{"id":"268d9c3e-4187-4893-9acf-f2160da87747","arxiv_id":"2411.18256","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A λ-ζ pair of thermodynamic variables built from the Rényi entropy parameter is introduced so that flat black holes acquire RPST-style thermodynamics with van der Waals-like phase transitions.","lead":"This paper proposes a thermodynamic framework for flat black holes that borrows the Restricted Phase Space Thermodynamics structure used for anti-de Sitter black holes, replacing the cosmological constant with a Rényi entropy parameter. The authors show that, in this reformulation, flat Reissner-Nordström and Kerr black holes display van der Waals-like and Hawking-Page-like phase transitions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The zeta d-lambda term and Euler relation are imposed by declaring lambda an extensive coordinate; absent an ensemble in which the Renyi parameter fluctuates, the RPST extension is formal, not physical.","rationale":"After re-deriving the mass and partial derivatives, the algebra in Eqs. (24)-(27) is internally consistent; the first law holds by construction once zeta is defined as the partial derivative of M with respect to lambda. The factor-of-4 discrepancy between Eq. (9) and Eq. (11) is a genuine issue for the motivational alpha-Lambda correspondence, but it does not affect the internal consistency of the RPST construction, so I do not base the verdict on it. The load-bearing step is the extensivity postulate for lambda. The reader identified the same weakest assumption; my critique sharpens it into a concrete scaling test. Since the authors themselves flag that the analogies may be purely mathematical, the conditional verdict is appropriate. No change to the reader's verdict is needed.","tokens_in":11876,"tokens_out":10770,"duration_ms":99389,"concrete_test":"Compute R(beta)=M_tilde(beta S, beta Q, lambda)/M_tilde(S,Q,lambda) from Eq. (24) with lambda held fixed. If R(beta) is not equal to beta for generic beta, then the first-order homogeneity and Euler relation rely on the extra postulate that the Renyi parameter lambda scales extensively; this settles that the zeta d-lambda term is a formal identity unless a physical ensemble with fluctuating alpha is provided.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Central claim depends on lambda=1/alpha being promoted from a fixed Renyi parameter to an independent, extensive thermodynamic coordinate. The homogeneity of Eq. (24) holds only under joint scaling S->beta S, Q->beta Q, lambda->beta lambda. In Renyi statistics, alpha labels the ensemble and is intensive; under the standard extensive scaling with lambda fixed, M(beta S, beta Q, lambda) is not beta M(S,Q,lambda) (direct substitution into Eq. (24) gives exponential or beta^{3/2} behavior instead). Therefore the Euler relation E=TS+phi Q+zeta lambda and the first law dM=T dS+phi dQ+zeta dlambda are consequences of the imposed rescaling G=kappa^2/lambda, not of Renyi thermodynamics. No statistical ensemble in which lambda can fluctuate is constructed, so zeta has no operational meaning as a response potential; the Hawking-Page-like structure in the zeta-lambda plane and the 'universality' in Fig. 3(b) are coordinate artifacts. The paper's conclusion that the analogy 'may be purely mathematical' concedes exactly this unsecured step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an RPST-inspired thermodynamic formalism for asymptotically flat black holes using Rényi entropy. It defines λ = 1/α as a deformation parameter analogous to the central charge C of AdS/CFT, introduces a conjugate 'response potential' ζ = ∂M/∂λ, and rescales Newton's constant as G → κ²/λ. For Reissner-Nordström and Kerr black holes, the authors derive explicit rescaled masses, temperatures, electric potential/angular frequency, response potentials, first laws, and Euler relations. They then study T–S, F–T, and ζ–λ processes, claiming van der Waals-like first-order phase transitions below a critical charge or angular momentum, and Hawking-Page-like transitions in the ζ–λ plane. The algebraic structure is self-consistent, but the physical status of λ as an independent fluctuating thermodynamic coordinate is not established, and the paper itself concedes that the analogies 'may be purely mathematical.'","tokens_in":12202,"tokens_out":6593,"duration_ms":61078,"significance":"If the λ-coordinate construction is accepted, the paper provides a way to extend restricted phase space thermodynamics to flat black holes, replacing the cosmological-constant/central-charge pair with