{"id":"2564ab35-16e2-4eb9-9825-9bca47f8dd48","arxiv_id":"2411.18712","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The BPS limit of the newly constructed hairy black holes has zero entropy and a diverging scalar field, so no regular supersymmetric hairy black holes exist in this sector.","lead":"Using numerical and analytical methods, this paper constructs rotating, charged hairy black holes in a five-dimensional supergravity dual to N=4 SYM, and finds that their supersymmetric (BPS) limit is a singular, horizonless configuration, not a new family of black holes. The result overturns a decade-old conjecture and sharpens the map of black hole phases in AdS5.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"BPS-singularity claim rests on numerical extrapolation from T L ~ 10^-7; a lower-temperature phase transition could restore finite entropy, and the BPS no-go proof explicitly permits oscillatory scalar divergences.","rationale":"The reader's weakest_assumption correctly identifies the extrapolation from T L ~ 10^-7 to T = 0 as the load-bearing premise. The paper provides three independent lines of evidence: very-low-temperature numerics, a near-horizon BPS no-go argument, and an explanation of the perturbative breakdown in Section 6.2. These are substantial and mutually supportive, and the numerical data down to 10^-7 are remarkably close to the BPS surface. However, none of these lines definitively rules out a phase transition or a new branch at even lower temperatures. The BPS analysis explicitly leaves open oscillatory scalar divergences, so it cannot exclude all exotic alternatives, and the low-T numerics, however impressive, are still finite-temperature data. The paper's own history with the 10^-3 data demonstrates the danger of extrapolating thermodynamic quantities to zero temperature in this system. For these reasons the conditional verdict is appropriate, and the concrete test of pushing to lower temperatures and checking convergence of the entropy would settle the concern. The stability caveat regarding other STU modes is acknowledged by the authors and affects the physical interpretation more than the mathematical claim about the specific hairy black hole family, so it is not the primary load-bearing concern.","tokens_in":1037,"tokens_out":779,"duration_ms":48043,"concrete_test":"Extend the numerical construction to T L ~ 10^-9 or lower with an adaptive mesh / arbitrary-precision spectral solver, and for the lowest-T solutions verify exponential convergence under doubling the number of Chebyshev points; then fit S(T) and Φ_horizon(T) over the entire range. If S(T) follows a single power law to the lowest T with S -> 0 and Φ_horizon diverging, the claim is supported; if S(T) shows a plateau or upturn, or if the first-law residual grows below 10^-7, the extrapolation fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the BPS limit of the hairy black holes has vanishing entropy and divergent scalar/curvature, contradicting earlier conjectures. The load-bearing premise is that the monotonic decrease of entropy observed down to T L ~ 10^-7 continues all the way to T = 0. This is an extrapolation, and the paper itself documents that data down to T L ~ 10^-3 led to a wrong finite-entropy conclusion, so there is precedent for misleading low-temperature extrapolation in this very system. Nothing in the analysis bounds the possibility of a new branch or a crossover below 10^-7. The near-horizon BPS analysis (Section 7.2) does not close this gap: its theorem in Appendix B applies only to monotonic φ, and the authors explicitly allow infinitely oscillating divergences such as x^-1 sin(1/x); such a solution would be singular, but the numerical family could in principle connect to a different finite-entropy branch at even lower temperatures. In addition, only a few (J, Q) slices are probed numerically, so the claimed universality is itself an extrapolation. Therefore the singular-BPS conclusion is not established with the certainty implied by the paper's phrasing.