{"id":"8504e1cb-115a-42c2-9dd2-4f47de2aba74","arxiv_id":"2501.01680","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"In a charged hairy black hole, increasing the boundary deformation parameter lowers the Lyapunov exponent relative to surface gravity but can push it above the axion Reissner-Nordström value, and generally shortens the scrambling time delay.","lead":"This paper computes how chaos measures (Lyapunov exponent, butterfly velocity, scrambling time delay) change with boundary deformation in a charged hairy black hole with axion and Einstein-Maxwell-scalar couplings. It finds the Lyapunov-to-surface-gravity ratio drops as deformation grows, while for large deformation the hairy black hole can scramble faster than the axion Reissner-Nordström case.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed λL/κ decrease may be an artifact of replacing the deformation-dependent unperturbed connected area by a constant normalization N; without the regulated A^(0)(ϕ0) the plotted ratio is not established as the Lyapunov exponent.","rationale":"The reader's verdict of CONDITIONAL is appropriate, but I would put the weight on the normalization step rather than on the near-singularity saddle. The reader's weakest_assumption is the rc→∞ branch; that concern is real but partially mitigated by Sec. III.C, where the authors show the near-singularity area vanishes and state that no exponential OTOC decay is expected there. The normalization issue, by contrast, is the step that converts a well-defined slope (Eq. 54) into the plotted λ_L (Eqs. 56–58). Because A^(0) is infinite and not computed, the deformation dependence of λ_L/κ is imposed by the ansatz N=constant rather than derived from the geometry of the unperturbed extremal surface. This is explicitly a limitation in the manuscript ('We also absorb the infinite length Ly into the definition of the normalization parameter'). The concrete check I propose is a regulated computation of A^(0)(ϕ0/T); it would settle whether the monotonic decrease in λ_L/κ survives. The near-singularity branch should be checked as a secondary item by scanning all roots of Eq. (51) and verifying the reality of the Kruskal integrands. If the check passes, the paper's qualitative claims stand; if it fails, the central claim of the abstract is an artifact of normalization. No change to the reader's CONDITIONAL verdict is needed: the concern is exactly the kind of unresolved gap that makes the verdict conditional rather than final.","tokens_in":21505,"tokens_out":20919,"duration_ms":222409,"concrete_test":"Compute the regulated unperturbed connected surface area A^(0)_{A∪B}(ϕ0/T) for the same backgrounds as Fig. 5 (ζ=2, ρ=0.2, γ=0, q=0.1), using a common UV cutoff in r and a fixed reference insertion time (α=O(1)). Then form λ_L^true = (Ly/2G_N) sqrt(-f(rc)e^{-χ(rc)}/r_c^4) / A^(0)(ϕ0/T), normalize by κ(ϕ0/T), and replot both panels of Fig. 5. If λ_L^true/κ is still monotonically decreasing and λ_L^true/λ_aRN still crosses unity, the normalization concern is settled; if the curves flatten or invert, Eqs. (56)–(58) do not define the physical Lyapunov exponent and the headline claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equations (55)–(58) define λ_L = (2/N) sqrt(-f(rc)e^{-χ(rc)}/r_c^4), with N fixed once at ϕ0→0 by the condition λ_L→κ. Equation (54), however, shows that the slope of the connected entropy in t_w is (Ly/2G_N) sqrt(-f(rc)e^{-χ(rc)}/r_c^4), independent of the unperturbed area. In the mutual-information method the Lyapunov scale is this slope divided by the unperturbed connected area A^(0)_{A∪B}; for the planar horizon A^(0) is divergent, and the paper replaces it by a ϕ0-independent constant N rather than computing the regulated A^(0)(ϕ0/T). Nothing in the paper shows that A^(0) is ϕ0-independent, and the manuscript explicitly says it 'absorb[s] the infinite length Ly into the normalization parameter' (Sec. III.B). If A^(0) decreases with deformation, the true ratio λ_L/κ could be flat or increasing, inverting the abstract's central claim. The separate near-singularity branch (Eq. 61, Sec. III.C) is also unresolved: the paper does not demonstrate that it is subdominant for real extremal surfaces, only that its area vanishes as K→0.