a Rényi-parameter/response-potential pair. The explicit first-law and Euler-relation derivations for RN and Kerr are a useful reference, and the T–S critical behavior at fixed λ is a genuine property of the Rényi mass function, consistent with earlier literature. The paper is also transparent in deriving the homogeneity that underlies the Euler relation. However, the physical significance is substantially weakened because no ensemble with fluctuating λ is constructed; the ζ dλ term and the associated phase structure are imposed by the chosen rescaling and coordinate promotion. The unresolved factor-of-4 mismatch in the α–Λ correspondence further undermines the motivational analogy. The paper's own closing caveat about the purely mathematical nature of the analogy correctly identifies the main uncertainty.","major_comments":[{"comment":"The motivation contains an unresolved factor-of-4 discrepancy. Comparing the small-α expansion of the Rényi Schwarzschild mass with the Schwarzschild-AdS mass gives α ≈ 4G/(πl²) in Eq. (9), while substituting Λ = −3/l² into the relation from Ref. [62] gives α ≈ G/(πl²) in Eq. (11). The manuscript notes both results but does not reconcile them. Since this relation is the primary motivation for replacing the AdS length with λ and for the rescaling in Eq. (22), the authors should either identify which derivation is correct, explain the discrepancy (e.g., an order-of-magnitude correspondence with the factor absorbed into κ), or explicitly state that the formalism does not depend on the precise coefficient.","section":"Sec. I, Eqs. (9)–(11)"},{"comment":"The central construction promotes λ = 1/α to an independent thermodynamic coordinate and rescales G to κ²/λ. The first-order homogeneity of Eq. (24) and the Euler relation (29) then follow by construction: under S → βS, Q̃ → βQ̃, λ → βλ, the exponential e^{S/λ} is invariant, so M̃ → βM̃. In Rényi statistics, α (hence λ) is a fixed parameter labeling the ensemble; the paper does not construct a statistical ensemble in which λ fluctuates. Consequently, ζ = ∂M̃/∂λ in Eq. (27) is a formal derivative, and the phase transitions in the ζ–λ plane (Fig. 3) are properties of the chosen coordinate system rather than demonstrated properties of the underlying Rényi thermodynamics. The concluding sentence that the analogies 'may be purely mathematical' concedes exactly this gap. The authors should either construct an ensemble with fluctuating λ and identify ζ as a physical response, or explicitly restrict the claims to a mathematical analogy and demonstrate which results (e.g., the T–S behavior at fixed λ) are independent of the coordinate promotion.","section":"Sec. I and Sec. II, Eqs. (22), (27), (29)"},{"comment":"Several load-bearing critical values are stated without derivation. For the RN case, Eq. (31) gives S_C = λ ln(2(√3 − 1)) and Q̃_C = √((7 − 4√3)/π) λ, and the temperature T_C = 0.256236 is quoted; the complicated free-energy expression in Eq. (34) is also presented without derivation. Similarly, the Kerr critical values in Eqs. (44) and (45) are asserted. These quantities underlie the phase-diagram claims, so the authors should provide the solution of the equations defining the critical point, or at least outline the calculation and state the assumptions, so that the results can be verified.","section":"Sec. II.1 and Sec. III, Eqs. (31), (34), (44), (45)"}],"minor_comments":[{"comment":"The expansion in Eq. (6) states the remainder is O(α^{3/2}), but the first correction is O(α²); the remainder should be O(α²).","section":"Sec. I, Eq. (6)"},{"comment":"The sentence 'Comparing eq.(8) with the first two terms in eq.(7)' appears to compare Eq. (6) with Eq. (8), not Eq. (7); please correct the cross-reference.","section":"Sec. I, text after Eq. (7)"},{"comment":"Eq. (18) is labeled as an expression for the event-horizon radius, but it actually gives the Rényi entropy S in terms of S₀ and λ; the surrounding text should be rephrased.","section":"Sec. II, Eq. (18)"},{"comment":"The phrase 'flat charged AdS black hole' should be 'flat charged black hole' or 'RN black hole'; the black hole is not AdS in this formalism.","section":"Sec. II.1, paragraph after Fig. 1"},{"comment":"The claimed universality of the iso-voltage ζ–λ plot is stated without proof; the text should explain that for fixed φ̃, ζ depends on S and λ only through the ratio S/λ, which makes the curves independent of S.","section":"Sec. II.1, Fig. 3(b)"},{"comment":"There are several typographical inconsistencies, including 'R´enyi' accents, 'isovoltage' versus 'iso-voltage', 'e-charge' versus 'Q̃', and a stray 'l' after 'respectively' in Sec. II; a careful proofreading pass is needed.