{"id":"511bb410-1469-48d8-be9d-8e1022d35a13","arxiv_id":"2610.08099","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"For a D>=4 family of charged regular black holes sourced by nonlinear electrodynamics, the ordinary first law with Hawking temperature and area entropy holds on the fixed-theory solution space, and the reported correction factor is a Jacobian of an invalid projection, not a new energy.","lead":"A class of regular black holes with no central singularity was reported to break the usual first law of thermodynamics. This paper shows that, for a broad family of such solutions, the standard law holds exactly once one distinguishes changes within a fixed theory from changes that move to a different theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"As typeset, Eq. (3) carries the wrong sign in the exponent: integrating the density from Eq. (2) gives [1+(r/lambda)^gamma]^{-(D-1)/gamma}, while the printed plus sign makes the mass function diverge, so Eq. (5) and the Xi analysis do not follow unless the missing minus is restored.","rationale":"The reader's conditional verdict is reasonable, but the load-bearing issue is not the one they emphasized. Their concern is that the Section V projection depends on the extended-phase-space treatment and on the normalization condition, and that one supporting expression is quoted from an unpublished companion paper. That concern is real but not decisive: Eq. (17) is explicitly Eq. (9) divided by Xi, so the non-closure result (18) can be checked directly from the metric family (5) without invoking the companion expression or the extended first law. The more serious issue visible in the manuscript as supplied is the sign inconsistency in Eq. (3). With Eq. (2) as written, the density decays at large r as [1+(r/lambda)^gamma]^{-(gamma+D-1)/gamma}, and integrating the Einstein equation produces a mass function with a negative exponent. The printed Eq. (3) has a positive exponent, which makes mhat(infinity) infinite and is inconsistent with the asymptotic flat metric (5) used throughout the paper. This could be a simple typesetting omission of a minus sign, and the analytic derivation (10)-(13) is independent of this formula, so the standard first law at fixed couplings is not threatened. But the paper's central explanatory claims about the meaning of Xi, the fraction of mass inside the horizon, and the non-existence of a local state function all depend on family (5). A one-line re-derivation of Eq. (3) from Eq. (2) settles the issue. I therefore recommend keeping the reader's CONDITIONAL verdict, with the condition upgraded to include this sign check; if the printed sign is confirmed, the verdict would need to be REJECT because the concrete family would be ill-defined.","tokens_in":12484,"tokens_out":30428,"duration_ms":454865,"concrete_test":"Differentiate Eq. (3) as printed, compare the result with mhat' = 16 pi rho r^{D-2}/(D-2) using rho derived from Eq. (2), set lambda = Q and m0 = 0, and check whether the metric (5) is reproduced. If the exponent must be -(D-1)/gamma, correct Eq. (3) and verify that Eqs. (4)-(18) are unchanged; if the plus sign is literal, derive the finite ADM mass and repeat the calculation of Eqs. (4)-(18).","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's concrete family is internally inconsistent as printed. Combining Eq. (2) with P=-2Q_ADM^2/r^{2(D-2)} and rho=-H/16pi gives rho = sigma/16pi [1+(r/lambda)^gamma]^{-(gamma+D-1)/gamma}. Integrating mhat' = 16 pi rho r^{D-2}/(D-2) yields mhat = m0 + [sigma r^{D-1}/((D-2)(D-1))] [1+(r/lambda)^gamma]^{-(D-1)/gamma}, i.e., the exponent in Eq. (3) needs a minus sign. With the printed positive exponent, mhat grows without bound, mhat(infinity) diverges, Eq. (4) does not follow, and the metric (5), f = 1 - M r^2/(r^gamma+Q^gamma)^{(D-1)/gamma}, is not obtained. Since Eqs. (8), (9), (17), and (18) are all computed on family (5), this sign is load-bearing for the advertised claims about Xi, the horizon mass fraction, and non-closure. The fixed-coupling first law (10)-(14) is independent of Eq. (3) and is sound. The companion-paper expression and the extended-phase-space normalization flagged by the reader are secondary, because Eqs. (17)-(18) also follow directly by dividing Eq. (9) by Xi.