{"id":"a706cc16-8789-4483-b57a-335b040f7f4e","arxiv_id":"2509.08005","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"EGUP corrections to the RN-AdS black hole surrounded by perfect fluid dark matter shift its critical pressure, temperature, and volume, and leave the first-order phase transition intact.","lead":"This paper adds quantum-gravity-inspired uncertainty corrections (EGUP) to the thermodynamics of a charged, dark-matter-surrounded black hole in anti-de Sitter space. It reports that the corrections shift the critical point slightly and mildly stabilize intermediate-sized black holes.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Entropy Eq. (44) is not conjugate to temperature Eq. (18): dM=T dS fails already at O(beta), so Eq. (53)'s Gibbs free energy and the swallowtail phase-transition plots are unsupported.","rationale":"The reader's weakest_assumption names the dA/dS substitution and the gamma calibration; these are heuristic modeling choices, but they are not the decisive problem. The decisive problem is internal: the entropy used to construct the Gibbs free energy is inconsistent with the temperature used to construct it. We verified this analytically in the alpha=0 limit where the paper's formulas are explicit enough to differentiate: Eq. (45) and Eq. (19) violate the first-law relation dM = T dS. Since Eq. (53) uses Eq. (44), the swallowtail and phase-equilibrium claims in the abstract and Sec. 5 inherit that inconsistency. This is not a disagreement with an outside consensus; it is an internal arithmetic failure. The gamma calibration is a modeling assumption that one could debate, but the entropy/temperature mismatch is a correctness error that can be checked directly. We therefore keep the reader's REJECT, with the note that the temperature and P-v criticality sections might be salvageable if the entropy were re-derived from the first law rather than from C/T dT. The concrete numerical test above would settle the issue immediately, especially if the authors provide their derivation or code.","tokens_in":13654,"tokens_out":7972,"duration_ms":68667,"concrete_test":"Set l_p = L = 1, Q = 0.3, lambda = 0.1, l = sqrt(3/(8 pi * 0.037)), alpha = beta = 0.03, and evaluate at r_+ = 1.0. Compute dM/dr_+ from Eq. (11), T_EGUP from Eq. (18), and dS_EGUP/dr_+ by numerically differentiating Eq. (44). If |dM/dr_+ - T_EGUP dS_EGUP/dr_+| / |dM/dr_+| exceeds 1%, Eq. (44) is not the integral of Eq. (18) and the G-T plots in Fig. 5 are built on an invalid free energy. A closed-form cross-check at alpha = 0 is the O(beta) disagreement between pi r_+(1 + sqrt(1 - beta l_p^2/(4r_+^2))) and 2 pi r_+ - pi beta l_p^2/(4r_+).","verdict_should_be":"UNCHANGED","load_bearing_attack":"Eq. (44) is not thermodynamically conjugate to Eq. (18). From the first law (13), consistency at fixed Q, P, lambda requires dM/dr_+ = T dS/dr_+. The paper's own alpha=0 results are explicit enough to test this. With F = 1 - Q^2/r_+^2 + 3r_+^2/l^2 + lambda/r_+, Eq. (11) gives dM/dr_+ = F/2, while Eq. (19) can be rewritten as T_GUP = F/[2 pi r_+ (1 + sqrt(1 - beta l_p^2/(4r_+^2)))]. Consistency therefore requires dS/dr_+ = pi r_+ (1 + sqrt(1 - beta l_p^2/(4r_+^2))). Differentiating the paper's Eq. (45) gives dS/dr_+ = 2 pi r_+ - pi beta l_p^2/(4r_+), which disagrees already at first order in beta: the small-beta expansion of the required derivative is 2 pi r_+ - pi beta l_p^2/(8r_+). The same defect carries into Eq. (44) and therefore into G = M - T S in Eq. (53), which drives the swallowtail curves of Fig. 5. Because G is not obtained by a Legendre transform from a consistent first law, the central phase-transition claim is not supported by the manuscript; the temperature and equation-of-state parts, which do not use S, might survive independently.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies extended generalized uncertainty principle (EGUP) corrections to the thermodynamics and P-v criticality of the Reissner-Nordström-AdS black hole surrounded by perfect fluid dark matter. The authors start from the first law in the extended phase space, use a heuristic relation between entropy-area change and position-momentum uncertainties to derive an EGUP-corrected Hawking temperature (Eq. (18)), and then compute the heat capacity, entropy, remnant quantities, equation of state, and Gibbs free energy. They analyze heat-capacity divergence and G-T 'swallowtail' diagrams. The central claims are that larger EGUP parameters shrink the unstable intermediate black hole branch, that the EGUP correction is beneficial for thermodynamic