{"id":"5e8ee46e-c652-41e7-9d34-73c2a007c3ba","arxiv_id":"2603.14872","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For regular BTZ black holes from an infinite Lovelock tower, scalar quasinormal modes bifurcate from complex BTZ branches into purely imaginary branches as the regularization length ℓ grows, producing mode switching and overtone reordering.","lead":"This paper computes how a massless scalar field vibrates on a regular version of the BTZ black hole, where the central singularity is replaced by a smooth core. As the core size grows, the vibration frequencies split into purely damped branches, a pattern that may guide studies of black-hole ringdown in theories that remove singularities.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The bifurcation claim rests on uncertified numerical eigenvalues: Leaver's stated convergence criterion omits the singular point at r=∞ (z=1), and the Horowitz–Hubeny recurrence is built from Eq. (34)'s rational functions while claiming polynomials.","rationale":"The reader's weakest assumption was that the Leaver/Horowitz–Hubeny eigenvalues are true QNMs, citing the missing convergence analysis and the HH polynomial/rational inconsistency. My stress-test confirms this as the load-bearing point: both numerical arms are series truncations of the same radial problem, and neither carries a residual/convergence certificate. The Leaver convergence statement is incomplete in a precise, checkable way (r=∞ maps to z=1 on the unit circle), and the HH recurrence as written is internally inconsistent. These are serious but addressable technical issues, not demonstrated fatal flaws; the BTZ benchmark in Appendix B and the analytic ℓ_crit formula give some independent support, and the reported two-method agreement is suggestive. Hence the appropriate disposition remains CONDITIONAL as the reader judged; no verdict change is needed.","tokens_in":26705,"tokens_out":10849,"duration_ms":104971,"concrete_test":"Reproduce the m=1, ℓ=0.6 fundamental mode (Table I: ω≈0.000−2.607i) with a Chebyshev pseudospectral discretization of Eq. (15) on r∈[r_h,R] for increasing N and R, imposing ingoing behavior at r_h and χ→0 at R. If the least-damped spectral eigenvalue differs from the Table I value by more than 0.1%, or if its imaginary part does not track the claimed ℓ-dependence, the bifurcation pattern is not robust.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central result—branch collisions and purely imaginary QNM branches in Table I and Figs. 2–4—is only as strong as the QNM certificate for the two numerical methods. Section IV's Leaver implementation asserts convergence because the singular points r−_h and r±_ℓ lie outside the unit circle in z=(r−r_h)/r. But the radial equation (19) also has a singular point at spatial infinity, which maps to z=1, on the unit circle. The prefactor (r_h/r)^{3/2} in (22) implements the desired r^{-3/2} falloff, yet it does not by itself prove the series Σ a_n z^n converges at z=1 for the ω returned by the continued fraction; no partial-sum convergence, residual, or N-truncation study is supplied. The Horowitz–Hubeny section does not repair this: Eq. (34) defines s(z), t(z), u(z) as rational functions with denominators [ℓ²(L²M z²−1)+L²] and its square, while the text states they are 'third-order polynomials' and uses Taylor coefficients (35)–(37) in recurrence (40). Unless those denominators cancel identically after substituting (6)—which the displayed forms do not show—the recurrence is an uncontrolled truncation, not an exact HH implementation. Agreement between two approximations that share the same leading near-horizon structure would then not establish that the reported eigenvalues are true QNMs.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies massless scalar quasinormal modes of a family of regular BTZ black holes obtained from an infinite tower of dimensionally regularized Lovelock corrections, focusing on the representative case c_n=1. The authors formulate the radial perturbation problem, note the presence of multiple singularities, and compute QNM frequencies using Leaver's continued-fraction method and the Horowitz–Hubeny power-series method. They report agreement to three decimal places and describe a sequence of spectral bifurcations as the regularization scale ℓ increases: for m=0 the purely imaginary BTZ mode splits into two branches; for m=1 the complex BTZ-like branch reaches the imaginary axis at ℓ_crit≈0.433 and splits into two purely imaginary branches; for m=2 a second collision produces a third purely imaginary branch. The paper claims linear stability (ω_I<0) throughout and connects the branch collisions to a change in the concavity of the effective potential at the horizon, deriving ℓ_crit(m)=√3 