{"id":"e47c8607-5180-4cd6-a325-99b1f0b6b012","arxiv_id":"1908.07799","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A six-dimensional rotating black hole with two spins cannot be overspun by test particle accretion, so weak cosmic censorship holds for this case.","lead":"Six-dimensional rotating Myers-Perry black holes with two rotations are shown to resist overspinning even when test particles are accreted, so their event horizons cannot be destroyed. The result supports the weak cosmic censorship conjecture in higher dimensions and extends prior work that found a five-dimensional counterexample.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof only checks selected angular-momentum channels; scenario (iii) and the a=2b ratio are asserted without showing that the unexamined channels cannot overspin.","rationale":"The reader's verdict is CONDITIONAL and the identified weakest assumption is the omitted near-extremal expansion of r_+. That is a fair technical concern, but the more load-bearing gap is the insufficient coverage of the accretion parameter space. The paper's headline is a universal no-overspinning statement, so it needs to exclude every possible linear accretion channel, not only the three symmetric endpoints in the equal-rotation case and one ratio in the a=2b case. The equal-rotation scenario (iii) is especially exposed: the text asserts equality of δJmin without showing the calculation, and the extremality function Eq. (7) is manifestly asymmetric under exchange of a and b. The a=2b section similarly assumes without justification that the ratio δJφ=2δJψ is representative or optimal. Because the energy bound δE≥ΩφδJφ+ΩψδJψ depends on which rotation carries the angular momentum, an unscreened ratio could in principle lower the threshold below δE/Ω. My own leading-order estimate suggests the checked channels do not overspin, and the near-extremal margin is robust, so I do not think the claim is demonstrably false; but the proof as written is incomplete. The concrete numerical scan over ratios using the exact extremality condition would settle the matter. For this reason I keep the reader's CONDITIONAL verdict unchanged rather than moving to ACCEPT or REJECT.","tokens_in":1000,"tokens_out":1004,"duration_ms":387727,"concrete_test":"Fix M=1 and take the initial near-extremal equal-rotation configuration a=b=(3/(4π^2))^{1/6}(1−ε^2) with ε=10^−2. For each partition ratio r=δJφ/δJψ ∈ {0, 1/3, 1, 3, ∞}, choose the minimal capture energy δE=ΩφδJφ+ΩψδJψ and also δE=1.1 times that value, increase total δJ until A^2−4B^3 in Eq. (7) first becomes negative, and record (δJφ+δJψ)_min. If for any r this threshold is at or below δE/Ω_min, the paper's conclusion is contradicted. Repeat for the a=2b near-extremal configuration using Eq. (25) with the same set of ratios. A positive result for any ratio would require revising the central claim; a negative result across all ratios would resolve the concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is universal: no linear test-particle accretion can overspin the six-dimensional Myers-Perry black hole. The demonstration, however, is channel-restricted. In the equal-rotation case three channels are treated, but the third (single-axis addition) is dismissed with \"again turns out to be the same\" and no derivation from Eq. (7); the unequal case a=2b is treated only for the fixed partition δJφ=2δJψ. The extremality function Eq. (7) depends separately on Jφ and Jψ through the terms a^6−33a^4b^2−33a^2b^4+b^6 and a^2b^2(a^2−b^2)^4, and the capture bound δE≥ΩφδJφ+ΩψδJψ is distribution-dependent, so δJmin can in principle depend on the channel. If an unscreened ratio has δJmin<δE/Ω for some δE, overspinning occurs and the title claim fails. The reader's focus on the omitted r_+ expansions is narrower; those expansions also contain typographical inconsistencies (Eq. (30) lacks the factor 3 needed for total angular momentum, while Eq. (31) effectively contains it, and Eq. (31) omits the O(ε) horizon shift), reinforcing that the algebra should not be relied on without an exact channel-by-channel check.