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The manuscript studies a Hamiltonian (P-frame) formulation of general relativity coupled to nonlinear electrodynamics in D>=4 dimensions, with the source density H(P) given by Eq. (2). It distinguishes three parameter spaces: solutions of a fixed theory, the regular subfamily with m0=0, and the two-parameter family of regular metrics obtained by treating M and Q as independent parameters. The central claims are: (i) on the two-parameter solution space of a fixed theory, the standard first law dM_ADM = T_h dS + Phi_h dQ_e holds, derived analytically in Eqs. (10)-(13) and checked numerically; (ii) the factor Xi that appears when M_ADM is differentiated along the regular metric family at fixed Q is the Jacobian of the projection from the extended first law onto that family and equals M_ADM/m_h; and (iii) Xi^{-1} dM_ADM is not a closed one-form on the two-parameter regular metric family, so it cannot be the differential of a local state function there. The paper also notes that the Legendre map between the F and P frames degenerates outside the horizon for near-extremal solutions in D>=5.","tokens_in":13110,"tokens_out":15576,"duration_ms":185972,"significance":"If the concrete family is corrected, the paper provides a useful clarification rather than a new physical law: it explicitly shows how the reported temperature mismatch arises from differentiating across different theories rather than within one theory, and it identifies the correction factor of Ref. [11] with a non-integrable one-form. The analytic derivation (10)-(14) is clean, self-contained, and independent of the specific form of H(P), and Eq. (18) gives an explicit check of non-closure. The authors also honestly acknowledge that the restoration of the first law at fixed couplings is not a new principle. The main advertised results, however, are tied to the concrete family (2)-(5), and that family is currently internally inconsistent as printed because of a sign error in Eq. (3).","major_comments":[{"comment":"The exponent in Eq. (3) is printed with the wrong sign, and the same sign error appears in the expression for rho immediately before it. Combining Eq. (2) with P=-2Q_ADM^2/r^{2(D-2)} gives rho = sigma/(16 pi) [1+(r/lambda)^gamma]^{-(gamma+D-1)/gamma}, whose integral is mhat(r) = m0 + [sigma r^{D-1}/((D-2)(D-1))] [1+(r/lambda)^gamma]^{-(D-1)/gamma}. With the printed positive exponents, mhat diverges at infinity, Eq. (4) does not follow, and the metric (5)---and hence Eqs. (9), (17), and (18)---are not valid as typeset. This is load-bearing for the advertised Xi analysis, although the fixed-theory first law (10)-(14) is independent of Eq. (3). The sign must be corrected and all formulas depending on Eq. (3) re-verified.","section":"Section II, Eq. (3) and preceding density formula","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The closed form for Phi_h Q_e is quoted from the companion paper, Ref. [14], which is listed as submitted rather than published; since this identity enters the extended-phase-space projection discussion, please either supply a short derivation in this paper or state more explicitly that the non-closure result (17)-(18) does not depend on it, as the text already notes that Eq. (17) is just Eq. (9) divided by Xi.","section":"Section IV, final paragraph","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"Ref. [14] is listed twice with the same arXiv identifier in the bibliography; the duplicate entry should be removed.","section":"References","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The numerical check of Eq. (14) for D=4-7 and gamma=1-3 is stated but not documented; a short table or a note on the sampling of the parameter space would make this check reproducible.","section":"Section IV, numerical check","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The sign error in Eq. (3) appears to be a typographical slip rather than a conceptual flaw: restoring the minus sign makes the construction (4)-(18) coherent. I would be willing to accept after a careful revision that corrects the sign, re-verifies the downstream equations, and makes the use of the companion paper self-contained or explicitly non-essential."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"Short version: this is a solid, honest clarification of a recurring confusion in regular black hole thermodynamics, and it deserves a serious referee, but there is one real typo-level flaw that must be fixed before publication.