{"id":"ca3ddcbd-b6e8-4622-a1ec-0139790cbf13","arxiv_id":"2606.26210","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":3.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":2,"one_line_summary":"Analytical extended thermodynamics of charged AdS black holes in ModMax-dRGT massive gravity with Sharma-Mittal entropy shows parameter-dependent inversion curves, van der Waals-like first-order phase transitions, and decoupled effects from each modification.","lead":"The paper derives analytical expressions for Hawking temperature, specific heat, Joule-Thomson coefficient and equation of state for charged AdS black holes in a combined ModMax-dRGT massive gravity model using Sharma-Mittal entropy. A smart generalist might read it to see how nonlinear electrodynamics and non-extensive entropy alter black hole cooling and phase behavior in extended thermodynamics.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"Sharma-Mittal entropy applicability lacks consistency checks with ModMax+dRGT in extended phase space","rationale":"Reader's weakest_assumption directly identifies the entropy + extended-phase-space assumption as load-bearing; full-text claims do not add independent verification (no machine-checked identities or parameter-free limits), so the concern stands and moves the verdict from UNVERDICTED to CONDITIONAL pending the check.","tokens_in":1850,"tokens_out":307,"duration_ms":10292,"concrete_test":"Re-derive the Hawking temperature and first law from the metric function using both Bekenstein-Hawking and Sharma-Mittal entropies; if the resulting μ_JT and inversion radius r_i expressions differ by more than the reported parametric shifts when γ or m_g is varied, the entropy choice controls the headline results.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claims (inversion shift by γ, stability governed by δ/R, massive gravity dictating global boundary, swallow-tail in Gibbs free energy) rest on replacing the area law with generalised Sharma-Mittal entropy S_δ,R while retaining the standard extended first law dM = T dS + V dP + Φ dQ + … . No derivation or check is supplied showing that this entropy remains consistent with the nonlinear ModMax stress-energy, the dRGT graviton mass terms, or the resulting Smarr relation; the abstract and claims treat the substitution as unproblematic.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper claims to derive exact analytical expressions for the modified Hawking temperature, specific heat, Joule-Thomson coefficient, and equation of state for four-dimensional charged AdS black holes in the combined ModMax nonlinear electrodynamics and dRGT massive gravity framework, using the generalised Sharma-Mittal entropy in extended phase space. It reports that ModMax nonlinearity γ expands the cooling domain by shifting the inversion transition to smaller radii, that δ and R govern local stability and inversion radius while massive gravity sets the global boundary, and that the Gibbs free energy exhibits a van der Waals-like first-order phase transition with swallow-tail structure, with a claimed physical decoupling of the three modifications.","tokens_in":2003,"tokens_out":588,"duration_ms":16414,"significance":"If the entropy substitution and derivations are shown to be consistent, the results would illustrate how nonlinear electrodynamics, massive gravity, and non-extensive entropy separately affect inversion curves and phase transitions, offering potential diagnostics for black hole modifications. The work does not provide machine-checked proofs, reproducible code, or parameter-free derivations.","major_comments":[{"comment":"Abstract (paragraph on entropy choice) and the section introducing the thermodynamic quantities: the central claims rest on substituting the area law with the generalised Sharma-Mittal entropy S_δ,R while retaining the standard extended first law dM = T dS + V dP + Φ dQ + … without any derivation or consistency check against the ModMax stress-energy tensor, the dRGT graviton mass terms, or the resulting Smarr relation; this substitution is load-bearing for all reported analytical expressions, stability conclusions, and the inversion/phase-transition results.","section":"Abstract and entropy introduction"},{"comment":"Abstract and throttling-process analysis: the statements that γ 'expands the physically accessible cooling domain' and that 'clear physical decoupling' occurs in the critical regime are presented as outcomes of the calculation, yet no explicit verification is supplied that δ and R are not adjusted post-hoc to produce the reported inversion radius and stability behaviors, raising the possibility that the results reduce to input choices by construction.","section":"Abstract and throttling-process analysis"}],"minor_comments":[{"comment":"Notation for the Sharma-Mittal parameters δ and R should be defined explicitly at first use with their physical interpretation, and any relation to the standard Bekenstein-Hawking entropy should be stated.","section":"Introduction or entropy section"}],"recommendation":"major_revision","confidential_remarks":"The manuscript combines three distinct modifications (ModMax, dRGT, Sharma-Mittal) whose mutual consistency is not demonstrated; this raises a scope concern for a journal focused on general relativity and quantum gravity, as the central novelty appears to hinge on an unverified entropy replacement rather than a new derivation within established frameworks."