m/(2√(m^2+3)).","tokens_in":27068,"tokens_out":13953,"duration_ms":127565,"significance":"If the numerical results are correct, the paper provides a new and controllable three-dimensional example of QNM spectral bifurcation in a regular black hole, complementing earlier examples in nearly extremal Kerr and Maxwell perturbations of AdS black holes. The authors include several positive elements: a benchmark of Leaver's method against the exact BTZ spectrum (Table II), two independent numerical schemes, and an analytic condition for the bifurcation threshold based on V''_eff(r_h)=0. However, the central claim rests on numerical eigenvalues whose certification is incomplete; the Horowitz–Hubeny implementation as written is internally inconsistent, and the Leaver convergence statement omits the singular point at spatial infinity. These issues must be resolved before the bifurcation pattern can be considered established.","major_comments":[{"comment":"The text states that s(z), t(z), u(z) in Eq. (34) are 'third-order polynomials in z', but the displayed forms are rational functions with denominator [ℓ²(L²M z²−1)+L²] and its square. The Taylor expansions (35)–(37) truncated at third order therefore discard an infinite tail, and the recurrence (40) is not an exact Horowitz–Hubeny recurrence for the actual ODE (33). As a result, the HH results are not a valid independent verification of the Leaver spectrum. The authors should either compute the exact Taylor coefficients of the rational functions to all orders and use them in a consistent recurrence, or replace the implementation with a clearly convergent spectral method, and show that the results converge as the truncation order is increased.","section":"Sec. V, Eqs. (34)–(37), (40)"},{"comment":"The Leaver convergence argument states that all singular points of the transformed radial equation lie outside the unit circle in z=(r−r_h)/r. This is incomplete: spatial infinity r=∞ maps to z=1, which lies on the unit circle and is a singular point of the ODE. The prefactor (r_h/r)^{3/2} removes the leading branch point, but because the indicial exponents at z=1 differ by an integer, a logarithmic term may be present in the decaying solution, and convergence of Σ a_n z^n at z=1 is not guaranteed by the location of the other singularities. The paper provides no partial-sum convergence test near z=1, no residual check, and no N-truncation study for the continued-fraction roots. I request a concrete convergence study of the Frobenius series at z=1 for representative (m,ℓ) values, including the behavior of |a_n| and the residual of Eq. (19).","section":"Sec. IV, Eq. (22) and convergence discussion"},{"comment":"The paper asserts that the condition V''_eff(r_h)=0 singles out the bifurcation point of the fundamental mode and leads to ℓ_crit(m)=√3 m/(2√(m^2+3)). This connection is stated without derivation and is load-bearing for the interpretation of the numerically observed branch collisions. Please either derive the relation from the perturbation equation (e.g., by analyzing the behavior of the QNM condition near ω_R=0), or explicitly compare the analytic ℓ_crit with the numerical value of ℓ at which ω_R crosses zero for each m and overtone in Table I. As written, the analytic formula appears to be transferred from Refs. [102,122] without a direct proof for the present potential.","section":"Sec. VI, text above Eq. (16) and ℓ_crit formula"}],"minor_comments":[{"comment":"For m=0 at ℓ=0, Table I lists a single purely imaginary value per overtone, while for ℓ>0 it lists two. The text says the BTZ value 'splits' as soon as ℓ is turned on. It would be helpful to clarify whether the two branches emanate from a degenerate point at ℓ=0 (and how the degeneracy is resolved for infinitesimal ℓ) in the figure and table.","section":"Sec. VI, Fig. 2 and Table I (m=0)"},{"comment":"No error estimates, convergence criteria, or code are provided. A statement of the Leaver truncation order N, the number of continued-fraction iterations, and the observed convergence as N grows (even for a few representative cases) would greatly increase confidence in the three-decimal agreement claimed in Table I.","section":"General numerical reporting"},{"comment":"The caption states the plot shows 'fundamental scalar quasinormal frequencies' for m=1 and m=2, while the main text (Sec. VI) says the fundamental and the first two overtones are plotted. Please make the caption consistent.","section":"Fig. 4 caption"}],"recommendation":"major_revision","confidential_remarks":"The main risk is numerical reliability. The Horowitz–Hubeny inconsistency is a clear internal flaw, and the Leaver convergence argument omits z=1. Both are fixable in a revision by adding convergence diagnostics and a corrected HH implementation. If the authors provide those, the paper could become acceptable; without