stability, and that a first-order phase transition persists for P<P_c with a slightly shifted equilibrium point and reduced critical pressure and temperature as alpha and beta increase.","tokens_in":14059,"tokens_out":14543,"duration_ms":126064,"significance":"Should the derivations hold, the paper would provide a self-contained extension of GUP-corrected black-hole thermodynamics to a PFDM background, with explicit analytic formulas and a parameter-free calibration of gamma from the HUP limit. The temperature and equation-of-state parts reduce correctly to the HUP case, and the graphical analysis is systematic. However, the central thermodynamic consistency condition dM = T dS fails for the derived entropy, and because the Gibbs free energy (and hence the swallowtail phase-transition curves) is built from that entropy, the paper's main phase-transition conclusions are not currently supported by the manuscript. The first-law failure is a mathematical error localized in the entropy integration; the temperature and P-v criticality results may survive a correction, but all S-dependent results need to be rederived before the phase-transition claims can be assessed.","major_comments":[{"comment":"The GUP entropy is not thermodynamically conjugate to the GUP temperature, so the Gibbs free energy and the swallowtail analysis are not supported. With F = 1 - Q^2/r_+^2 + 3r_+^2/l^2 + lambda/r_+, Eq. (11) gives dM/dr_+ = F/2, and Eq. (19) is equivalent to T_GUP = F/[2 pi r_+ (1 + sqrt(1 - beta l_p^2/(4r_+^2)))]. The first law (13) at fixed Q, P, lambda then requires dS/dr_+ = pi r_+ (1 + sqrt(1 - beta l_p^2/(4r_+^2))) = 2 pi r_+ - pi beta l_p^2/(8r_+) + O(beta^2). Differentiating the paper's Eq. (45) gives dS_GUP/dr_+ = 2 pi r_+ - pi beta l_p^2/(4r_+), disagreeing already at first order in beta. This inconsistency propagates into the full EGUP entropy Eq. (44) and into G = M - T S in Eq. (53), which is the quantity whose swallowtail curves are shown in Fig. 5. The phase-transition conclusions drawn from G are therefore not established; the temperature and equation-of-state results, which do not use S, are not affected by this particular error.","section":"Section 3, Eqs. (19), (45), and (53)"},{"comment":"The entropy integral is not written correctly, and the stated derivation cannot produce a consistent S. The formula S = Integral (partial M / T)_{Q,P,lambda} should read S = Integral (1/T)(partial M/partial r_+) d r_+ at fixed Q, P, lambda, and the replacement by Integral C dT/T is only meaningful if C is computed as (partial M/partial T)_P. As written, the display is formally ambiguous. More importantly, the failure of T and S to satisfy dM = T dS shows that Eq. (44) is not the result of a correct integration; the authors should recompute S directly from dS = dM/T and then rederive all S-dependent quantities, including Eq. (53) and Fig. 5.","section":"Section 3, Eq. (43)"},{"comment":"The entire EGUP correction is fixed by the heuristic dA/dS ~ (gamma/ln2) Delta X Delta P with Delta X ~ 2 r_+, and gamma = 4 ln2 is chosen to reproduce the HUP limit. This is an imported assumption rather than a derivation, and the HUP limit is enforced by construction rather than obtained from the model. The functional form of every corrected quantity, including the critical pressure and temperature in Table 1, depends on this choice. The authors should test the robustness of their qualitative conclusions by, for example, repeating the analysis for Delta X = k r_+ with k a free parameter of order one, and discuss whether the stability and phase-transition conclusions are independent of k.","section":"Section 2 and Eq. (15)"}],"minor_comments":[{"comment":"Logarithms of dimensionful quantities appear (L^4, r_+, and combinations with l_p^2). Since the figures set L = l_p = 1, this is hidden, but in a dimensionful formulation the arguments of the logarithms should be made dimensionless.","section":"Eqs. (44), (45), and (53)"},{"comment":"The paper defines kappa as f'(r_+), whereas the standard surface gravity is f'(r_+)/2. The subsequent conventions are internally consistent after fixing gamma, but the nonstandard definition should be stated explicitly to avoid confusion.","section":"Eq. (12) and surrounding text"},{"comment":"The remnant-mass expression appears to contain typos, including 'alpha beta 12 p' in the denominator and an inconsistent power of l_p. Please check and correct the formula.","section":"Eq. (37)"},{"comment":"The roots r_+1 and r_+2 are displayed with plus-minus signs, but only the physical positive root is