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading of our manuscript and the constructive comments. We respond to the major comments point by point below.","responses":[{"response":"The extended first law is postulated in the extended phase space as the fundamental relation, with temperature defined by T = (∂M/∂S) holding the other extensive variables fixed. Substituting the Sharma-Mittal form for S modifies the resulting T and equation of state in a manner consistent with this definition. The Smarr relation then follows directly from Euler homogeneity applied to the scaling properties of the extended thermodynamic variables, independent of the explicit functional form of S. The metric and mass function are determined by the field equations involving the ModMax stress-energy and dRGT terms, after which the thermodynamic quantities are obtained from the first law; no re-derivation of the stress-energy tensor is needed for the entropy substitution, which is a standard phenomenological step in the literature on generalized entropies. We maintain that the procedure is internally consistent and do not plan to alter the derivations.","revision_made":"no","referee_comment":"[Abstract and entropy introduction] Abstract (paragraph on entropy choice) and the section introducing the thermodynamic quantities: the central claims rest on substituting the area law with the generalised Sharma-Mittal entropy S_δ,R while retaining the standard extended first law dM = T dS + V dP + Φ dQ + … without any derivation or consistency check against the ModMax stress-energy tensor, the dRGT graviton mass terms, or the resulting Smarr relation; this substitution is load-bearing for all reported analytical expressions, stability conclusions, and the inversion/phase-transition results."},{"response":"The inversion temperature, Joule-Thomson coefficient, and critical points are obtained from the closed-form analytical expressions in which γ, δ, R, and the massive gravity parameters appear as independent variables. The reported effects are demonstrated by explicit differentiation and by comparing families of curves in which one class of parameters is varied while the others are held fixed (including the limiting cases that recover the standard entropy). The functional dependence itself produces the observed expansion of the cooling domain under γ and the separation of influences in the critical regime; no auxiliary tuning of δ or R is performed to force these outcomes.","revision_made":"no","referee_comment":"[Abstract and throttling-process analysis] Abstract and throttling-process analysis: the statements that γ 'expands the physically accessible cooling domain' and that 'clear physical decoupling' occurs in the critical regime are presented as outcomes of the calculation, yet no explicit verification is supplied that δ and R are not adjusted post-hoc to produce the reported inversion radius and stability behaviors, raising the possibility that the results reduce to input choices by construction."}],"tokens_in":1514,"tokens_out":586,"duration_ms":22596,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper combines ModMax nonlinear electrodynamics, dRGT massive gravity, and Sharma-Mittal entropy in charged AdS black holes, then works out the extended thermodynamics and throttling process. It produces analytical expressions for temperature, specific heat, Joule-Thomson coefficient, and the equation of state, and it tracks how the ModMax parameter shifts the inversion curve while the massive gravity term sets the overall phase boundary.\n\nWhat stands out is the reported decoupling: ModMax affects the critical boundary by weakening electromagnetic contributions, Sharma-Mittal parameters control local stability, and dRGT sets the global landscape. The Gibbs free energy plot shows the usual swallowtail for a first-order transition. These are the concrete outputs.\n\nThe central move is replacing the area law with Sharma-Mittal entropy while keeping the standard extended first law. The abstract gives no derivation showing that this entropy remains consistent with the ModMax stress-energy tensor or the dRGT graviton mass terms, nor does it check the resulting Smarr relation. The parameters δ and R are introduced to govern stability and inversion radius; once they are free to adjust, the claimed decoupling risks becoming partly a consequence of those choices rather than an independent result. The stress-test note on this point holds up from the abstract alone.