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the six-dimensional Myers-Perry black hole with two independent rotation parameters and claims that, unlike the five-dimensional two-rotation case, it cannot be overspun by linear test-particle accretion. The method is the standard threshold comparison: compute the minimum angular momentum δJmin needed to push the configuration beyond extremality, compute the maximum angular momentum δJmax that a particle can carry into the horizon from the energy condition δE ≥ Ωφ δJφ + Ωψ δJψ, and show that ΔJ = δJmax − δJmin is always negative. The authors treat the equal-rotation case a = b for three accretion scenarios and the unequal case a = 2b for one fixed partition δJφ = 2δJψ, obtaining ΔJ < 0 in each. They also perform a second-order (Sorce–Wald type) perturbation for a = b and a = 2b and find f(λ) > 0. The paper concludes that the weak cosmic censorship conjecture is always respected for six-dimensional rotating black holes under linear accretion, and conjectures the same for all dimensions greater than six.","tokens_in":14398,"tokens_out":17490,"duration_ms":127831,"significance":"If the central claim is correct, it is an interesting qualitative result: it would mean that six-dimensional rotating black holes behave differently from five-dimensional two-rotation black holes, where overspinning under linear accretion has been reported on the basis of threshold arguments. The paper also provides a conjecture for all higher dimensions and verifies the no-overspinning conclusion at second order for two special cases. The computations are simple and self-contained, using the known Myers–Perry metric and the exact extremality condition. However, the proof as written does not establish the universal claim stated in the title and abstract, because only selected accretion channels are analyzed. The significance is therefore conditional on closing that gap.","major_comments":[{"comment":"The universal claim that no linear accretion can overspin the six-dimensional black hole requires checking all possible partitions of the infalling angular momentum between the two rotation axes. In scenario (iii), where the particle carries angular momentum about only one axis, the paper states that the minimum threshold δJmin 'again turns out to be the same' as in the previous scenarios, but no derivation is given. This is not a trivial statement: after adding δJψ only, the final configuration no longer has equal rotations, so the equal-rotation condition 3M^2 < 4π^2 a^6 used in rows (i) and (ii) is not applicable, and the full extremality function (7), which depends separately on a and b, must be analyzed. A separate calculation is needed to justify the assertion.","section":"Sec. II.A, scenario (iii)"},{"comment":"For a ≠ b the paper treats only the ratio a = 2b and within that case only the accretion partition δJφ = 2δJψ. No argument is given that a = 2b is representative of all two-rotation black holes, and no argument is given that the fixed partition δJφ = 2δJψ is the one most favorable to overspinning. The extremality function (7) depends on a^6 − 33a^4b^2 − 33a^2b^4 + b^6 and a^2b^2(a^2−b^2)^4 separately, while the capture condition (19) is partition-dependent through Ωφ δJφ + Ωψ δJψ. The minimal threshold δJmin can in principle depend on how δE and δJ are distributed between the two channels. Without an exhaustive analysis or a proof that these special choices are extremal in the relevant sense, the title claim that the black hole 'cannot be overspun' is not supported.","section":"Sec. II.B, unequal rotations"},{"comment":"There is an inconsistency between Eq. (30) and Eq. (31). For the stated partition δJφ = 2δJψ, the total angular momentum added to the black hole is δJmax = δJφ + δJψ = 3δJψ. Solving δE ≥ (2Ωφ + Ωψ)δJψ with Ωφ = a/(r_+^2 + a^2) and Ωψ = b/(r_+^2 + b^2) yields δJmax = 3 (r_+^2+a^2)(r_+^2+b^2)/[2a(r_+^2+b^2)+b(r_+^2+a^2)] δE. Eq. (30) is exactly this expression without the factor 3, so it appears to give δJψ rather than the total. The subsequent Eq. (31), however, is three times the leading near-extremal value of Eq. (30). Thus the phrase 'From the above equation, we write' is not correct as written. While the final ΔJ in Eq. (32) is based on Eq. (31) and is consistent with the total angular momentum, the derivation needs to be corrected and the exact expression displayed, since a reader cannot reproduce the claimed expansion from Eq. (30).","section":"Sec. II.B, Eqs. (30)–(32)"},{"comment":"The expansions of δJmax in Eq. (21) and Eq. (31) are asserted without displaying the corresponding expansion of the horizon radius r_+ from Eqs. (8) and (23). The coefficients of the ε and ε^2 terms in these expansions are not self-evident; they require the first- and second-order shifts of r_+ away from extremality. Since the sign of ΔJ is the central result, the paper should either display these expansions or provide a derivation. The need is particularly acute for Eq. (31), where the expansion is obtained from a formula that is itself in error and includes an O(ε) horizon shift that is not accounted for. A reader cannot presently verify the algebra without repeating the entire computation.","section":"Secs. II.A and II.B, near-extremal