{"id":"ca7202be-ce30-4f9c-a95d-8ca4e364a20e","arxiv_id":"2505.13555","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Stability and the inverse isoperimetric inequality impose 2Λr_+^2/((d-1)(d-2)) < g^2 < -2Λr_+^2/((d-1)(d-2)) on gravity's rainbow functions, and super-entropic black holes occur exactly when f>1.","lead":"This paper computes thermodynamic quantities for black holes in gravity's rainbow and derives constraints on the rainbow functions from stability conditions and the inverse isoperimetric inequality. It concludes that one common rainbow model is ruled out, another is allowed, and super-entropic black holes appear when the rainbow function f is greater than one.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section III's exclusion of model (iii) for λ>0 contradicts Eq. (26): model (iii) with λ>0 satisfies the paper's own stability inequalities for sufficiently large r_+, so the f>1 instability claim rests on an unproved conjecture.","rationale":"The most load-bearing issue is not the algebra leading to (26) or (29), which is correct; it is the use of the super-entropic-instability conjecture in Section III as a theorem. The reader flagged this as the weakest assumption, and I agree it is central, but the problem is sharper than a missing proof: the paper's own stability conditions invalidate the exclusion. For model (iii) with λ>0, Eq. (26) is satisfied for all sufficiently large r_+, so the solution is locally and globally stable by Eqs. (16) and (18), while f>1 gives R<1. The Section III assertion that such a solution is thermodynamically unstable therefore contradicts Section II, unless the unproved conjecture is adopted as a theorem. This is also why the separate claim that g^2 cannot be constant is a non-sequitur: a constant g^2 can satisfy (26) for a range of r_+, so model (i) is not excluded by that argument alone. I recommend keeping the reader's conditional verdict: derivations (26) and (29) remain valid, but the model-selection conclusions must be removed, explicitly conditioned on the conjecture, or replaced by a proof. My disagreement with the reader's exact framing is only that the problem is not just an unproven conjecture; the paper's own equations provide a direct counterexample to its Section III conclusion.","tokens_in":7825,"tokens_out":20292,"duration_ms":207777,"concrete_test":"Take d=4, Λ<0, model (iii) with λ=0.5 and ε=0.1, so f=g=1/(1-0.05)≈1.0526 and g^2≈1.108. Choose r_+ with r_+^2 > 3.324/|Λ|, so that g^2 < -2Λr_+^2/6. Evaluate the stability conditions: temperature (10), mass (14), heat capacity (16), free energy (18), and C_P from (24). Each has the required sign (T>0, M>0, C>0, F<0, C_P>0), giving a thermodynamically stable black hole with R=1/f^{1/3}<1. This directly contradicts the Section III claim that f>1 implies thermodynamic instability, and shows that the exclusion of model (iii) for λ>0 depends on the unproved super-entropic-instability conjecture rather than on the paper's own stability analysis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's model-selection conclusion is internally inconsistent. Section II derives inequality (26), 2Λr_+^2/((d-1)(d-2)) < g^2 < -2Λr_+^2/((d-1)(d-2)), as the combined thermal-stability condition (positive temperature, mass, heat capacities, and negative free energy). For model (iii) with λ>0, f=g=1/(1-λε)>1 (for 0<λε<1). Fix any such λ and ε and take r_+ large enough; the upper bound in (26) grows as |Λ|r_+^2, so the inequality is satisfied. By the paper's own criteria, these black holes are thermodynamically stable. Yet Section III then declares that f>1 gives R=1/f^{1/(d-1)}<1 and that the black holes are 'thermodynamically unstable systems', ruling out model (iii) for λ>0. This conclusion relies entirely on the unproven conjecture from refs [46-49] that super-entropic black holes are unstable, and it directly contradicts the Section II stability analysis for the same solutions. The paper cannot have it both ways: either the Section II criteria define stability, in which case model (iii) with λ>0 admits stable large AdS black holes and the f>1 exclusion fails, or the super-entropic conjecture is silently promoted to a blanket theorem, which is not established. The same issue affects the claim that g^2 cannot be constant: for fixed g^2, (26) holds for a range of r_+, so model (i) is not excluded by that argument alone.