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper is a formal construction whose physical interpretation hinges on whether λ can be treated as a fluctuating thermodynamic variable. The mathematical derivations of the first law and Euler relation are internally consistent, but the lack of an ensemble and the unresolved factor-of-4 mismatch are significant. I recommend major revision rather than rejection because the T–S phase behavior at fixed λ is a real feature of Rényi black hole thermodynamics and the formalism could be reframed as an explicit mathematical analogy. The authors should also provide derivations for all quoted critical values."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you the short version: the algebra in this paper is fine, the construction is new in a narrow sense, and the authors are appropriately cautious about what it means. The problem is that the central result—the extended first law with ζ dλ and the Euler relation—is true by construction, not by physics. The homogeneity of the mass (eq. 24) holds only if you scale S, Q, and λ together. But λ is the inverse Rényi parameter; in Rényi statistics it labels the ensemble and is intensive. The authors never construct an ensemble in which λ fluctuates, so ζ = ∂M/∂λ is a formal derivative. The paper's own concluding sentence—that the analogy may be purely mathematical—concedes exactly this.\n\nWhat is genuinely new: the explicit λ-ζ pair as an RPST-style extension for flat black holes, with phase diagrams for RN and Kerr drawn in the T-S and F-T planes. I haven't seen this specific construction before. And the paper is openly honest: they flag the possible formality, and they don't hide the derivation gaps.\n\nSoft spots, in proportion: first, the motivation has a factor-of-4 mismatch between eq (9) and eq (11) for α in terms of l. The authors notice it but don't resolve it; that's a hole in the cosmological-constant analogy. Second, the critical values in eqs (31), (44), (45) are stated without derivation. For a paper whose main output is phase structure, that's a noticeable gap. Third, the 'universality' in Fig 3(b) is likely a coordinate artifact: once λ and ζ are fixed by the rescaling, the iso-voltage curve collapses to a universal form by construction. The stress-test note is right about this.\n\nOverall the paper is what it says: a formalism, not a prediction. If you take it as a repackaging of known Rényi masses into an RPST-like language, it is internally consistent and might be useful to people working on Rényi black hole thermodynamics. It does not resolve any long-open question and offers no testable new physics.\n\nI'd send this to a serious referee if it lands in an appropriate journal—the algebra deserves checking and the critical values should be derived. But as a reader, I wouldn't cite it for any physical result, and I wouldn't bring it to a reading group as a must-read. It's a competent formal exercise with honest caveats.","headline":"A self-consistent formal extension of RPST to flat Rényi black holes, but the physical content is limited by the fact that the new coordinate λ is imposed rather than derived.","tokens_in":12699,"tokens_out":2132,"would_cite":false,"duration_ms":19640,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C57","80A10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper argues that flat black holes with Rényi entropy can be given a restricted phase space thermodynamics in which the inverse Rényi parameter and a response potential act as a new variable pair, yielding van der Waals-like and…","keywords":["restricted phase space thermodynamics","Rényi entropy","flat black holes","Reissner-Nordström black hole","Kerr black hole","van der Waals phase transition","Hawking-Page transition","response potential"],"falsifier":"Recompute the canonical ensemble for a flat RN black hole with fixed physical charge $Q$ and the standard Bekenstein-Hawking entropy $S_0=\\pi r_+^2/G$, and check whether $F(T)$ has a swallowtail; the paper's transition appears only after promoting $\\lambda$ to a fluctuating variable, so a calculation that fixes $\\alpha$ and never differentiates with respect to it should show no van der Waals critical point. A direct numerical search for the predicted critical exponents in the heat capacity near $T_C$ for fixed $\\tilde Q<\\tilde Q_C$ would also settle whether the transition is genuine or an artifact of the coordinate