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies charged rotating hairy black holes in the S^3-invariant sector of U(1)^3 gauged supergravity, with equal charges Q and equal angular momenta J, dual to thermal states of N=4 SYM. Using matched asymptotic expansions and high-accuracy numerics, the authors construct the non-supersymmetric hairy black hole family that emerges from the scalar condensation instability of the Cvetič-Lü-Pope (CLP) black holes, confirm that these solutions dominate the microcanonical ensemble where they coexist with CLP, and follow them to temperatures as low as T L ~ 10^{-7}, several orders of magnitude below previous work. The central claim is that the BPS limit E = 3Q + 2J/L is singular: the entropy vanishes, while the scalar field and curvature invariants diverge at the horizon. The authors argue that this contradicts the earlier conjectures of Bhattacharyya-Minwalla-Papadodimas and Markeviciute-Santos, and support their conclusion with three lines of evidence: low-temperature numerics that pass first-law checks at the 0.1% level, an analytic explanation of why the perturbative expansion breaks down near the BPS surface, and a near-horizon analysis of the BPS equations that excludes regular solutions under stated assumptions.","tokens_in":59929,"tokens_out":4430,"duration_ms":43623,"significance":"If the conclusion is correct, the paper resolves a long-standing question about the existence of a two-parameter family of supersymmetric hairy black holes extending the one-parameter Gutowski-Reall solution, and it provides a concrete example of a phase of SYM whose gravitational dual is a hairy black hole that becomes singular at the BPS bound. The manuscript is notable for its transparency: numerical data are checked against the first law, the perturbative series (4.13) is compared with independent numerics, and the near-horizon BPS no-go argument is an analytic derivation with clearly stated assumptions. The paper also makes a falsifiable prediction about the low-temperature behavior of the entropy, which can in principle be checked with further numerical or analytic work. However, the strongest form of the claim, namely that the BPS limit is singular with certainty, rests on an extrapolation from finite-temperature data and on a BPS theorem that explicitly leaves open oscillatory scalar divergences; the gap between the evidence and the categorical phrasing is the main weakness.","major_comments":[{"comment":"The statement that S -> 0 at E = E_BPS is based on extrapolating numerical data from T L ~ 10^{-7} to exactly T = 0, for a small set of (J,Q) slices: J/N^2 = 0.005 with QL/N^2 = 0.05 and 0.1, and J/N^2 = 0.05 with QL/N^2 = 0.15. The paper itself documents that data down to T L ~ 10^{-3} in refs. [2,3] suggested a finite entropy and were misleading, so this system already exhibits a low-temperature regime that changes qualitative behavior well below the previously explored range. Nothing in the present analysis excludes a further branch, crossover, or oscillatory approach below T L ~ 10^{-7} that could restore finite entropy at the BPS limit. To make the central claim conclusive, the authors should either provide a quantitative bound (for example from the near-horizon BPS equations) that rules out such behavior below the numerically reached temperature, or explicitly state that the conclusion is a strong numerical inference rather than a demonstrated theorem, and soften the wording accordingly.","section":"Section 6.1, Figs. 15-16"},{"comment":"The near-horizon BPS analysis proves that no regular BPS black hole exists if the scalar field is finite at the horizon, or if it diverges monotonically. As the authors explicitly acknowledge, the theorem does not exclude infinitely oscillating divergences such as x^{-1} sin(1/x). This is not a purely technical loophole: such an oscillating behavior could in principle connect to a finite-entropy branch at even lower temperatures, which is precisely the scenario the numerical extrapolation cannot rule out. The theorem should be strengthened, for example by adding physical assumptions such as boundedness of curvature invariants, analyticity of the horizon, or a condition on the oscillation frequency, or the paper should explicitly state that the no-go argument leaves this case open and therefore does not by itself close the gap left by the numerics.","section":"Section 7.2 and Appendix B"},{"comment":"The perturbative-breakdown argument convincingly explains why the series (4.13) and the non-interacting thermodynamic model fail close to the BPS surface. However, this argument does not by itself prove that S -> 0 at E = E_BPS; the reinterpretation via a simultaneous 'double BPS limit' y_+ -> 0, delta -> 0 is, as the authors admit, hand-waved. The two roles of this section should be cleanly separated: as a diagnosis of the failure of perturbation theory it is sound, but as an independent derivation of the singular limit it needs a controlled statement about the commutativity of the limits y_+ -> 0, delta -> 0 and the numerical T -> 0 limit. In the absence of such a statement, the numerical extrapolation remains the load-bearing evidence for the central claim.","section":"Section 6.2, Eqs. (6.3)-(6.4)"}],"minor_comments":[{"comment":"The phrase 'à priori' should be 'a priori' throughout.","section":"Section 