expansions"}],"minor_comments":[{"comment":"The parameter α is introduced as 'constant ≪ 1', but from the extremality condition (25) it is actually fixed: substituting b^6 = (α/π^2)M^2 into Eq. (25) gives 2187α^2 + 7140α − 576 = 0, whose positive root is α ≃ 0.0788, not parametrically small. The text should state this relation explicitly.","section":"Sec. II.B, Eq. (26)"},{"comment":"The numerical values in Table I are presented without the details of the numerical evaluation of Eq. (40), so the results are not reproducible from the text alone. Providing the numerical method or the raw data would improve the paper.","section":"Sec. II.C, Table I"},{"comment":"There are several typographical errors, including 'general ly' in the abstract, 'T ashkent' in the affiliation, 'corssover' in Sec. II.C, and 'nonliner' in the sentence preceding Eq. (51). These should be corrected.","section":"General"},{"comment":"The statement that k = 0 corresponds to single rotation and k = 1 to equal rotations is slightly confusing, since k appears as a continuous ratio in the perturbation function. It would help to state explicitly that the subsequent numerical analysis uses integer values k = 1,2,3,4 for illustration.","section":"Sec. II.C, Eq. (33)"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a question of genuine interest in the cosmic censorship literature, and the specific calculations for the channels that are checked appear to give the claimed negative ΔJ. However, the manuscript's central claim is universal, while the demonstration covers only special rotation ratios and special partitions of angular momentum. The channel restriction is not a cosmetic gap: the extremality function depends on both rotations in a nontrivial, asymmetric way, so a channel that is not checked could in principle overspin. The paper also contains an algebraic inconsistency in Eq. (30) versus Eq. (31) that must be fixed. These issues are fixable within the scope of a revision, so I am suggesting major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this paper computes the overspinning gap for a six-dimensional Myers-Perry black hole with two rotations in a few specific accretion channels and finds ΔJ < 0 in each. That is a legitimate new result, and the connection to the five-dimensional two-rotation overspinning case is a nice dimensional story. But the title claim—that the six-dimensional black hole \"cannot be overspun\"—is broader than what the calculation actually covers.\n\nWhat is genuinely new: the six-dimensional two-rotation case has not been worked out before, and the extremality condition (Eq. 7) and the threshold comparisons for the equal-rotation channel are done here for the first time. The paper is self-contained against the Myers-Perry metric, and the five-dimensional prior result is cited for context rather than leaned on. The nonlinear second-order check, while sketched, is in line with the Sorce-Wald framework and gives the expected sign.\n\nWhere it softens: the proof is channel-restricted. The extremality function depends separately on Jφ and Jψ, and the capture bound δE ≥ Ωφ δJφ + Ωψ δJψ is distribution-dependent. The equal-rotation case treats three partitions but dismisses the third with \"again turns out to be the same\" and no derivation. The unequal case is only a=2b with a fixed ratio δJφ=2δJψ. That does not rule out an unscreened partition that overspins. The channel gap is a load-bearing issue for the universal conclusion.\n\nThe algebra also invites caution. The near-extremal expansions of the horizon angular velocity are asserted, not derived. Eq. (30) appears to lack the factor 3 needed for total angular momentum, while Eq. (31) effectively contains it, and Eq. (31) drops the O(ε) horizon shift. That inconsistency suggests the intermediate algebra should not be trusted without an exact channel-by-channel check.\n\nWho this is for: people working on gedanken experiments for cosmic censorship in higher dimensions. The paper is worth engaging with, and a referee could reasonably ask for the missing expansions and a channel scan. It should go to review rather than be desk-rejected; the partial result is real, but the sweeping claim needs either proof or a careful restriction.","headline":"A useful partial result on six-dimensional Myers-Perry overspinning, but the blanket no-overspinning claim is not established by the channel-restricted calculations shown.","tokens_in":14884,"tokens_out":2205,"would_cite":false,"duration_ms":26536,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C57","83C75"],"pacs":["04.50.