\n\nThe work is incremental. It applies three previously studied pieces to one more model and recovers expected qualitative behavior. No new technique or falsifiable prediction appears.\n\nThis is for specialists already tracking modified black hole thermodynamics in AdS who want the combined case catalogued. It does not resolve open questions or supply reproducible checks that would make the claims robust. I would not send it for peer review; the entropy consistency needs to be shown before the rest can be evaluated.","headline":"Routine substitution of three known ingredients into AdS black hole thermodynamics, with the entropy replacement left unexamined.","tokens_in":2501,"tokens_out":418,"would_cite":false,"duration_ms":13567,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"ModMax nonlinearities expand the cooling domain for charged AdS black holes in massive gravity by shifting inversion to smaller radii.","keywords":["AdS black holes","extended thermodynamics","Joule-Thomson expansion","ModMax electrodynamics","massive gravity","Sharma-Mittal entropy","phase transitions"],"falsifier":"A direct computation of the inversion radius as a function of the ModMax parameter γ to check whether it decreases and enlarges the cooling domain.","tokens_in":2758,"feed_emoji":"","tokens_out":592,"duration_ms":26468,"temperature":0.7,"pith_summary":"This paper studies the extended thermodynamics of charged AdS black holes combining ModMax nonlinear electrodynamics and dRGT massive gravity, using Sharma-Mittal entropy to capture non-extensive effects. It shows that the ModMax parameter enlarges the region of cooling during the throttling process by moving the inversion transition to smaller horizon radii. Sharma-Mittal parameters control local stability while the massive gravity background fixes the global phase boundary. The Gibbs free energy displays a van der Waals-like first-order phase transition with a swallow-tail structure. The work separates the influences of each modification on thermodynamic observables.","feed_headline":"ModMax nonlinearities expand black hole cooling domain","feed_subtitle":"Inversion transition shifts inward while massive gravity sets global phase boundary in throttling process.","key_machinery":"The inversion curve of the Joule-Thomson expansion in extended phase space, modified by the ModMax nonlinearity parameter γ and Sharma-Mittal entropy parameters.","core_discovery":"The conformal nonlinearities of the ModMax field expand the physically accessible cooling domain by shifting the inversion transition to smaller horizon radii. While the Sharma-Mittal parameters critically govern local thermodynamic stability and the inversion radius, the global inversion phase boundary remains fundamentally dictated by the massive graviton background. An analysis of the Gibbs free energy uncovers a van der Waals-like first-order phase transition characterized by a distinct swallow-tail structure.","pith_inferences":["Thermodynamic observables can act as diagnostics to distinguish the separate contributions of nonlinear electrodynamics, non-extensive entropy, and massive gravity.","The observed decoupling of effects suggests that similar combined models may allow independent tuning of different thermodynamic features."],"forward_implications":["The physically accessible cooling domain enlarges with increasing ModMax nonlinearity.","Local thermodynamic stability and inversion radius are controlled by Sharma-Mittal parameters δ and R.","The global inversion phase boundary is set by the massive graviton background.","A first-order phase transition appears with swallow-tail structure in the Gibbs free energy."],"fun_headline_variants":["ModMax shifts black hole inversion to smaller radii","Sharma-Mittal governs local stability and radius","Massive gravity dictates global inversion boundary","Van der Waals transition in AdS black hole phases"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The generalised Sharma-Mittal entropy correctly accounts for non-extensive statistical correlations in this black hole system.","fun_headline_variants_meta":{"raw":{"variants":["ModMax shifts black hole inversion to smaller radii","Sharma-Mittal governs local stability and radius","Massive gravity dictates global inversion boundary","Van der Waals transition in AdS black hole phases"]},"model":"grok-4.3","cost_usd":0.006236,"raw_usage":{"total_tokens":2958,"prompt_tokens":712,"num_sources_used":0,"completion_tokens":56,"cost_in_usd_ticks":62362000,"prompt_tokens_details":{"text_tokens":712,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2190,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":712,"tokens_out":56,"duration_ms":11341,"temperature":1.0,"reasoning_tokens":2190,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T01:27:58.388164+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A direct computation of the inversion radius as a function of the ModMax parameter γ to check whether it decreases and enlarges the cooling domain.","supporting_citations":[],"review_version":1}