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies d-dimensional Schwarzschild-AdS black holes in gravity's rainbow, with energy-dependent metric functions f(ε) and g(ε), and uses thermodynamic stability to constrain the rainbow functions. The authors compute the Hawking temperature, entropy, total mass, heat capacity, and Helmholtz free energy, and combine the positivity conditions for temperature, mass, heat capacity, and negative free energy into the window (26) for g²(ε). In extended phase space they write M(S,P), the thermodynamic volume, and a Smarr relation. They then use the inverse isoperimetric inequality to compute R = 1/f^{1/(d-1)}, concluding that f>1 makes the black holes super-entropic and thermodynamically unstable. On this basis they select among the three rainbow-function models: model (ii) is allowed, while model (i) and model (iii) with λ>0 are excluded. The central algebraic derivation of (26) is internally consistent, but the model-selection conclusions are not supported by the paper's own stability criteria.","tokens_in":8216,"tokens_out":11484,"duration_ms":108429,"significance":"If fully supported, the paper would offer a concrete thermodynamic selection among rainbow-function models and connect the super-entropic condition to gravity's rainbow. The derivation of Eq. (26) is transparent and largely self-contained once the metric function and AMD mass formula are imported from refs. [50-52]; the computation of R in Eq. (29) follows correctly from the paper's own volume and area formulas, and there is no parameter fitting. However, the central model-selection claims rely on an unproven conjecture and on an unjustified 'for all r_+' reading of Eq. (26), so the significance of the paper as a constraint on rainbow functions is not yet established.","major_comments":[{"comment":"The exclusion of model (iii) for λ>0 is inconsistent with the paper's own stability criterion. For model (iii), f(ε)=g(ε)=1/(1-λε), and for λ>0 with 0<λε<1 one has g²>1. For AdS (Λ<0), the lower bound in Eq. (26) is negative while the upper bound grows like -2Λr_+²/((d-1)(d-2)), so for every fixed λ>0 there exist sufficiently large horizon radii r_+ for which Eqs. (11), (15), (17), and (19), and hence Eq. (26), are all satisfied. By the Section II criteria these are thermodynamically stable black holes. The later assertion that f>1 gives R<1 and therefore thermodynamic instability depends entirely on the 'suggested' conjecture of refs. [46-49], which the manuscript itself describes as unconfirmed; the assertion is used as a blanket theorem and directly contradicts the Section II stability analysis for the same solutions. This contradiction is load-bearing because it is the basis for rejecting model (iii) with λ>0; the paper must either prove the super-entropic instability link or reconcile the two criteria.","section":"Section III (text after Eq. (29))"},{"comment":"The conclusion that g² cannot be constant does not follow from Eq. (26) as stated. Equation (26) is an r_+-dependent interval; for a fixed constant value of g², and fixed Λ<0 and d, all four underlying conditions are satisfied for an interval of r_+ (in particular, for sufficiently large r_+). To infer that a constant g² is forbidden, one must additionally assume that thermodynamic stability is required for every allowed horizon radius r_+, an assumption that is neither stated nor justified. Without that assumption, model (i) with g(ε)=1 is not excluded by the thermodynamic analysis.","section":"Section II (text after Eq. (26))"},{"comment":"Eq. (28) states M=2(TS-PV) for d-dimensional black holes, but the scaling relation for the mass function (21) gives (d-3)M=(d-2)TS-2PV; equivalently, M=((d-2)/(d-3))TS-(2/(d-3))PV. Equation (28) is therefore valid only in d=4. Since the paper presents this as a d-dimensional result, it should be corrected or explicitly restricted to d=4.","section":"Section II (Eq. (28))"}],"minor_comments":[{"comment":"There is a typo in the text: 'two follwoing constrains' should be 'two following constraints'.","section":"Section II, below Eq. (17)"},{"comment":"The phrase 'evaluating the three conditions in Eqs. (11), (15), (17), and (19)' is inconsistent, since four constraints are listed and Eq. (17) itself contains two inequalities.","section":"Section II, before Eq. (26)"},{"comment":"The text refers to 'the thermodynamic volume (V) of bumblebee AdS black holes'; this paper is not about bumblebee gravity, so 'bumblebee' should be removed.","section":"Section II, paragraph containing Eq. (22)"},{"comment":"The parenthetical 'i.e., V∝S' is inaccurate: from Eqs. (22) and (12), V ∝ S^{(d-1)/(d-2)}, not V ∝ S. The conclusion that C_V=0 only requires V to be a function of S independent of P.","section":"Section II, paragraph containing