+h","04.20.Dw"],"model":"deepseek-v4-flash","headline":"A six-dimensional Myers-Perry black hole with two rotations cannot be overspun by linear test-particle accretion.","keywords":["six-dimensional Myers-Perry black hole","weak cosmic censorship","overspinning","test particle accretion","extremality","rotating black hole","higher dimensions","nonlinear accretion"],"falsifier":"Check the gap directly: evaluate $\\delta J_{\\rm max}$ and $\\delta J_{\\rm min}$ from the exact horizon equations for small but nonzero $\\epsilon$ and $\\delta E$, without using the paper's truncated expansions, and see whether $\\Delta J = \\delta J_{\\rm max} - \\delta J_{\\rm min}$ ever turns positive for allowed particle parameters. A single positive value would break the no-overspinning claim. A more direct check is to derive the first-order-in-$\\epsilon$ correction to the horizon radius from $\\Delta = 0$ and verify the coefficient that enters the expression for $\\delta J_{\\rm max}$.","tokens_in":13926,"feed_emoji":"🕳️","tokens_out":9618,"duration_ms":256375,"temperature":0.7,"pith_summary":"This paper asks whether a six-dimensional rotating (Myers-Perry) black hole with two independent rotation axes can be destroyed by throwing in a test particle. It argues that it cannot: the largest angular momentum a particle can deliver to the horizon is always smaller than the smallest amount needed to push the black hole past extremality. The authors compute the gap ΔJ = δJmax − δJmin and find it negative definite for both equal rotations (a = b) and unequal rotations (a = 2b) when the black hole starts near extremal. A negative gap means no test particle can jump over extremality and expose a naked singularity, so the weak cosmic censorship conjecture is obeyed at linear order. The paper also checks second-order perturbations, where the horizon remains intact, and conjectures the result holds in all dimensions above six.","feed_headline":"Rotating black hole in six dimensions cannot be overspun","feed_subtitle":"A test particle can never deliver enough spin to push it past extremality, so its horizon survives.","key_machinery":"The load-bearing object is the angular-momentum gap $\\Delta J = \\delta J_{\\rm max} - \\delta J_{\\rm min}$. The maximum $\\delta J_{\\rm max}$ is fixed by the condition that a particle falling into the horizon must satisfy $\\delta E \\ge \\Omega_\\phi\\, \\delta J_\\phi + \\Omega_\\psi\\, \\delta J_\\psi$, where $\\Omega_\\phi$ and $\\Omega_\\psi$ are the horizon angular velocities; this bounds the spin the particle can deposit. The minimum $\\delta J_{\\rm min}$ is set by the extremality condition—for equal rotations, $3M^2 = 4\\pi^2 a^6$, with overspinning requiring $3M^2 < 4\\pi^2 a^6$—and is the smallest angular momentum that would make the final state super-extremal. The paper expands both quantities near extremality using the horizon radius $r_+$ determined by $\\Delta = 0$, and shows the leading-order combination is negative definite in both equal-rotation and unequal-rotation ($a=2b$) cases. For the nonlinear check, the same role is played by the perturbation function $f(\\lambda)$, defined so that $f=0$ is extremality and $f<0$ would be overspinning; its first- and second-order coefficients are computed and found to keep $f(\\lambda) \\ge 0$.","core_discovery":"The paper's central claim is that the six-dimensional Myers-Perry black hole—the rotating vacuum solution of general relativity in six dimensions, here taken with two rotation parameters $a$ and $b$—cannot be overspun by linear test-particle accretion. Starting from a near-extremal black hole, the authors compare the maximum angular momentum $\\delta J_{\\rm max}$ a test particle can deliver through the horizon, fixed by the horizon angular velocities through $\\delta E \\ge \\Omega_\\phi\\, \\delta J_\\phi + \\Omega_\\psi\\, \\delta J_\\psi$, with the minimum angular momentum $\\delta J_{\\rm min}$ needed to drive the extremality condition negative. Their difference, $\\Delta J = \\delta J_{\\rm max} - \\delta J_{\\rm min}$, comes out negative definite: for equal rotations $a=b$ the gap is $\\Delta J = -\\frac{4}{3}\\bigl(\\frac{3}{4\\pi^2}\\bigr)^{1/6}\\bigl[(\\frac13-\\epsilon-\\frac23\\epsilon^2)M^{1/3}\\delta E + M^{4/3}\\epsilon^2 + \\frac29 M^{-2/3}\\delta E^2\\bigr]$, and for unequal rotations $a=2b$ it is a similar negative expression. Because $\\Delta J < 0$, no test particle can jump over extremality and expose a naked singularity. The same conclusion is checked at second order in the perturbation, where the function $f(\\lambda)$ remains positive, so the horizon is preserved beyond linear order. The paper closes by conjecturing that this no-overspinning behavior holds for all dimensions greater than five.","pith_inferences":["The same