them, the bifurcation claim is not sufficiently supported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is the first quasinormal-mode computation for the Fernandes regular BTZ black hole with c_n = 1, and it finds a clean bifurcation pattern as ℓ grows: complex branches collide with the imaginary axis and split into purely imaginary branches, with an extra second bifurcation for m = 2. The phenomenon is not new—Maxwell–AdS and Coulomb-like AdS black holes show the same—but the specific background and the three-branch m = 2 case are new. The BTZ limit reproduces the analytic spectrum to three decimals, and Leaver and Horowitz–Hubeny agree to that precision. The critical ℓ values come from an analytic concavity condition, not from fitting the QNM data, which is good. Stability (ω_I < 0 throughout) is supported by Table I. The paper is clearly written and the recurrence reduction is spelled out in detail.\n\nThe main weakness is that the numerical eigenvalues are not certified. For Leaver, the singular point at spatial infinity sits at z = 1 on the unit circle of the Frobenius expansion; the text says all singular points must lie outside the unit circle but does not discuss this one. The prefactor (r_h/r)^{3/2} likely handles the falloff, but without a convergence or residual study the continued-fraction roots remain plausible rather than proven. The Horowitz–Hubeny section is worse: Eq. (34) defines s, t, u as rational functions with denominator ℓ²(L²M z²−1)+L², yet the text calls them third-order polynomials. That is not a polynomial truncation; either the denominator cancels (it does not, as written) or the Taylor expansion is infinite. As described, recurrence (40) is not an exact HH implementation. Agreement between two approximate methods that share the near-horizon structure does not by itself certify the eigenvalues.\n\nAlso, the conclusion that regularization “generally reduces |ω_I|” is not supported by Table I—the more damped branches have larger |ω_I| than BTZ. And the results are only for c_n = 1, which the authors acknowledge. These are minor compared to the numerical certification issue.\n\nWho is this for? QNM practitioners and people working on regular black holes in AdS/CFT. The qualitative bifurcation story is likely right—the pattern matches known cases—so the paper is worth engaging. But it needs a revised version with convergence studies, residual checks, and a corrected HH section, ideally with code or data. I would send it to peer review rather than desk-reject; the issues are fixable and the result, if confirmed, is a useful addition. I would bring it to reading group, but flag the numerical caveats.","headline":"First QNM computation for the regular BTZ background, with a plausible bifurcation story; the numbers are probably right, but the numerical evidence is not fully certified and the Horowitz–Hubeny section has an internal inconsistency.","tokens_in":27559,"tokens_out":6219,"would_cite":true,"duration_ms":54685,"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":"For a regularized BTZ black hole, the scalar quasinormal spectrum remains stable but its complex branches collide with the imaginary axis as the core size grows, splitting into purely imaginary branches and reordering overtones.","keywords":["Quasinormal modes","Regular BTZ black holes","Spectral bifurcations","Lovelock gravity","Black hole perturbations","Continued fraction method","Horowitz-Hubeny method","Massless scalar field"],"falsifier":"Recompute the modes with a pseudospectral or other discretization that does not rely on the rational-to-polynomial truncation, and check whether (i) both numerical methods still agree, (ii) the real part crosses zero at approximately 0.433 for m=1 and 0.655 for m=2, and (iii) the three m=2 branches persist at higher resolution. A discrepancy in any of these three checks would falsify the bifurcation claim.","tokens_in":1780,"feed_emoji":"🕳️","tokens_out":3561,"duration_ms":73278,"temperature":0.7,"pith_summary":"The paper claims that the massless scalar quasinormal spectrum of the regular BTZ black hole is linearly stable for all core sizes below the AdS radius: every mode has negative imaginary frequency. More strikingly, as the regularization scale grows, the standard BTZ complex branches are driven toward the imaginary axis and bifurcate into purely imaginary, overdamped branches. For the m=1 harmonic there are two such branches after a critical core size; for m=2 a second collision produces three. The frequencies are computed with Leaver's continued-fraction and Horowitz-Hubeny power-series methods, which agree to three decimal places. A sympathetic reader would care because this places regular three-dimensional black holes in the same family of bifurcating quasinormal spectra known from