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies chaos in a charged hairy black hole solution of Einstein-Maxwell-scalar theory with axion and EMS couplings, using the holographic mutual information of two boundary strips perturbed by charged shock waves. The authors numerically construct the bulk solutions, extract a 'quantum Lyapunov exponent' from the late-time slope of the connected extremal surface, and compute the butterfly velocity and the scrambling time delay as functions of the dimensionless deformation phi0/T and the model parameters zeta, gamma, rho. The central quantitative claims are that lambda_L/kappa decreases with increasing deformation, that for large deformation lambda_L can exceed the axion-Reissner-Nordstrom value, and that the scrambling time delay decreases with deformation, with a strong gamma dependence. The paper also studies the relation between these chaotic quantities and the interior Kasner exponent p_t.","tokens_in":21785,"tokens_out":16273,"duration_ms":156853,"significance":"The paper applies a standard holographic shock-wave framework to a phenomenologically motivated charged hairy black hole and provides a broad numerical survey of how the chaotic diagnostics depend on the UV deformation and on the couplings. The exploration of the connection between boundary deformation and the Kasner interior, and the finding that the scrambling delay can be strongly suppressed by the EMS coupling, are potentially interesting. The manuscript is clearly written and the numerical integrations appear careful. However, the main quantitative claim about the monotonic decrease of lambda_L/kappa rests on a normalization prescription that has not been justified, so the significance of the reported trends is currently conditional.","major_comments":[{"comment":"The definition of the Lyapunov exponent in Eq. (56) uses a normalization constant N fixed by the condition lambda_L(phi0->0)=kappa. This implicitly assumes that the unperturbed connected area A^(0)_{A union B} appearing in Eq. (55) is proportional to N and hence independent of phi0 (up to the factor Ly). The paper explicitly states that it does not compute A^(0) and instead absorbs the infinite length Ly into N. However, A^(0) is the area of an extremal surface in the background, so it should in general depend on the bulk geometry, hence on phi0. Consequently, the plotted ratio lambda_L/kappa reduces to the ratio of the slope function sqrt(-f(r_c)e^{-chi(r_c)}/r_c^4) to its value at phi0=0, which is not necessarily the ratio of the true Lyapunov exponents defined by Eq. (55). Unless the authors compute the regulated finite part of A^(0)(phi0) (for example by subtracting the disconnected contributions and taking the large-lx limit) or give a physical argument that A^(0)/Ly is deformation-independent, the central claim that lambda_L/kappa decreases with phi0/T (Figs. 5, 7, 9) is not established.","section":"III.B, Eqs. (55)-(58)"},{"comment":"The paper notes in Section III.B (after Eq. (61)) that r_c -> infinity is also a critical radius and says it will return to this limit. Section III.C analyzes the area functional for K -> 0 and concludes that the near-singularity area vanishes, but it does not determine the Lyapunov exponent in that limit, nor does it compare the on-shell actions of the near-horizon and near-singularity saddles. Without such a comparison, it is not shown that the near-horizon root is the relevant saddle for the extremal surfaces used to extract lambda_L. If the near-singularity branch dominates in some part of the parameter space, the extracted lambda_L and its monotonicity in phi0/T could change. Please provide a numerical or analytic check that the near-horizon branch gives the global minimum of the area functional.","section":"III.B, Eq. (61) and III.C"},{"comment":"The comparison lambda_L/lambda_{aRN} in Section III.B (Eqs. (59)-(60)) is presented as evidence that the hairy black hole can become more chaotic than the axion-Reissner-Nordstrom black hole. But N_{aRN} is fixed by the same condition lambda_L(phi0->0)=kappa, so the ratio again equals the ratio of the slope functions. The aRN solution is the phi=0 limit of the hairy solution, and the unperturbed area in that limit should be evaluated with the same regularization as the hairy case. The paper does not show that this ratio is insensitive to the normalization issue raised in the first comment, so the claim of exceeding the aRN bound is subject to the same caveat.","section":"III.B, Eqs. (59)-(60)"}],"minor_comments":[{"comment":"The rescaling in Eq. (24) sets a_1 = e^{-chi(0)/2}, but chi(0) is not known until after the rescaling; please clarify the iterative procedure or define a_1 through the boundary value of chi before the rescaling.","section":"II.A, Eq. (24)"},{"comment":"In Eq. (59), the parentheses are unbalanced; please check the formula for lambda_{aRN}.","section":"III.B, Eq. (59)"},{"comment":"The sentence ending with 'We also see how the coupling parameter q between the scalar field and the gauge field affect Finally' is incomplete; the word 'Finally' appears to be a