choice.","tokens_in":2179,"feed_emoji":"🕳️","tokens_out":2244,"duration_ms":81604,"temperature":0.7,"pith_summary":"The paper claims that asymptotically flat black holes, when described by Rényi entropy with deformation parameter $\\lambda=1/\\alpha$, can be given a restricted phase space thermodynamics once $\\lambda$ is promoted to an independent thermodynamic coordinate paired with a response potential $\\zeta$. For flat Reissner-Nordström and Kerr black holes the authors verify a first law of the form $d\\tilde M = T\\,dS + \\tilde\\phi\\,d\\tilde Q + \\zeta\\,d\\lambda$ (with $\\Omega\\,dJ$ replacing the charge term for Kerr) together with the Euler relation, and they show that the $T$--$S$ and $F$--$T$ curves exhibit van der Waals-like first-order phase transitions below a critical charge or angular momentum. They also find Hawking-Page-like transitions in the $\\zeta$--$\\lambda$ plane. If correct, this would mean the phase structure usually attributed to a cosmological constant can emerge from the entropy statistics alone.","feed_headline":"Flat black holes get van der Waals transitions via Rényi entropy","feed_subtitle":"A deformation parameter and response potential recreate AdS-style thermodynamics for flat black holes.","key_machinery":"The load-bearing object is the Rényi entropy rewritten as $S=\\lambda\\ln(1+S_0/\\lambda)$ with $\\lambda=1/\\alpha$, combined with the rescaling $G\\to\\kappa^2/\\lambda$ and $\\tilde Q\\to\\kappa Q/\\sqrt G$. This combination makes the rescaled black hole mass a homogeneous degree-one function of $S$, the charge or angular momentum, and $\\lambda$, which is exactly what guarantees the first law and the Euler relation; the response potential $\\zeta=\\partial\\tilde M/\\partial\\lambda$ is the conjugate force generated by that homogeneity. The phase structure then comes from the non-monotonic $T(S)$ produced by the exponentials $e^{S/\\lambda}$ in the mass formulas.","core_discovery":"The central claim is that flat black holes in Rényi statistics obey the same thermodynamic form as AdS restricted phase space thermodynamics, with the inverse Rényi parameter $\\lambda=1/\\alpha$ playing the role of the central charge and the response potential $\\zeta=\\partial\\tilde M/\\partial\\lambda$ playing the role of the chemical potential. The proof mechanism is a rescaling $G\\to\\kappa^2/\\lambda$ and $\\tilde Q\\to\\kappa Q/\\sqrt G$, under which the rescaled mass $\\tilde M=M\\kappa$ becomes a first-order homogeneous function of $S$, $\\tilde Q$ (or $J$), and $\\lambda$; homogeneity then forces the first law and the Euler relation $\\tilde M = T S + \\tilde\\phi \\tilde Q + \\zeta\\lambda$ (or $TS+\\Omega J+\\zeta\\lambda$) to hold. From the explicit mass functions the authors compute critical points: for RN, $\\tilde Q_C = \\sqrt{(7-4\\sqrt3)/\\pi}\\,\\lambda$ and $T_C\\approx0.256236$, below which the isocharge $T$--$S$ curve is non-monotonic and $F$--$T$ has a swallowtail; for Kerr, $S_C=0.483833\\lambda$ and $J_C=0.0193724\\lambda$ with the same swallowtail structure. In the $\\zeta$--$\\lambda$ plane they identify Hawking-Page-like transitions, and in the RN iso-voltage case they report a universal $\\zeta$--$\\lambda$ curve independent of $S$ and $\\phi$.","pith_inferences":["If the correspondence is physical rather than formal, $\\lambda$ may encode the effective number of microscopic degrees of freedom of a flat-space black hole, and the sign of $\\zeta$ would then serve as a macroscopic order parameter for whether those degrees of freedom interact attractively or repulsively.","The reported universality of the iso-voltage $\\zeta$--$\\lambda$ curve for RN is striking; repeating the construction with a third conserved charge or in higher dimensions would test whether this universality reflects a hidden scaling symmetry of the Rényi ensemble.","Because the small-$\\alpha$ expansion of the Schwarzschild mass reproduces the Schwarzschild-AdS mass with $\\alpha\\propto 1/l^2$, one extension is to compute the critical exponents of the flat-space transitions and compare them with the AdS restricted phase space values; matching exponents would strengthen the case that $\\lambda$ truly substitutes for a central charge."],"forward_implications":["Flat Reissner-Nordström and Kerr black holes acquire a first law and Euler relation of the same form as AdS restricted phase space thermodynamics, with $\\lambda$ and $\\zeta$ replacing the central charge and chemical potential.","Isocharge $T$--$S$ and $F$--$T$ curves for flat RN black holes show a first-order van der Waals-like phase transition for $0<\\tilde Q<\\tilde Q_C$, turning