3.2"},{"comment":"There is a typo in 'the scalr field' that should read 'the scalar field'.","section":"Section 6.1"},{"comment":"The sentence 'We conclude that the latter always has higher entropy' is ambiguous: if 'the latter' refers to the CLP black hole, it contradicts the preceding claim that the hairy black hole dominates the microcanonical ensemble; if it refers to the hairy black hole, the antecedent should be made explicit.","section":"Section 6.1"},{"comment":"The spelling 'Breitlohner-Freedmann bound' is inconsistent with the standard 'Breitenlohner-Freedman' used in the references; please standardize.","section":"Sections 2.3, 8, and references"},{"comment":"The right panels plot entropy against E - E_BPS on a log-log scale rather than against temperature; the captions should state this explicitly to avoid confusion.","section":"Figs. 15-16 captions"},{"comment":"The thermodynamic series (4.13a)-(4.13g) are given to high order with logarithmic terms, but the precise conditions of validity (for example delta >> y_+ and delta >> epsilon) are stated only later in Section 6.2; a brief note or footnote near (4.13) would prevent readers from applying the expansion outside its regime of validity.","section":"Section 4, Eq. (4.13)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a serious and careful contribution, and the central claim may well be correct, but the categorical wording ('unequivocally demonstrates', 'bullet-proof evidence', 'we have demonstrated') exceeds what the extrapolative evidence supports. A revision that (i) tempers the central claim to 'strong numerical evidence' or closes the extrapolation gap with a quantitative bound, and (ii) strengthens or explicitly delimits the BPS no-go theorem, would be suitable for publication. The citation pattern includes several self-citations, but the key evidence is checked against independent numerics and the first law, so I do not see that as a concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper argues, with three independent lines of evidence, that the BPS limit of the equal-charge, equal-spin hairy black holes in AdS5×S5 is singular: entropy goes to zero, the scalar field and curvature invariants diverge at the horizon. That directly contradicts the Bhattacharyya–Minwalla–Papadodimas conjecture and the Markeviciute–Santos interpretation of their numerics. I think the authors are probably right, and more importantly, they are unusually honest about where they cannot be sure.\n\nWhat is genuinely new: the double perturbative construction of rotating charged hairy black holes in Section 4 (expansion in horizon radius and scalar condensate), which is checked against independent numerics with first-law violations stated order by order in (4.14). The numerical campaign reaching T L ~ 10^-7, four orders of magnitude lower than the previous work, is the key empirical contribution: it shows the entropy trend down to T ~ 10^-3 had appeared to extrapolate to a finite BPS value, then bends sharply. That is a real finding, and it vindicates the effort to push numerics further. The near-horizon BPS analysis in Section 7.2 is parameter-free and consistent with the numerics, and the Appendix B theorem is stated with its assumptions clear.\n\nWhere the soft spots are, in proportion: the central claim is an extrapolation. Numerics cannot prove S→0 at exactly T = 0, and the BPS analysis explicitly leaves the oscillatory-divergence loophole open (x^-1 sin(1/x) type behavior) — the authors say this themselves. Only a few (J,Q) slices are probed numerically, so the universality of the statement is itself an extrapolation over parameter space. And stability of the hairy solutions against the remaining STU modes is unverified, which the paper flags in Section 8; if the family is not the true endpoint, the conclusion about what happens on the BPS surface could be moot. The stress-test note worries about a lower-temperature phase transition restoring finite entropy; that is a legitimate worry, but I would not call it a flaw, because the paper never claims a proof — it argues, and it tells you exactly what would falsify the claim.