typo.","section":"I. Introduction"},{"comment":"In Section V, 'paramter' should be 'parameter'.","section":"V. Summary and Discussions"},{"comment":"Figure 4 shows only a limited range of r; to support the statement that there is no other root in the deep interior, it would be helpful to plot d/dr(f e^{-chi}/r^4) over a wider range, especially in light of Eq. (61).","section":"III.B, Fig. 4"}],"recommendation":"major_revision","confidential_remarks":"The normalization issue is the main obstacle to accepting the central claim. If the authors can compute the regulated unperturbed area or reframe the claim in terms of the slope function, the paper could be publishable. The near-singularity saddle question is also worth addressing with a concrete numerical check. The manuscript fits the journal's scope and the numerical work appears solid."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, this is a clean numerical application of the charged shock-wave mutual-information method to the Sword-Vegh charged hairy black hole with axion and EMS couplings; the new results are the parametric trends for λ_L, butterfly velocity, and scrambling time delay as functions of ϕ0/T, ζ, γ, ρ. The numerics look careful and the authors are appropriately honest about limitations, including a non-invertible relation between λ_L and the Kasner exponent and the near-singularity saddle which they set aside because its area vanishes.\n\nThe second thing is the serious soft spot. The paper's central claim—that λ_L/κ decreases with ϕ0/T—depends on identifying λ_L through Eq. (56) with a constant normalization N fixed once at ϕ0=0. The true λ_L in the mutual-information method would require dividing the slope of I by the unperturbed connected area per unit Ly, A^(0)/Ly. The paper never checks whether that quantity depends on ϕ0. If it does, the plotted λ_L/κ is not the physical Lyapunov exponent. The stress-test note is right: nothing in the paper rules out a ϕ0-dependent A^(0) that could flatten or even invert the claimed trend. The authors' phrase about absorbing Ly into N is an admission of the ambiguity, not a resolution. The reader's note that N cancels in λ_L/λ_{aRN} is true only if N is truly constant; but the physical ratio would require the respective unperturbed areas to be compared.\n\nA smaller issue is the unresolved near-singularity critical radius from Eq. (61). The paper says it will return to this limit, but Section III.C only analyzes the area functional and does not directly show that the near-singularity saddle is subdominant for the Lyapunov extraction. The vanishing area argument is suggestive but not definitive.\n\nOverall, this is a competent extension that deserves a serious referee, especially because the central trend may be correct. The referee should ask for either an explicit computation of A^(0)(ϕ0) or a derivation of λ_L from the OTOC two-point function. I would not cite it as it stands, but it is a useful benchmark for people working on holographic superconductor chaos. For you: read it if you work in that area; otherwise, passing is fine.\n\nRecommendation: send to peer review—conditional accept with a request to address the normalization or reframe the claims.","headline":"Competent numerical extension of holographic chaos to a specific charged hairy black hole, but the headline λ_L/κ claim rests on a normalization whose ϕ0-dependence is not established.","tokens_in":22324,"tokens_out":6940,"would_cite":false,"duration_ms":65641,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C57","83C47","81T40"],"pacs":["04.70.Dy","11.25.Tq"],"model":"deepseek-v4-flash","headline":"Under a scalar boundary deformation, charged hairy black holes show a decreasing ratio of quantum Lyapunov exponent to surface gravity, yet at sufficiently large deformation their Lyapunov exponent can exceed the axion Reissner-Nordström…","keywords":["quantum Lyapunov exponent","charged hairy black hole","Kasner interior","holographic chaos","mutual information","scrambling time delay","butterfly velocity","Einstein-Maxwell-Scalar coupling"],"falsifier":"Take the same charged-hairy-black-hole background, push the entangling surface turning point to the deep-interior root $r_c\\to\\infty$ of $\\frac{d}{dr}(f e^{-\\chi}/r^4)=0$, and extract $\\lambda_L$ from the coefficient of $t_w$ in the area; if $\\lambda_L/\\kappa$ does not decrease monotonically with $\\phi_0/T$, or if $\\lambda_L/\\lambda_{\\rm aRN}$ does not exceed one at large deformation, the paper's central numerical claim fails.","tokens_in":21297,"feed_emoji":"🕳️","tokens_out":7475,"duration_ms":67871,"temperature":0.7,"pith_summary":"This paper asks how a scalar-driven boundary deformation