\n\nI checked the stress-test sign issue: it is correct. The density printed after Eq. (2) carries the wrong exponent, and integrating the correct H(P) density gives [1+(r/lambda)^gamma]^{-(D-1)/gamma} in Eq. (3), not the printed positive exponent. As typeset, the mass function diverges and Eq. (5) does not follow. The rest of the calculation uses the correct metric, and the fixed-coupling first law (10)-(14) is independent of Eq. (3), so the conceptual message survives. But an arXiv reader trying to follow the derivation will be stopped cold, and the authors need to correct both the density and the mass function.\n\nWhat is actually new is stated fairly. The authors do not claim that restoring the first law at fixed couplings is new; they credit Rasheed, Yang-Gao, and Bokulic et al. The genuinely new pieces are the clean separation of the three parameter spaces, the Jacobian interpretation of Xi^{-1} as the mass fraction inside the horizon, the general formula Xi = (d-hat-m/dM)^{-1}, and the non-closure argument ruling out the Ma-Zhao corrected internal energy as a local state function. I checked the central derivation (10)-(13) and the non-closure calculation (18); both are clean and self-contained, and the numerical checks are reasonable supporting evidence.\n\nThe soft spots are proportionate. The non-closure statement is established on the extended phase space and with the authors' specific normalization of beta; the authors say this explicitly, so it is a scope condition rather than a hidden assumption. The quote from the unpublished companion paper for the closed form of Phi_h Q_e is a real weakness in principle, but it is not load-bearing: Eq. (17) follows directly from Eq. (9) by dividing by Xi, so the companion expression can be treated as a convenience, not a needed input. The D>=5 Legendre degeneracy issue is acknowledged and correctly flagged as a separate dynamical question.\n\nCitation practice looks fine. The self-citation to the companion paper is the only weak link, and it is minor because the derivation can be made independent.\n\nRecommendation: send it to peer review. After the sign typo is fixed and the companion-paper dependence is either removed or the companion paper appears, I would accept it. It is a useful paper for anyone working on regular black holes or NED thermodynamics, and it clears up a persistent apparent violation without overclaiming.","headline":"A mostly correct and genuinely clarifying paper that deserves refereeing, but Eq. (3) as typeset has a real sign error that must be fixed: the printed positive exponent makes the mass function diverge instead of giving the Bardeen/Hayward metric used in the rest of the paper.","tokens_in":14200,"tokens_out":738,"would_cite":true,"duration_ms":41662,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["83C57","83C22"],"pacs":["04.70.-s","04.70.Bw"],"model":"deepseek/deepseek-v4-flash","headline":"This paper proves that regular black holes in nonlinear electrodynamics obey the standard first law within any fixed theory in D≥4, and that the reported factor Ξ is only the Jacobian of a projection across theories, not a failure of…","keywords":["regular black holes","nonlinear electrodynamics","first law of thermodynamics","Hawking temperature","area entropy","Bardeen and Hayward metrics","extended phase space","non-closed one-form"],"falsifier":"Integrate the one-form $T_h\\,dS+\\Phi\\,dQ$ around a closed loop in the $(r_h,Q)$ plane of the regular-metric family: Eq. (18) predicts a strictly positive loop integral proportional to the loop's area for $Q>0$ and $D\\ge4$, so any closed loop with vanishing integral would refute the claim that $\\Xi^{-1}dM_{\\rm ADM}$ is not closed, and exhibiting a smooth function $U(r_h,Q)$ with $dU=\\Xi^{-1}dM_{\\rm ADM}$ would refute it as well. An independent check of the base claim is to verify numerically, for $D=4$–$7$ and $\\gamma=1$–$3$, that $(\\partial M_{\\rm ADM}/\\partial S)_{Q_{\\rm ADM}}=T_h$ and $(\\partial M_{\\rm ADM}/\\partial Q_e)_S=\\Phi_h$ hold as asserted.","tokens_in":13983,"feed_emoji":"🕳️","tokens_out":20029,"duration_ms":116134,"temperature":0.7,"pith_summary":"This paper addresses a recurring controversy over regular black holes — nonsingular spacetime geometries obtained by coupling general relativity to nonlinear electrodynamics. Several authors had reported that differentiating the mass along the family of regular metrics at fixed length parameter $Q$ gives a temperature that differs from the Hawking temperature by a factor $\\Xi$, and had proposed modified entropies or a corrected internal energy to rescue