²|Q|) (Eq. 52). The threshold α²|Q| = 1 is obtained by setting this leading coefficient to zero, not by fitting to the subsequent numerical examples. The topological stability dictionary C > 0 ↔ w = +1 follows from the standard Jacobian identity J0 = ∂SφS = T/C (Eqs. 47–51), which is cited to Duan and Wei–Liu and then applied, not assumed as a prediction. The Ruppeiner curvature is computed from the same mass function through Eqs. (58)–(61). Self-citations (e.g., refs. [13, 25, 69]) appear as contextual applications of the same thermodynamic-topology and geometrothermodynamic tools and are not load-bearing for the main claim. The skeptic's algebraic concern about whether Eq. (10) actually solves f(rh)=0 is a correctness or consistency issue, not a circularity: even if the coefficient in Eq. (52) were wrong, the derivation would still be an independent consequence of the stated inputs rather than a reduction of the conclusion to the premise. The extrapolation from the small-r expansion to the full phase structure (IBH branch, vertical tangencies) is an inferential gap, but it is not circular. Overall, no step reduces by construction to its own output; the score of 1 reflects only the presence of minor, non-load-bearing self-citations.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No parameters are fitted to data; the contour amplitudes S1, S2 in Eq. (39) are conventional and do not affect the central results. The paper introduces no new entities, particles, or forces. Its main load-bearing inputs are the external RegMax solution from [49] and the standard interpretive frameworks of Duan topology and Ruppeiner geometry, plus the implicit assumption that small-r behavior governs global topology.

axioms (5)
  • domain assumption RegMax-AdS solution and its first law/Smarr relation from [49] are correct.
    The entire analysis builds on the lapse (2), mass (10), temperature (12), and first law (17) taken from Hale et al. [49].
  • standard math Duan's topological current: zeros of φ correspond to thermodynamic critical points with winding numbers.
    Background method from [26–28]; accepted in the black-hole topology literature.
  • domain assumption Ruppeiner metric in the (S,P) canonical ensemble with g22 = 0 (since M is linear in P).
    Choice of fluctuation variables and ensemble; a standard but non-unique construction in thermodynamic geometry.
  • ad hoc to paper Small-r asymptotic expansion ln(1 + √Q/(αr)) ≈ ln(√Q/(αr)) is sufficient to classify global phase structure.
    The transition at α²Q = 1 follows from the leading coefficient in F(rh), but its use as a global classifier is assumed, not proven.
  • domain assumption The sign of Ruppeiner curvature indicates attractive/repulsive microscopic interactions.
    Standard interpretative assumption of Ruppeiner geometry, not derived in this paper.

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read the original abstract

We study the thermodynamic topology and microscopic interaction properties of charged black holes in RegMax gravity, focusing on the role of the coupling parameter $\alpha$. Using the Duan topological current method together with Ruppeiner geometry, we show that $\alpha$ controls a sharp change in phase structure. Above a certain critical threshold, we find that the Duan defect curve develops an intermediate branch and vertical tangency points, producing continuous (second-order) critical behaviour. Furthermore, the Ruppeiner curvature becomes negative at very small horizon radii before turning positive and progressively vanishing at larger radii. By contrast, below the critical value of the coupling, the intermediate black hole phase disappears, and the system shows a simpler small/large first-order/coexistence behaviour driven by free-energy competition. In this regime, the Ruppeiner curvature remains predominantly positive. Overall, increasing $\alpha$ enriches the thermodynamic topology (allowing for second-order criticality) while simultaneously reducing the domain in which classical energy conditions (ECs) are satisfied, thus linking exotic thermodynamic behaviour to more severe violations of standard energy conditions.

Figures

Figures reproduced from arXiv: 2510.20983 by G.G. Luciano, I. Ibragimov, J. Rayimbaev, M.K. Jasim, S.K. Maurya, S. Muminov, Y. Sekhmani.

Figure 1
Figure 1. Figure 1: FIG. 1. Embedding-style surfaces [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Plot of NEC using the fixed benchmark [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Plot of SEC using the fixed benchmark [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Defect curve [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Thermodynamic topology for RegMax-AdS BHs using [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Plots of [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Plots of [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗

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