second-order at $\\tilde Q_C=\\sqrt{(7-4\\sqrt3)/\\pi}\\,\\lambda$ with $T_C\\approx0.256236$.","Flat Kerr black holes show the same structure: below $J_C=0.0193724\\lambda$ the $T$--$S$ curves are non-monotonic with a swallowtail in $F$--$T$, and the transition becomes second-order at the critical point.","In the $\\zeta$--$\\lambda$ plane the formalism predicts Hawking-Page-like transitions, with $\\zeta=0$ marking the Hawking-Page temperature in the iso-voltage RN case.","Because the framework is extensive despite starting from nonextensive Rényi entropy, all of this phase structure can be analyzed with standard thermodynamic machinery."],"supporting_citations":[{"why":"Supplies the Rényi entropy formula $S=\\frac{1}{\\alpha}\\ln(1+\\alpha S_0)$ that the entire construction starts from.","marker":"[36]"},{"why":"Showed that flat black holes in Rényi entropy mimic AdS black hole thermodynamics, which is the motivation for this formalism.","marker":"[39]"},{"why":"Defines restricted phase space thermodynamics for AdS black holes via holography, the framework that this paper adapts to flat spacetime.","marker":"[48]"},{"why":"Earlier attempt to make black hole thermodynamics extensive by varying Newton's constant; the present paper contrasts its $\\lambda$--$\\zeta$ pair with that approach.","marker":"[53]"},{"why":"Relates the Rényi parameter $\\alpha$ to the cosmological constant $\\Lambda$, motivating $\\lambda=1/\\alpha$ as a cosmological-constant-like coordinate.","marker":"[62]"},{"why":"Provides the $P$--$V$ criticality analysis of charged AdS black holes whose van der Waals behavior the flat-space transitions are modeled on.","marker":"[28]"},{"why":"Defines the Hawking-Page transition used to characterize the $\\zeta$-plane transitions.","marker":"[13]"},{"why":"Identifies Davies-type heat-capacity phase transitions, the baseline that the new first-order transitions are contrasted with.","marker":"[12]"}],"fun_headline_variants":["Flat black holes mimic AdS thermodynamics via Rényi entropy","Rényi entropy brings van der Waals transitions to flat black holes","Deformation parameter unlocks AdS-like phase structure for flat black holes","RPST-inspired formalism extends AdS thermodynamics to flat black holes"],"cache_read_input_tokens":14848,"weakest_assumption_plain":"Everything rests on promoting $\\lambda=1/\\alpha$ to an independent thermodynamic coordinate and on rescaling $G$ to $\\kappa^2/\\lambda$; Rényi statistics alone does not require the deformation parameter to fluctuate, so if $\\lambda$ is a fixed model parameter the response potential is a formal derivative and the phase transitions are artifacts of the chosen variables.","fun_headline_variants_meta":{"raw":{"variants":["Flat black holes mimic AdS thermodynamics via Rényi entropy","Rényi entropy brings van der Waals transitions to flat black holes","Deformation parameter unlocks AdS-like phase structure for flat black holes","RPST-inspired formalism extends AdS thermodynamics to flat black holes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001085,"raw_usage":{"total_tokens":4599,"prompt_tokens":1075,"completion_tokens":3524,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":691,"completion_tokens_details":{"reasoning_tokens":3450}},"tokens_in":691,"tokens_out":3524,"duration_ms":22571,"temperature":1.0,"reasoning_tokens":3450,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:22:59.671381+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the canonical ensemble for a flat RN black hole with fixed physical charge $Q$ and the standard Bekenstein-Hawking entropy $S_0=\\pi r_+^2/G$, and check whether $F(T)$ has a swallowtail; the paper's transition appears only after promoting $\\lambda$ to a fluctuating variable, so a calculation that fixes $\\alpha$ and never differentiates with respect to it should show no van der Waals critical point. A direct numerical search for the predicted critical exponents in the heat capacity near $T_C$ for fixed $\\tilde Q<\\tilde Q_C$ would also settle whether the transition is genuine or an artifact of the coordinate choice.","supporting_citations":[{"cited_title":"R´ enyi, Acta Math","cited_arxiv_id":null,"evidence_quote":"Supplies the Rényi entropy formula $S=\\frac{1}{\\alpha}\\ln(1+\\alpha S_0)$ that the entire construction starts from."},{"cited_title":"Wang and L","cited_arxiv_id":null,"evidence_quote":"Earlier attempt to make black hole thermodynamics extensive by varying Newton's constant; the present paper contrasts its $\\lambda$--$\\zeta$ pair with that approach."}],"review_version":1}