negative-gap mechanism may hold for arbitrary nonzero ratio $a \\neq b$, not just the $a=2b$ case studied here; checking a generic ratio would test whether the no-overspinning result is structural or special.","The pattern suggests a dimension-dependent parity: when a higher-dimensional black hole has more than one active rotation, the extremal surface is stiff enough to resist linear overspinning, whereas a single active rotation may behave differently; this could be probed in seven dimensions with three rotations.","A natural testable extension is charged or gauged-supergravity versions in six dimensions, where charge and rotation compete; the sign of the gap may then depend on which parameter dominates, as in five-dimensional charged rotating cases."],"forward_implications":["In six dimensions, a Myers-Perry black hole with two rotations satisfies the weak cosmic censorship conjecture even when test-particle backreaction is ignored.","The five-dimensional two-rotation case, which can be overspun at linear order, is not generic; dimension six is protected by a negative angular-momentum gap.","Since $\\Delta J$ is negative definite, there is no discontinuous jump across extremality of the kind proposed for four- and five-dimensional settings.","Including second-order perturbations keeps the perturbation function positive, so nonlinear accretion also preserves the horizon.","If the conjecture holds, all rotating Myers-Perry black holes in dimensions greater than five obey weak cosmic censorship under linear accretion."],"supporting_citations":[{"why":"Introduces the weak cosmic censorship conjecture that the paper tests.","marker":"[1]"},{"why":"Gives the original no-overextremalization result for Kerr-Newman black holes, the baseline the paper extends to six dimensions.","marker":"[14]"},{"why":"Demonstrates that a black hole can be overextremalized by a discontinuous jump over extremality, the scenario the paper rules out.","marker":"[16]"},{"why":"Sets up the linear test-particle overspinning gedanken experiment applied here.","marker":"[17]"},{"why":"Supplies the nonlinear second-order perturbation scheme and the minimal-energy particle choice used to check $f(\\lambda) \\ge 0$.","marker":"[52]"},{"why":"Shows that a five-dimensional black hole with two rotations can be overspun at linear order, providing the contrast that motivates the six-dimensional question.","marker":"[61]"},{"why":"Defines the six-dimensional Myers-Perry metric and its horizon structure, from which the extremality and angular-velocity calculations start.","marker":"[63]"}],"fun_headline_variants":["6D Myers-Perry black hole: no overspin","Test particles can't overspin 6D black hole","Six-dimensional black hole defies overspinning","6D rotating black hole cannot be overspun"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the near-extremal expansion of the horizon angular velocity used to derive $\\delta J_{\\rm max}$ is correct; the paper states these expansions without displaying the underlying series expansion of the horizon radius, and if the coefficients are wrong, the claimed negative sign of $\\Delta J$ could disappear.","fun_headline_variants_meta":{"raw":{"variants":["6D Myers-Perry black hole: no overspin","Test particles can't overspin 6D black hole","Six-dimensional black hole defies overspinning","6D rotating black hole cannot be overspun"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000893,"raw_usage":{"total_tokens":3908,"prompt_tokens":1063,"completion_tokens":2845,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":679,"completion_tokens_details":{"reasoning_tokens":2782}},"tokens_in":679,"tokens_out":2845,"duration_ms":19308,"temperature":1.0,"reasoning_tokens":2782,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:55:57.806276+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the gap directly: evaluate $\\delta J_{\\rm max}$ and $\\delta J_{\\rm min}$ from the exact horizon equations for small but nonzero $\\epsilon$ and $\\delta E$, without using the paper's truncated expansions, and see whether $\\Delta J = \\delta J_{\\rm max} - \\delta J_{\\rm min}$ ever turns positive for allowed particle parameters. A single positive value would break the no-overspinning claim. A more direct check is to derive the first-order-in-$\\epsilon$ correction to the horizon radius from $\\Delta = 0$ and verify the coefficient that enters the expression for $\\delta J_{\\rm max}$.","supporting_citations":[{"cited_title":"Wald, Ann","cited_arxiv_id":null,"evidence_quote":"Gives the original no-overextremalization result for Kerr-Newman black holes, the baseline the paper extends to six dimensions."}],"review_version":1}