Eq. (22)"}],"recommendation":"major_revision","confidential_remarks":"The core inequality (26) is a useful and correct result, but the paper's advertised conclusions about which rainbow-function model is viable depend on two unsupported steps: reading Eq. (26) as requiring stability for all r_+, and promoting the super-entropic instability conjecture to a theorem in Section III. These are fixable in revision, but the model-selection conclusions need to be substantially reworded or additionally justified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new piece here is the inverse isoperimetric inequality applied to gravity's rainbow: the identity R = f^{-1/(d-1)} follows directly from their volume formula, and combining positivity of temperature, mass, heat capacities, and free energy into the interval (26) is a sensible consistency filter for rainbow functions. The derivation of (26) is short but correct for AdS, and the paper is honest about where the thermodynamic scaffolding comes from (ref [50], largely overlapping authors). No parameter fitting, no circularity. That is worth something.\n\nThe soft spots are real, though. The Smarr relation (28) is stated without the dimension-dependent prefactor that makes it valid only for d=4; in other dimensions it is simply wrong. The claim that g^2 cannot be constant relies on an unstated 'for all r_+' reading of (26) — for any fixed g^2, the inequality holds on a range of horizon radii, so model (i) is not excluded by that argument alone. Most seriously, the Section III exclusion of model (iii) with λ>0 directly contradicts the Section II stability analysis. Under (26), model (iii) with fixed λ>0 satisfies the stability window for sufficiently large r_+; by the paper's own criteria those are stable black holes. Then Section III declares them unstable because f>1 implies R<1, leaning entirely on the unproven conjecture that super-entropic implies unstable. The paper itself calls that link 'suggested' and says confirming it 'would be of great importance' — then silently promotes it to a theorem. You cannot have it both ways. Either (26) defines stability and model (iii) λ>0 survives, or the super-entropic conjecture is the operative criterion, in which case the Section II constraints are irrelevant to model selection. This is an internal inconsistency in the load-bearing argument, not a minor quibble.\n\nWho is this for? People working on rainbow-gravity black hole thermodynamics will find the inequality and the identity useful, and the paper deserves a serious referee — it is not desk-reject material. But the referee should push for the model-exclusion claims to be either proven from the stated stability conditions or explicitly conditioned on accepting the super-entropic instability conjecture. As it stands, the conclusions outrun the evidence.\n\nMy recommendation: send it out, but expect major revision. The core inequality can stand; the exclusion arguments need to be rebuilt.","headline":"A correct inequality and a neat identity, but the model-exclusion claims contradict the paper's own stability criteria and need revision.","tokens_in":8707,"tokens_out":1621,"would_cite":false,"duration_ms":18830,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Thermodynamic stability in gravity's rainbow forces the rainbow function $g^2$ into a horizon-radius-dependent interval and, through the inverse-isoperimetric ratio $R = 1/f^{1/(d-1)}$, identifies $f>1$ rainbow black holes as…","keywords":["gravity's rainbow","rainbow functions","black hole thermodynamics","super-entropic black holes","inverse isoperimetric inequality","thermodynamic stability","AdS black holes","heat capacity"],"falsifier":"Evaluate the heat capacity (16) for model (iii) with $\\lambda>0$ and $f=g=1/(1-\\lambda\\varepsilon)$ in AdS. If any horizon radius $r_+$ gives $T>0$, $M>0$, and $C>0$ while $R<1$, then a super-entropic rainbow black hole is thermodynamically stable, contradicting the paper's instability claim. A simpler check is to exhibit any super-entropic black hole with positive specific heat in any theory of gravity; the instability lemma used in Section III would then be false.","tokens_in":7659,"feed_emoji":"🕳️","tokens_out":17430,"duration_ms":148186,"temperature":0.7,"pith_summary":"In gravity's rainbow, spacetime is modified by rainbow functions $f(\\varepsilon)$ and $g(\\varepsilon)$ of the probing particle's energy. This paper asks which of the three standard rainbow models can describe AdS black holes that are thermodynamically stable, in both ordinary and extended phase space. Computing the Hawking temperature, total