the first law. The paper proves that within any one fixed theory the standard first law $dM_{\\rm ADM}=T_h\\,dS+\\Phi_h\\,dQ_e$ holds exactly on the whole two-parameter family of static solutions, in any dimension $D\\ge4$, and that the mismatch arises only because the regular-metric variation moves through different theories and changes the conserved charge. The authors state plainly that restoring the law at fixed couplings is not a new principle; their contribution is the explicit mechanism, the interpretation of $\\Xi$ as the Jacobian of a projection across theories, and the proof that the proposed correction is not integrable. The point matters because it dissolves the apparent paradox without any modification of the entropy or of the energy.","feed_headline":"Regular black holes obey the first law","feed_subtitle":"The mismatch factor is a projection across theories — entropy and temperature stay standard.","key_machinery":"The machinery is a strict separation of three parameter spaces that earlier work had mixed: the two-parameter solution space of a fixed theory, labelled by the integration constants $m_0$ and $Q_{\\rm ADM}$, on which the first law is proved; the one-parameter subfamily of regular solutions with $m_0=0$, on which the mass is determined by the charge; and the two-parameter family of regular metrics parametrized by $(M,Q)$, whose two members generically belong to two different theories. The load-bearing calculation is the integral representation $M_{\\rm ADM}=m_h+\\Omega_{D-2}\\int_{r_h}^{\\infty}\\rho\\,r^{D-2}dr$, which yields $(\\partial M_{\\rm ADM}/\\partial S)_{Q_{\\rm ADM}}=T_h$ and $(\\partial M_{\\rm ADM}/\\partial Q_e)_S=\\Phi_h$, together with the projection of the extended first law — in which the couplings $\\sigma,\\beta$ are treated as thermodynamic variables — onto the metric family; that projection produces the prefactor $m_h/M_{\\rm ADM}=1/\\Xi$ and the one-form of Eq. (17). The non-integrability of $\\Xi^{-1}dM_{\\rm ADM}$ is then settled by the direct mixed-derivative computation of Eq. (18), whose right-hand side is strictly positive for $Q>0$ and $D\\ge4$.","core_discovery":"The central claim is that the thermodynamics of regular black holes is standard once the variations are taken inside a fixed theory. For the theory defined by the Hamiltonian density $H(P)$ of Eq. (2), whose couplings $\\sigma,\\beta,\\gamma$ are fixed and contain no solution parameters, the paper proves analytically that $dM_{\\rm ADM}=T_h\\,dS+\\Phi_h\\,dQ_e$ holds on the whole two-parameter family of static solutions in any dimension $D\\ge4$, with the Hawking temperature, the area entropy, and the electrostatic potential of the horizon; the derivation uses only the Einstein equations, the area entropy, and the theory-fixed energy density, and the authors note it reproduces the general first law for nonlinear electrodynamics already derived elsewhere. The reported factor $\\Xi$ appears only when the mass is differentiated along the one-parameter subfamily of regular metrics at fixed $Q$ — a variation that moves through the space of theories, changes the coupling constants and the conserved charge $Q_e$, and so is not a variation within one theory at all. The paper identifies $\\Xi^{-1}$ with $m_h/M_{\\rm ADM}$, the fraction of the ADM mass inside the horizon, gives its general form $\\Xi=(\\partial\\hat m(r_h;M,Q)/\\partial M)^{-1}$ for an arbitrary regular mass function, and proves that $\\Xi^{-1}dM_{\\rm ADM}$ is not a closed one-form on the two-parameter family of regular metrics, so it cannot be the differential of any local internal energy there; the horizon mass $m_h$ is a state function, but its differential is not $\\Xi^{-1}dM_{\\rm ADM}$, while $dM_{\\rm ADM}$ itself is exact and needs no correction. As a separate point, the paper observes — and flags as a limitation — that the Legendre map between the $F$ and $P$ frames degenerates at a radius that lies outside the horizon of near-extremal solutions in $D\\ge5$, leaving the dynamical status of those backgrounds open.","pith_inferences":["The same projection logic should transfer to other regular metrics built by prescribing $f$ and deriving the source: whenever the source can be rewritten with fixed couplings, the factor $\\Xi=(\\partial\\hat m(r_h;M,Q)/\\partial M)^{-1}$ should again equal the inverse mass fraction inside the horizon, a relation one could check for each new regular-metric proposal.","A practical consequence