retained for the constraint analysis. Please specify which branches are used and why.","section":"Eqs. (23)-(26)"},{"comment":"Table 1 shows that the ratio P_c v_c/T_c decreases from 0.389 to 0.284 as alpha and beta increase, a change of about 27 percent. The text says the phase-equilibrium point changes only slightly; this quantitative shift in the critical parameters should be mentioned and interpreted.","section":"Section 4 and Table 1"}],"recommendation":"major_revision","confidential_remarks":"I confirmed the first-law inconsistency independently by expanding both sides of dM = T dS in powers of beta, so this is not a stylistic concern. The temperature and equation-of-state sections are separable and could form the basis of a revised manuscript, but the entropy and all S-dependent quantities must be recomputed. If the authors cannot restore dM = T dS, the phase-transition claims should be withdrawn."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know: this is a routine extension of the EGUP-based black hole thermodynamics program to the PFDM background. The temperature and P-v criticality sections are plausible; the entropy, however, fails the first law with the stated temperature, so the Gibbs free energy swallowtail plots and the first-order phase transition conclusion are not supported.\n\nThe genuinely new bit is the specific background: no one has done EGUP corrections for this RN-AdS+PFDM line element before. The temperature in Eq. (18), the EOS in Eq. (50), and the critical-point table are consistent with the uncorrected limits and follow the same path as Tan's quintessence paper. The heat capacity analysis, Figs. 2 and 6, is based on C = (∂M/∂T) and does not depend on the entropy, so the claim that EGUP shrinks the unstable intermediate branch may be right.\n\nThe problem is the entropy. Take the alpha=0 limit. From Eq. (19), T_GUP can be written as F/(2π r_+ (1+sqrt(...))), with F the metric function combination. The first law dM = T dS then requires dS/dr_+ = π r_+ (1+sqrt(...)) = 2π r_+ - π β l_p^2/(8 r_+) to leading order. Differentiating the paper's Eq. (45) gives 2π r_+ - π β l_p^2/(4 r_+), off by a factor of two. So dM = T dS fails already at O(β). That error carries into the full EGUP entropy Eq. (44) and into G = M - T S in Eq. (53). The swallowtail curves in Fig. 5 are therefore not Legendre transforms of a consistent thermodynamic system. This is a load-bearing flaw, not a typo.\n\nThe heuristic input—Delta X ≈ 2r_+ and γ fitted to the HUP limit—is imported from prior work. That is a model assumption, not a derivation, and makes the whole framework phenomenological. I would not call it fatal, but it is worth noting.\n\nWho gets value: readers working on GUP/EUP-corrected black hole thermodynamics will find the temperature and P-v parts useful, and the entropy error is instructive. It deserves a serious referee, because the mistake is fixable and the rest is standard. Recommendation: send to review, but the current version should not be accepted; the entropy and G must be redone before any reliable phase-transition statement.","headline":"Routine EGUP extension to PFDM; temperature and P-v parts are plausible, but the entropy fails the first law, so the swallowtail phase-transition claim is unsupported.","tokens_in":14523,"tokens_out":5306,"would_cite":false,"duration_ms":42557,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that extended generalized uncertainty principle (EGUP) corrections to the thermodynamics of a charged, dark-matter-surrounded anti-de Sitter black hole shrink the unstable intermediate branch, lower the critical pressure…","keywords":["extended generalized uncertainty principle","perfect fluid dark matter","Reissner-Nordstrom anti-de Sitter black hole","black hole thermodynamics","P-v criticality","first-order phase transition","Hawking temperature","Gibbs free energy"],"falsifier":"Recompute the EGUP-corrected Hawking temperature using the full position-uncertainty bounds of Eq. (6) instead of the simplifying assumption $\\Delta X \\simeq 2r_+$, then re-derive the heat capacity; if the negative-heat-capacity region does not shrink with $\\alpha$ and $\\beta$, the stability conclusion is an artifact of the simplification. A direct semiclassical calculation of $dA/dS$ for the PFDM metric would also either confirm or overturn the substitution at the heart of the paper.","tokens_in":13447,"feed_emoji":"🕳️","tokens_out":7215,"duration_ms":56575,"temperature":0.7,"pith_summary":"The paper sets out to graft the extended generalized uncertainty principle (EGUP) onto the thermodynamics of a Reissner-Nordström anti-de Sitter black hole surrounded by perfect fluid dark matter (PFDM), producing explicit corrected formulas for Hawking temperature, heat capacity, entropy, and Gibbs free energy. It argues that the corrections favor thermodynamic stability by shrinking the negative-heat-capacity region of the intermediate black hole branch, and that the familiar small-to-large black hole phase transition survives with a slightly moved equilibrium point. A sympathetic reader should care because EGUP is a proposed bridge between quantum gravity and black hole physics, and this work tests whether that bridge preserves the van der Waals-like phase structure of AdS black holes. The paper also derives remnant mass and temperature, giving a concrete prediction for where black hole evaporation stops.","feed_headline":"Uncertainty corrections shrink the unstable black hole branch","feed_subtitle":"EGUP-modified thermodynamics keeps the swallowtail phase transition while lowering critical pressure and temperature.","key_machinery":"The load-bearing device is the EGUP uncertainty relation, $\\Delta x_i \\Delta p_j \\geq (\\hbar/2)\\delta_{ij}[1 + \\beta l_p^2 (\\Delta p_j)^2/\\hbar^2 + \\alpha (\\Delta x_i)^2/L^2]$, combined with the information-theoretic substitution $dA/dS \\simeq (\\gamma/\\ln 2)\\Delta X\\Delta P$ used to convert surface gravity into a corrected Hawking temperature. With $\\Delta X \\simeq 2r_+$ and $\\gamma = 4\\ln 2$ fixed by demanding the HUP limit reproduce $T = 1/(4\\pi r_+)$, this machinery injects the EGUP parameters into every subsequent quantity: the heat capacity follows from $C_P = (\\partial M/\\partial T)_P$, the entropy from integrating $C/T$, the equation of state from inverting the corrected temperature for $P$, and the Gibbs free energy from $G = M - TS$. The same substitution also produces the minimum horizon radii $r_{\\rm rem(EGUP)}$ and $r_{\\rm rem(GUP)}$ that define black hole remnants.","core_discovery":"The paper claims that applying the EGUP to the RN-AdS black hole surrounded by PFDM yields a consistent set of corrected thermodynamic quantities: the Hawking temperature in Eq. (18), heat capacity in Eq. (33), entropy in Eq. (44), and Gibbs free energy in Eq. (53). Using these, the paper argues that the negative-heat-capacity region corresponding to the intermediate black hole shrinks as the EGUP parameters $\\alpha$ and $\\beta$ increase, so the correction favors thermodynamic stability. It further claims that the first-order phase transition signaled by a swallowtail in the $G$-$T$ diagram survives for $P < P_c$, with the equilibrium point shifting only slightly, and that for $P = P_c$ and $P > P_c$ the phase-transition behavior mirrors the uncorrected case, even though the critical pressure, critical temperature, and the ratio $P_c v_c / T_c$ all decrease with the EGUP parameters.","pith_inferences":["Editorial extension: the same $dA/dS$ substitution with $\\Delta X \\simeq 2r_+$ could be applied to other AdS black holes, such as rotating or higher-dimensional ones; the paper's logic suggests the qualitative stabilization and preservation of the phase transition would persist, but the size of the shifts would depend on the metric details.","Editorial extension: because $\\gamma$ is calibrated to recover the HUP limit, the correction is forced to vanish at large radii; a sharper test of the framework would probe intermediate radii where the correction peaks, for instance by computing quasinormal mode frequencies or lensing signatures that depend on the corrected horizon temperature.","Editorial extension: the monotonic decrease of $P_c$ with $\\alpha$ and $\\beta$ in Table 1 hints at a systematic relation that could be extracted by scanning more parameter values; such a relation would let future observations or analogue experiments bound the EGUP parameters."],"forward_implications":["For fixed charge, dark matter parameter, and pressure, increasing $\\alpha$ and $\\beta$ shrinks the interval of horizon radii with negative heat capacity, so the small and large black hole branches become stable over a wider range.","The critical pressure $P_c$, critical specific volume $v_c$, and the ratio $P_c v_c / T_c$ all decrease as $\\alpha$ and $\\beta$ grow, shifting the phase diagram toward lower pressure and temperature.","The swallowtail in the Gibbs free energy versus temperature plot persists for $P < P_c$, meaning the first-order small-black-hole/large-black-hole transition survives EGUP corrections.","For $P = P_c$, a second-order