mass, heat capacity, Helmholtz free energy, and constant-pressure heat capacity, the authors find that stability requires $g^2(\\varepsilon)$ to satisfy a two-sided bound set by the cosmological constant, the spacetime dimension, and the horizon radius $r_+$; hence $g^2$ cannot be a constant, ruling out model (i). Using the inverse isoperimetric inequality, they show $R = 1/f(\\varepsilon)^{1/(d-1)}$, so the rainbow function $f$ decides whether the hole is super-entropic, and they combine this with the suggested super-entropic-instability link to exclude model (iii) with $\\lambda>0$. The surviving model is (ii), with $f=1$ and $g=\\sqrt{1-\\eta\\varepsilon^n}$, which satisfies all stability conditions and gives $R\\ge1$.","feed_headline":"Stability check leaves only one rainbow gravity model","feed_subtitle":"Thermodynamics plus the isoperimetric ratio forces g² to vary with horizon size, ruling out constant-g and f>1 cases.","key_machinery":"The load-bearing object is the inverse-isoperimetric ratio $R = ((d-1)V/\\omega_{d-2})^{1/(d-1)}(\\omega_{d-2}/A)^{1/(d-2)}$, which compares a black hole's thermodynamic volume with that of a round ball of equal area; $R<1$ is the super-entropic regime. In this rainbow spacetime the identity $R=1/f(\\varepsilon)^{1/(d-1)}$ collapses the whole isoperimetric content into the rainbow function $f$. The second mechanism is the inequality chain from $T>0$, $M>0$, $C>0$, $F<0$, and $C_P>0$, which compresses into the two-sided bound on $g^2$ that forces $g$ to be horizon-dependent and eliminates model (i). Together these two pieces convert thermodynamic-stability conditions into concrete constraints on the rainbow functions $f$ and $g$.","core_discovery":"The central claim is a selection rule for rainbow functions. For $d$-dimensional AdS black holes in gravity's rainbow, positivity of the temperature, total mass, heat capacity, negative Helmholtz free energy, and constant-pressure heat capacity together imply $$\\frac{2\\Lambda(\\varepsilon) r_+^2}{(d-1)(d-2)} < $g^{2}$(\\varepsilon) < -\\frac{2\\Lambda(\\varepsilon) r_+^2}{(d-1)(d-2)},$$ which, for $\\Lambda<0$, leaves the operative bound $g^2(\\varepsilon) < -2\\Lambda(\\varepsilon) r_+^2/((d-1)(d-2))$ and makes $g^2$ depend on the horizon radius; no constant $g$ can satisfy this at every $r_+$, so model (i) is excluded. Separately, the inverse-isoperimetric ratio for these spacetimes is exactly $R=1/f(\\varepsilon)^{1/(d-1)}$, so $f(\\varepsilon)>1$ means the black hole violates the isoperimetric bound and is super-entropic. Taking the suggested correspondence between super-entropic black holes and thermodynamic instability as given, the paper concludes that $f>1$ holes are thermodynamically unstable, ruling out model (iii) with $\\lambda>0$; model (ii), with $f=1$, obeys $R\\ge1$ and passes every stability condition, making it the only admissible model among the three.","pith_inferences":["A natural next test is to run the same inequality chain for charged or rotating rainbow black holes; if the $g^2$ window shifts, the excluded models could be revived in those sectors.","The identity $R=1/f^{1/(d-1)}$ is independent of $g$ and of the details of the metric function, so if the instability correspondence is ever proven, the stability classification of rainbow black holes becomes a property of $f$ alone, extending the paper's model-by-model verdict to any rainbow theory.","The paper's 'no constant $g$' conclusion is the all-radii version. If stability is required only over a bounded range of horizon radii, a constant-$g$ model such as model (i) could still describe stable black holes in that window, a weaker reading the paper does not address."],"forward_implications":["Among the three standard rainbow models, only model (ii), with $f=1$ and $g=\\sqrt{1-\\eta\\varepsilon^n}$, passes every thermodynamic stability condition and satisfies $R\\ge1$.","A viable rainbow function must have $g^2(\\varepsilon) < -2\\Lambda(\\varepsilon) r_+^2/((d-1)(d-2))$ for AdS black holes, so $g^2$ has to depend on the horizon radius; constant-$g$ rainbow deformations of Schwarzschild-AdS cannot be stable at every radius.","If the super-entropic instability correspondence holds, $f(\\varepsilon)>1$ is a direct instability marker: rainbow black holes with $f>1$ are super-entropic and thermodynamically unstable, which forbids model (iii) for $\\lambda>0$ and any analogous $f>1$ rainbow model.","In the extended phase space the same conditions become a lower bound on the thermodynamic pressure, $P>(d-2)(d-3)(\\omega_{d-2}/4S)^{2/(d-2)}/16\\pi$, tying the allowed rainbow