\n\nClearly serious thinking, honest engagement with the literature, and the earlier incorrect conjecture by one of the same authors is handled gracefully. Who this is for: anyone working on AdS5 black hole microstate counting or hairy black hole phase diagrams. It deserves a serious referee, and in my view the right outcome is refereed publication with the caveats kept front and center.","headline":"A careful, probably-correct negative result on the BPS limit of hairy AdS5 black holes; the central claim rests on an acknowledged numerical extrapolation, but the paper is honest about exactly that.","tokens_in":60440,"tokens_out":3200,"would_cite":true,"duration_ms":28369,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper argues that the supersymmetric limit of rotating charged hairy black holes in AdS$_5\\times S^5$ is singular: entropy vanishes, the scalar field diverges at the horizon, and no new regular two-parameter BPS family exists.","keywords":["hairy black holes","AdS5/CFT4","gauged supergravity","BPS limit","scalar condensation instability","microcanonical ensemble","Gutowski-Reall black hole","near-horizon BPS equations"],"falsifier":"A regular solution of the BPS equations with a smooth horizon, finite entropy, finite scalar field, and $\\Delta_{\\rm KLR}\\neq 0$ would refute the claim; so would numerical detection of a new finite-entropy branch below $T L\\sim 10^{-7}$.","tokens_in":59464,"feed_emoji":"🕳️","tokens_out":8583,"duration_ms":88970,"temperature":0.7,"pith_summary":"This paper revisits a long-standing puzzle in holography: whether the known one-parameter family of supersymmetric black holes in AdS$_5\\times S^5$ (the Gutowski-Reall family) can be extended by adding scalar hair into a two-parameter family. Earlier work conjectured that such a regular two-parameter family exists, citing a thermodynamic model and numerical solutions that seemed to approach finite entropy as the temperature dropped. The authors construct the hairy black holes both in matched perturbation theory and by direct numerical solution, pushing the temperature down to $T L\\sim 10^{-7}$, about four orders of magnitude colder than earlier studies. Their central claim is that the BPS limit of these hairy black holes is singular: the entropy goes to zero while the charged scalar and curvature invariants diverge at the horizon, contradicting the earlier conjecture. If correct, the gravitational phase space near the BPS bound does not contain a new regular hairy family, and the one-parameter Gutowski-Reall solution remains the only regular supersymmetric black hole in this sector.","feed_headline":"Hairy black holes hit a singular supersymmetric limit","feed_subtitle":"New low-temperature data contradict the long-expected BPS black-hole family.","key_machinery":"The main machinery is a matched asymptotic expansion with three overlapping regions (far, intermediate, near) that constructs hairy black holes in a double expansion in horizon radius and scalar condensate, cross-checked against direct numerical integration down to $T L\\sim 10^{-7}$. On the BPS side the decisive object is the near-horizon analysis of the BPS equations in the coordinates (7.3)--(7.5): using the monotonicity theorem that $f'=o(g')$ implies $f$ is either a constant plus $o(g-g(0))$ or $o(g)$, the authors show a regular horizon forces the charged scalar to vanish identically, which would reduce the theory to the bald Gutowski-Reall solution.","core_discovery":"In the equal-charge, equal-angular-momentum sector of $U(1)^3$ gauged supergravity, the paper finds that as the mass approaches the BPS bound $E = 3Q + 2J/L$, the hairy black hole reaches $T\\to 0$, $\\mu\\to 1$, and $\\Omega_H L\\to 1$, but its entropy vanishes and the scalar field at the horizon diverges, so curvature invariants diverge there. The BPS limit is therefore a singular, horizonless configuration rather than a new two-parameter family of regular supersymmetric black holes with $\\Delta_{\\rm KLR}\\neq 0$. This directly contradicts the conjectures based on the non-interacting thermodynamic model and on previous numerical data that had only reached $T L\\sim 10^{-3}$. Three independent routes support the conclusion: a perturbative construction whose regime of validity is shown to break down precisely near the BPS surface, an improved numerical analysis at much lower temperature, and a near-horizon study of the BPS equations that finds no regular solution with finite scalar and finite horizon entropy.","pith_inferences":["If the scalar-condensation channel cannot produce a regular BPS family, the natural candidates to fill the missing region of the phase diagram are the time-periodic black resonators and the graviton-gas/dual-giant endpoints discussed in the paper; the singular-limit result sharpens the case for those alternatives.","The near-horizon BPS analysis leaves open a scalar that diverges while oscillating infinitely often (for example like $x^{-1}\\sin(1/x)$); testing whether such exotic solutions exist is a concrete next step beyond this paper.","The no-go is proven only for the equal-charge, equal-spin sector; an immediate extension would be to repeat the low-temperature numerical analysis for unequal charges or angular momenta, where a regular two-parameter