phase transition remains, signaled by a single heat-capacity divergence and the disappearance of the intermediate branch, while for $P > P_c$ no phase transition occurs.","The EGUP-corrected entropy is smaller than the area-law entropy for the same horizon radius, and the deficit grows with the EGUP parameters."],"supporting_citations":[{"why":"Supplies the EGUP relation and the information-theoretic $dA/dS$ substitution used to derive the corrected Hawking temperature.","marker":"[50]"},{"why":"Provides the $\\Delta X \\simeq 2r_+$ assumption connecting particle size to the black hole horizon radius.","marker":"[58]"},{"why":"Gives the $dA/dS \\simeq (\\gamma/\\ln 2)\\Delta X\\Delta P$ relation linking area change to information loss.","marker":"[56]"},{"why":"Gives the RN-AdS black hole solution surrounded by PFDM and its first law in extended phase space.","marker":"[28]"},{"why":"Introduces the perfect-fluid dark matter energy-momentum tensor and the logarithmic metric term used in the line element.","marker":"[18]"},{"why":"Introduces the extended uncertainty principle that is linearly combined with the GUP to form the EGUP.","marker":"[45]"},{"why":"Supplies the linear combination of EUP and GUP that gives the EGUP form used in Eq. (4).","marker":"[46]"},{"why":"Sets up the $P$-$v$ criticality and van der Waals comparison used in Section 4.","marker":"[61]"}],"fun_headline_variants":["EGUP correction shrinks unstable black hole branch","Uncertainty principle stabilizes black holes","EGUP lowers critical pressure in black hole","Swallowtail phase transition persists with EGUP","Modified uncertainty shrinks black hole instability"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire chain rests on treating the uncertainty-product substitution $dA/dS \\simeq (\\gamma/\\ln 2)\\Delta X\\Delta P$ with $\\Delta X \\simeq 2r_+$ as valid for a charged, dark-matter background; if that heuristic fails, every corrected quantity—temperature, heat capacity, entropy, and free energy—changes.","fun_headline_variants_meta":{"raw":{"variants":["EGUP correction shrinks unstable black hole branch","Uncertainty principle stabilizes black holes","EGUP lowers critical pressure in black hole","Swallowtail phase transition persists with EGUP","Modified uncertainty shrinks black hole instability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000187,"raw_usage":{"total_tokens":1342,"prompt_tokens":971,"completion_tokens":371,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":587,"completion_tokens_details":{"reasoning_tokens":304}},"tokens_in":587,"tokens_out":371,"duration_ms":3744,"temperature":1.0,"reasoning_tokens":304,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:19:41.859229+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the EGUP-corrected Hawking temperature using the full position-uncertainty bounds of Eq. (6) instead of the simplifying assumption $\\Delta X \\simeq 2r_+$, then re-derive the heat capacity; if the negative-heat-capacity region does not shrink with $\\alpha$ and $\\beta$, the stability conclusion is an artifact of the simplification. A direct semiclassical calculation of $dA/dS$ for the PFDM metric would also either confirm or overturn the substitution at the heart of the paper.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the EGUP relation and the information-theoretic $dA/dS$ substitution used to derive the corrected Hawking temperature."},{"cited_title":"Hassanabadi, E","cited_arxiv_id":null,"evidence_quote":"Provides the $\\Delta X \\simeq 2r_+$ assumption connecting particle size to the black hole horizon radius."},{"cited_title":"Xiang and X","cited_arxiv_id":null,"evidence_quote":"Gives the $dA/dS \\simeq (\\gamma/\\ln 2)\\Delta X\\Delta P$ relation linking area change to information loss."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the RN-AdS black hole solution surrounded by PFDM and its first law in extended phase space."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the perfect-fluid dark matter energy-momentum tensor and the logarithmic metric term used in the line element."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the extended uncertainty principle that is linearly combined with the GUP to form the EGUP."},{"cited_title":"Bolen and M","cited_arxiv_id":null,"evidence_quote":"Supplies the linear combination of EUP and GUP that gives the EGUP form used in Eq. (4)."},{"cited_title":"Kubiznak and R","cited_arxiv_id":null,"evidence_quote":"Sets up the $P$-$v$ criticality and van der Waals comparison used in Section 4."}],"review_version":1}