parameters to an admissible pressure range.","The first law $dM=T\\,dS+V\\,dP$ and the Smarr relation $M=2(TS-PV)$ hold for all three models, so the selection among rainbow functions takes place inside a thermodynamically consistent framework."],"supporting_citations":[{"why":"Introduces gravity's rainbow and the energy-dependent action and metric on which the calculation is built.","marker":"[1]"},{"why":"Supplies the d-dimensional rainbow black hole metric function and the temperature, entropy, and mass formulas used in Section II.","marker":"[50]"},{"why":"Defines the inverse-isoperimetric ratio R that the paper evaluates to 1/f^{1/(d-1)}.","marker":"[53]"},{"why":"Suggested that super-entropic black holes are thermodynamically unstable, the premise behind excluding f>1 models.","marker":"[46]"},{"why":"Provides additional support for the super-entropic instability correspondence used in Section III.","marker":"[47]"},{"why":"Earlier work connecting super-entropic black holes to thermodynamic instability, cited as part of the suggested correspondence.","marker":"[48]"},{"why":"Most recent discussion of the super-entropic instability conjecture, completing the cited basis for the paper's instability step.","marker":"[49]"},{"why":"Establishes the extended phase space treatment in which the cosmological constant acts as thermodynamic pressure, used for CP.","marker":"[26]"},{"why":"Develops the extended phase space thermodynamics that the constant-pressure heat capacity and Smarr analysis rely on.","marker":"[27]"}],"fun_headline_variants":["Rainbow gravity: only one model survives stability","Super-entropic black holes rule out most rainbow models","Thermodynamics picks a single rainbow function","Rainbow gravity: two models fail, one survives"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the tentative link between entropy-exceeding (super-entropic) black holes and thermodynamic instability is actually true, together with the implicit requirement that stability hold for every horizon radius; if either assumption fails, the exclusions of model (iii) with positive λ and of model (i) no longer follow.","fun_headline_variants_meta":{"raw":{"variants":["Rainbow gravity: only one model survives stability","Super-entropic black holes rule out most rainbow models","Thermodynamics picks a single rainbow function","Rainbow gravity: two models fail, one survives"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000814,"raw_usage":{"total_tokens":3589,"prompt_tokens":986,"completion_tokens":2603,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":602,"completion_tokens_details":{"reasoning_tokens":2543}},"tokens_in":602,"tokens_out":2603,"duration_ms":18261,"temperature":1.0,"reasoning_tokens":2543,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:27:40.999644+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the heat capacity (16) for model (iii) with $\\lambda>0$ and $f=g=1/(1-\\lambda\\varepsilon)$ in AdS. If any horizon radius $r_+$ gives $T>0$, $M>0$, and $C>0$ while $R<1$, then a super-entropic rainbow black hole is thermodynamically stable, contradicting the paper's instability claim. A simpler check is to exhibit any super-entropic black hole with positive specific heat in any theory of gravity; the instability lemma used in Section III would then be false.","supporting_citations":[{"cited_title":"Magueijo, L","cited_arxiv_id":null,"evidence_quote":"Introduces gravity's rainbow and the energy-dependent action and metric on which the calculation is built."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the d-dimensional rainbow black hole metric function and the temperature, entropy, and mass formulas used in Section II."},{"cited_title":"Cvetic, G","cited_arxiv_id":null,"evidence_quote":"Defines the inverse-isoperimetric ratio R that the paper evaluates to 1/f^{1/(d-1)}."},{"cited_title":"Cong, and R","cited_arxiv_id":null,"evidence_quote":"Suggested that super-entropic black holes are thermodynamically unstable, the premise behind excluding f>1 models."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides additional support for the super-entropic instability correspondence used in Section III."},{"cited_title":"Eslam Panah, Phys","cited_arxiv_id":null,"evidence_quote":"Earlier work connecting super-entropic black holes to thermodynamic instability, cited as part of the suggested correspondence."},{"cited_title":"Kubiznak, and R","cited_arxiv_id":null,"evidence_quote":"Develops the extended phase space thermodynamics that the constant-pressure heat capacity and Smarr analysis rely on."}],"review_version":1}