BPS family could still hide.","The authors note they have not checked stability against condensation of the remaining scalar fields that were set to zero; if those channels become unstable at very low temperature, the true endpoint could differ from the singular branch studied here."],"forward_implications":["Where hairy and CLP black holes coexist at fixed energy, charge, and angular momentum, the hairy black holes have larger entropy; if the central claim holds, they are the dominant microcanonical phase and describe a new thermodynamic phase of ${\\cal N}=4$ SYM.","The BPS surface is not a new branch of regular solutions: the one-parameter Gutowski-Reall family remains the only regular supersymmetric black hole in this sector, and the proposed missing gravitational parameter is not scalar hair.","The non-interacting thermodynamic model and the perturbative expansions are excellent away from the BPS surface but fail extremely close to it; their finite-entropy prediction at $T=0$ is an artifact of the expansion.","Earlier low-temperature data stopping at $T L\\sim 10^{-3}$ were insufficient: the entropy curve changes slope and plunges to zero only below that range.","Because the scalar field diverges at the horizon in the BPS limit, the limiting configuration cannot serve as a regular endpoint in the AdS/CFT phase diagram, so other solutions are needed in parts of the phase diagram where CLP black holes do not exist."],"supporting_citations":[{"why":"Conjectured a regular supersymmetric hairy black hole and supplied the non-interacting thermodynamic model whose BPS prediction the present paper refutes.","marker":"[1]"},{"why":"Provided numerical hairy-black-hole data down to $T L\\sim 10^{-3}$ that was interpreted as finite entropy in the BPS limit; this paper argues that extrapolation was misleading.","marker":"[2]"},{"why":"Constructed rotating hairy black holes at moderate temperatures and identified the critical-charge behaviour that the new data extends to much lower temperature.","marker":"[3]"},{"why":"Derived the BPS equations and ansatz used in the near-horizon no-go analysis.","marker":"[15]"},{"why":"Constructed the Gutowski-Reall black hole, the one-parameter regular BPS family that the conjectured two-parameter family would have extended.","marker":"[21]"},{"why":"Derived the $\\Delta_{\\rm KLR}=0$ charge constraint that frames the missing-parameter puzzle for supersymmetric AdS5 black holes.","marker":"[26]"},{"why":"Supplied the pseudospectral Newton-Raphson numerical methods used to reach temperatures $T L\\sim 10^{-7}$.","marker":"[48]"}],"fun_headline_variants":["Hairy black holes end in singularity, not BPS family","BPS limit of hairy black holes is singular, no new family","No new BPS black holes: hairy limit is singular","Singular end for hairy black holes at BPS bound","Hairy black holes contradict BPS family conjecture"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the numerical family seen at $T L\\sim 10^{-7}$ continues all the way to $T=0$ with nothing new appearing in between; if a colder branch with finite entropy existed, the central claim would fail.","fun_headline_variants_meta":{"raw":{"variants":["Hairy black holes end in singularity, not BPS family","BPS limit of hairy black holes is singular, no new family","No new BPS black holes: hairy limit is singular","Singular end for hairy black holes at BPS bound","Hairy black holes contradict BPS family conjecture"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000339,"raw_usage":{"total_tokens":1984,"prompt_tokens":1173,"completion_tokens":811,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":789,"completion_tokens_details":{"reasoning_tokens":729}},"tokens_in":789,"tokens_out":811,"duration_ms":8044,"temperature":1.0,"reasoning_tokens":729,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T10:57:12.681115+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A regular solution of the BPS equations with a smooth horizon, finite entropy, finite scalar field, and $\\Delta_{\\rm KLR}\\neq 0$ would refute the claim; so would numerical detection of a new finite-entropy branch below $T L\\sim 10^{-7}$.","supporting_citations":[{"cited_title":"Rotating Hairy Black Holes in AdS$_5\\times$S$^5$","cited_arxiv_id":"1809.04084","evidence_quote":"Constructed rotating hairy black holes at moderate temperatures and identified the critical-charge behaviour that the new data extends to much lower temperature."},{"cited_title":"New supersymmetric solutions of N=2, D=5 gauged supergravity with hyperscalars","cited_arxiv_id":"0705.2234","evidence_quote":"Derived the BPS equations and ansatz used in the near-horizon no-go analysis."}],"review_version":1}