for the field is reporting hygiene: a 'first law of regular black holes' is meaningful only after declaring which space the variation lives in, and adopting the paper's three-space distinction as a convention would remove a whole class of apparent paradoxes from future papers.","If $1/\\Xi\\,dM_{\\rm ADM}$ is genuinely non-closed, then any corrected first law built on that prefactor is path-dependent: the value of the putative internal energy at a given $(r_h,Q)$ would depend on the history of the process, a signature that the one-form is a projection artifact rather than new physics.","The branch-point result for $D\\ge5$ suggests a concrete probe: electromagnetic perturbations on near-extremal regular backgrounds should feel the singular effective metric at $r_b>r_{\\rm ext}$, which could be searched for once the dynamics across $r_b$ is understood."],"forward_implications":["For the whole family of theories (2) in any dimension $D\\ge4$, the entropy remains the area entropy, the temperature remains the Hawking temperature, and no modified entropy or corrected internal energy is needed.","The correction proposed in Ref. [11] — putting $1/\\Xi$ in front of $dM_{\\rm ADM}$ and calling the result a new internal energy — is shown to be non-integrable: $1/\\Xi\\,dM_{\\rm ADM}$ is not closed, so no local energy function has that differential, whereas $dM_{\\rm ADM}$ itself is exact and its first law (14) needs no correction.","Reported temperature mismatches for regular black holes are explained as variations that leave the fixed-theory solution space; keeping $Q$ fixed instead of $Q_e$ changes the mismatch's value and even its sign, which shows the mismatch is a property of the chosen variation, not of the solutions.","On the regular subfamily, $S$ and $Q_e$ are not independent, so no independent derivative $(\\partial M_{\\rm ADM}/\\partial S)_{Q_e}$ exists within it; the thermodynamic temperature of a regular black hole must be read from the larger space of neighboring solutions, which is the usual situation for an equilibrium first law.","In $D\\ge5$ the $F\\leftrightarrow P$ Legendre map degenerates at a radius outside the horizon for near-extremal solutions, so those backgrounds require a separate analysis of their characteristics before they can be declared acceptable dynamical solutions."],"supporting_citations":[{"why":"Fan and Wang's construction of regular black holes in general relativity; supplies the P-frame formulation and the four-dimensional first law with Lagrangian parameters that this paper generalizes to arbitrary dimension.","marker":"[7]"},{"why":"Rasheed's general first law of black hole mechanics for nonlinear electrodynamics, which the paper's Eq. (14) reproduces as the static electric case.","marker":"[8]"},{"why":"Zhang and Gao's covariant phase-space derivation of the first law and Smarr formula in nonlinear gauge theories, including conjugate potentials for dimensionful parameters.","marker":"[9]"},{"why":"Bokulic, Juric and Smolic's first law for black holes in the presence of nonlinear electromagnetic fields, the basis for treating field-theory parameters as thermodynamic variables.","marker":"[10]"},{"why":"Ma and Zhao's proposal of a corrected first law with the prefactor 1/Ξ, the specific claim the paper interprets as a projection Jacobian and shows to be non-integrable.","marker":"[11]"},{"why":"Bokulic, Juric and Smolic's proof that mass and charge of regular NED black holes cannot be independent, which grounds the paper's mass–charge relation (4) on the regular subfamily.","marker":"[13]"},{"why":"The companion paper supplying the solution family, horizons, energy conditions, Smarr formula, and the closed form of Φ_h Q_e used in the projection calculation.","marker":"[14]"},{"why":"Kastor, Ray and Traschen's extended first law with the cosmological constant as a thermodynamic variable, the template for the σ and β terms in Eq. (15).","marker":"[17]"}],"fun_headline_variants":["Regular black holes pass the first law of thermodynamics","First law holds with Hawking temperature for regular black holes","No modified entropy needed for regular black holes","Mismatch factor is just a projection between theories","Thermodynamic mismatch in regular black holes resolved"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The non-integrability claim for $\\Xi^{-1}dM_{\\rm ADM}$ rests on the extended phase-space treatment in which the coupling constants $\\sigma$ and $\\beta$ are promoted to thermodynamic variables with the specific normalization $\\beta=(\\gamma+D-1)\\sigma/2(D-2)$; if that treatment is rejected, the no-local-state-function statement is not established, although the fixed-theory first law (14) is independent of this choice.","fun_headline_variants_meta":{"raw":{"variants":["Regular black holes pass the first law of thermodynamics","First law holds with Hawking temperature for regular black holes","No modified entropy needed for regular black holes","Mismatch factor is just a projection between theories","Thermodynamic mismatch in regular black holes resolved"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000321,"raw_usage":{"cost":0.0003213,"is_byok":false,"cost_details":{"upstream_inference_cost":0.0003213,"upstream_inference_prompt_cost":0.0001897,"upstream_inference_completions_cost":0.0001316},"total_tokens":1825,"prompt_tokens":1355,"completion_tokens":470,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":0,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":410}},"tokens_in":1355,"tokens_out":470,"cost_source":"openrouter_usage","duration_ms":5466,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":410,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-07T16:14:56.708362+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Integrate the one-form $T_h\\,dS+\\Phi\\,dQ$ around a closed loop in the $(r_h,Q)$ plane of the regular-metric family: Eq. (18) predicts a strictly positive loop integral proportional to the loop's area for $Q>0$ and $D\\ge4$, so any closed loop with vanishing integral would refute the claim that $\\Xi^{-1}dM_{\\rm ADM}$ is not closed, and exhibiting a smooth function $U(r_h,Q)$ with $dU=\\Xi^{-1}dM_{\\rm ADM}$ would refute it as well. An independent check of the base claim is to verify numerically, for $D=4$–$7$ and $\\gamma=1$–$3$, that $(\\partial M_{\\rm ADM}/\\partial S)_{Q_{\\rm ADM}}=T_h$ and $(\\partial M_{\\rm ADM}/\\partial Q_e)_S=\\Phi_h$ hold as asserted.","supporting_citations":[{"cited_title":"Construction of Regular Black Holes in General Relativity","cited_arxiv_id":"1610.02636","evidence_quote":"Fan and Wang's construction of regular black holes in general relativity; supplies the P-frame formulation and the four-dimensional first law with Lagrangian parameters that this paper generalizes to arbitrary dimension."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Rasheed's general first law of black hole mechanics for nonlinear electrodynamics, which the paper's Eq. (14) reproduces as the static electric case."},{"cited_title":"First law and Smarr formula of black hole mechanics in nonlinear gauge theories","cited_arxiv_id":"1610.01237","evidence_quote":"Zhang and Gao's covariant phase-space derivation of the first law and Smarr formula in nonlinear gauge theories, including conjugate potentials for dimensionful parameters."},{"cited_title":"Black hole thermodynamics in the presence of nonlinear electromagnetic fields","cited_arxiv_id":"2102.06213","evidence_quote":"Bokulic, Juric and Smolic's first law for black holes in the presence of nonlinear electromagnetic fields, the basis for treating field-theory parameters as thermodynamic variables."},{"cited_title":"Corrected form of the first law of thermodynamics for regular black holes","cited_arxiv_id":"1411.0833","evidence_quote":"Ma and Zhao's proposal of a corrected first law with the prefactor 1/Ξ, the specific claim the paper interprets as a projection Jacobian and shows to be non-integrable."},{"cited_title":"Conundrum of regular black holes with nonlinear electromagnetic fields,","cited_arxiv_id":null,"evidence_quote":"Bokulic, Juric and Smolic's proof that mass and charge of regular NED black holes cannot be independent, which grounds the paper's mass–charge relation (4) on the regular subfamily."},{"cited_title":"Electrically charged regular black holes in higher dimensions: Maxwellian limit, branch point of the Lagrangian, and thermodynamics","cited_arxiv_id":"2609.29097","evidence_quote":"The companion paper supplying the solution family, horizons, energy conditions, Smarr formula, and the closed form of Φ_h Q_e used in the projection calculation."},{"cited_title":"Enthalpy and the mechanics of AdS black holes","cited_arxiv_id":"0904.2765","evidence_quote":"Kastor, Ray and Traschen's extended first law with the cosmological constant as a thermodynamic variable, the template for the σ and β terms in Eq. (15)."}],"review_version":1}