changes chaos in a charged hairy black hole. By injecting charged shock waves and computing the holographic mutual information, it extracts a quantum Lyapunov exponent and finds that the ratio $\\lambda_L/\\kappa$ decreases as the deformation parameter $\\phi_0/T$ grows. For sufficiently large deformation, the Lyapunov exponent can exceed the axion Reissner-Nordström value, meaning the deformed geometry can be more chaotic than its un-hairy counterpart. The paper also finds that boundary deformation generally reduces the scrambling time delay, with the Einstein-Maxwell-Scalar coupling having a strong suppressing effect. These results show that boundary chaos data carry partial information about the Kasner interior, but the non-invertible relation between the Lyapunov exponent and the Kasner exponent means the interior geometry is not fully determined by these boundary observables.","feed_headline":"Hairy black holes scramble slower as boundary deformation grows","feed_subtitle":"Charged, deformed black holes can still beat axion Reissner-Nordström chaos at large deformation.","key_machinery":"The central object is the area functional of the entangling surface that connects the two asymptotic boundaries through the black hole interior, together with the conserved quantity $K$ fixed at the turning point. The Lyapunov exponent is read from the coefficient of $t_w$ in the area when the turning point approaches the critical radius $r_c$, which is the near-horizon root of $\\frac{d}{dr}(f e^{-\\chi}/r^4)=0$, through $\\lambda_L = \\frac{2}{N}\\sqrt{-f(r_c)e^{-\\chi(r_c)}/r_c^4}$; the normalization $N$ is chosen so that $\\lambda_L\\to\\kappa$ as $\\phi_0\\to 0$. This machinery converts the exponential growth of the shock-wave parameter into a boundary-observable Lyapunov exponent, while the same area functional in the $K\\to 0$ limit probes the near-singularity Kasner region.","core_discovery":"The paper studies a charged hairy black hole whose bulk scalar field acts as a relevant deformation in the boundary theory, driving the deep interior to a Kasner spacetime. Using charged gravitational shock waves and the holographic mutual information of two boundary regions, it extracts a quantum Lyapunov exponent $\\lambda_L$ normalized so that $\\lambda_L\\to\\kappa$ when the deformation vanishes. Its central finding is that $\\lambda_L/\\kappa$ decreases monotonically as $\\phi_0/T$ grows, while $\\lambda_L/\\lambda_{\\rm aRN}$ first dips and then rises, exceeding one for sufficiently large deformation. The paper also reports that the butterfly velocity generally decreases with deformation and with the axion and charge-density parameters, while the scrambling time delay shrinks as $\\phi_0/T$ increases and is strongly suppressed by the Einstein-Maxwell-Scalar coupling $\\gamma$. It further shows that the relation between $\\lambda_L$ and the Kasner exponent $p_t$ is non-invertible, so boundary chaos diagnostics alone do not uniquely fix the interior Kasner geometry.","pith_inferences":["Editorial inference: If the deep-interior branch $r_c\\to\\infty$ were the dominant saddle, the same calculation would make $\\lambda_L$ a direct probe of Kasner data, offering a sharper test of interior reconstruction than the near-horizon normalization used here.","Editorial inference: The near independence of $\\lambda_L/\\kappa$ from $\\gamma$, alongside the strong $\\gamma$ sensitivity of the scrambling delay, suggests that different chaos diagnostics encode different near-horizon or boundary data, a separation that may persist in other scalar-hairy models.","Editorial inference: The suppression of the scrambling delay with $\\phi_0/T$ hints that in holographic superconductor duals, driving the system deeper into the trans-IR flow could effectively shorten the scrambling window, which would be observable as a sharper decay of mutual information in the boundary theory."],"forward_implications":["The ratio $\\lambda_L/\\kappa$ falls monotonically as $\\phi_0/T$ increases, so the boundary deformation makes the black hole scramble less efficiently relative to its surface gravity.","For sufficiently large deformation, $\\lambda_L/\\lambda_{\\rm aRN}>1$, meaning the hairy deformed geometry can be more chaotic than the axion Reissner-Nordström black hole.","The scrambling time delay shrinks as $\\phi_0/T$ grows, and the Einstein-Maxwell-Scalar coupling $\\gamma$ can drive it to near zero, effectively turning off the charged-shock-wave bounce.","The butterfly velocity decreases with the axion parameter $\\zeta$ and the charge density $\\rho$, and it deviates from the Schwarzschild value once deformation and axion charge are turned on.","Because $\\lambda_L$ and the Kasner exponent $p_t$ are related non-invertibly, boundary chaos diagnostics do not uniquely reconstruct the interior Kasner geometry; additional near-singularity data are needed."],"supporting_citations":[{"why":"Supplies the charged hairy black hole model with Einstein-Maxwell-Scalar coupling and axion field whose interior becomes Kasner.","marker":"[23]"},{"why":"Introduces the charged shock-wave bounce inside the horizon and the scrambling time delay that this paper computes.","marker":"[15]"},{"why":"Establishes the shock-wave and holographic RG-flow setup for extracting chaos in hairy black hole interiors, which this work extends to charged waves.","marker":"[28]"},{"why":"Provides the mutual-information method for extracting the Lyapunov exponent from the area of an entangling surface in a shock-wave background.","marker":"[12]"},{"why":"Shows how the instantaneous Lyapunov exponent and scrambling delay are extracted in a dyonic rotating AdS black hole, serving as the template for the normalization used here.","marker":"[19]"},{"why":"States the universal chaos bound whose saturation fixes the normalization of the Lyapunov exponent in the undeformed limit.","marker":"[11]"},{"why":"Provides the holographic superconductor interior and the collapse of the inner horizon that motivates the Kasner flow studied here.","marker":"[21]"},{"why":"Introduces the shock-wave and butterfly-effect calculation for black holes that underlies the out-of-time-ordered correlation interpretation.","marker":"[7]"}],"fun_headline_variants":["Charged hairy black holes: slower scrambling, but can beat axion RN chaos","Hairy black hole chaos: deformation slows scrambling but can flip Lyapunov","Deformation tames black hole scrambling; large deformation can beat axion RN","Kasner interior reveals how boundary deformation alters black hole chaos","Slower scrambling in hairy black holes, but chaos can beat axion RN at large deformation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The reported Lyapunov exponent is read from the near-horizon root $r_c$ of a derivative condition, and the paper itself notes that $r_c\\to\\infty$, a surface probing the singularity, also satisfies the condition; if that deep-interior branch controls the extremal surface instead, the monotonic behavior of $\\lambda_L/\\kappa$ could change.","fun_headline_variants_meta":{"raw":{"variants":["Charged hairy black holes: slower scrambling, but can beat axion RN chaos","Hairy black hole chaos: deformation slows scrambling but can flip Lyapunov","Deformation tames black hole scrambling; large deformation can beat axion RN","Kasner interior reveals how boundary deformation alters black hole chaos","Slower scrambling in hairy black holes, but chaos can beat axion RN at large deformation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000899,"raw_usage":{"total_tokens":3879,"prompt_tokens":957,"completion_tokens":2922,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":573,"completion_tokens_details":{"reasoning_tokens":2820}},"tokens_in":573,"tokens_out":2922,"duration_ms":22709,"temperature":1.0,"reasoning_tokens":2820,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:22:35.059723+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the same charged-hairy-black-hole background, push the entangling surface turning point to the deep-interior root $r_c\\to\\infty$ of $\\frac{d}{dr}(f e^{-\\chi}/r^4)=0$, and extract $\\lambda_L$ from the coefficient of $t_w$ in the area; if $\\lambda_L/\\kappa$ does not decrease monotonically with $\\phi_0/T$, or if $\\lambda_L/\\lambda_{\\rm aRN}$ does not exceed one at large deformation, the paper's central numerical claim fails.","supporting_citations":[{"cited_title":"Jahnke, K","cited_arxiv_id":null,"evidence_quote":"Introduces the charged shock-wave bounce inside the horizon and the scrambling time delay that this paper computes."},{"cited_title":"Leichenauer, Disrupting entanglement of black holes, Physical Review D - Particles, Fields, Gravitation and Cosmology 90 (4) (2014)","cited_arxiv_id":null,"evidence_quote":"Provides the mutual-information method for extracting the Lyapunov exponent from the area of an entangling surface in a shock-wave background."},{"cited_title":"Malvimat, R","cited_arxiv_id":null,"evidence_quote":"Shows how the instantaneous Lyapunov exponent and scrambling delay are extracted in a dyonic rotating AdS black hole, serving as the template for the normalization used here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the holographic superconductor interior and the collapse of the inner horizon that motivates the Kasner flow studied here."}],"review_version":1}