nearly extremal Kerr and AdS black holes, and because it ties the bifurcation to a geometric, near-horizon feature of the potential.","feed_headline":"BTZ black hole's smoothed core splits its ringdown spectrum","feed_subtitle":"As the core size grows, complex oscillation modes become purely imaginary and overtone order changes.","key_machinery":"The central object is the radial Klein-Gordon equation for a massless scalar on the regular BTZ metric, whose effective potential has six singular points and is not reducible to a hypergeometric or Heun equation. The computation rests on two numerical methods: Leaver's continued fraction method, using a Frobenius ansatz and an eight-term recurrence reduced by Gaussian elimination to a three-term recurrence, and the Horowitz-Hubeny power-series method. The proposed physical mechanism is the near-horizon concavity of the effective potential: the condition V''_eff(r_h)=0 singles out the critical core size for the fundamental mode, yielding the closed-form threshold in terms of m.","core_discovery":"For the regular BTZ black hole from an infinite tower of dimensionally regularized Lovelock corrections with c_n=1, the massless scalar quasinormal spectrum is qualitatively reshaped while remaining linearly stable. At the critical values for m=1 and m=2, the complex BTZ frequency branches reach the imaginary axis and split into purely imaginary branches; for m=2 a second collision creates a third branch. The paper identifies the bifurcation locus of the fundamental mode with the vanishing second derivative of the effective potential at the horizon, giving the closed-form threshold. It also shows that the leading near-horizon equation and the Hawking temperature are unchanged from BTZ, so th","pith_inferences":["Since the bifurcation threshold saturates as m grows, an analytic eikonal derivation of the threshold from the potential should be possible; the paper does not provide one.","The m=2 secondary bifurcation resembles exceptional-point collision dynamics; if the same mechanism holds for other fields, the pattern may be generic to regularized cores.","Because the Hawking temperature is unchanged even though the QNM pole structure shifts, holographic two-point functions might carry a core signature not visible in thermodynamics.","If such regular cores exist, ringdown templates that assume a single damped sinusoid may need to include purely imaginary branches above the critical core size."],"forward_implications":["Scalar perturbations of these regular BTZ black holes remain linearly stable for all core sizes below the AdS radius.","Regularization reduces the decay rate of the least damped mode, lengthening ringdown, while also creating overdamped branches.","The overtone ladder is reordered as the core size grows, so the mode that dominates late-time decay changes.","The bifurcation threshold saturates at large harmonic index, suggesting a high-frequency geometric mechanism independent of the core details.","The leading near-horizon conformal structure is BTZ-universal, so the global spectral change arises from subleading near-horizon corrections."],"fun_headline_variants":["Regular BTZ black hole ringdown splits into multiple imaginary branches","Core size drives bifurcations in BTZ quasinormal spectra","Lovelock regularized BTZ shows spectral branching","Mode switching in regular BTZ from core-size tune","Bifurcating quasinormal modes in core-corrected BTZ"],"cache_read_input_tokens":28800,"weakest_assumption_plain":"The numerical roots are assumed to be true quasinormal modes: the Frobenius series is asserted to converge for all core sizes below the AdS radius because additional singular points lie outside the unit circle, but no convergence or residual check is shown, and the Horowitz-Hubeny recurrence treats rational coefficient functions as polynomial truncations; if those roots are numerical artifacts, the bifurcation picture collapses.","fun_headline_variants_meta":{"raw":{"variants":["Regular BTZ black hole ringdown splits into multiple imaginary branches","Core size drives bifurcations in BTZ quasinormal spectra","Lovelock regularized BTZ shows spectral branching","Mode switching in regular BTZ from core-size tune","Bifurcating quasinormal modes in core-corrected BTZ"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000431,"raw_usage":{"total_tokens":2050,"prompt_tokens":768,"completion_tokens":1282,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":512,"completion_tokens_details":{"reasoning_tokens":1195}},"tokens_in":512,"tokens_out":1282,"duration_ms":10407,"temperature":1.0,"reasoning_tokens":1195,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T18:08:04.576030+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the modes with a pseudospectral or other discretization that does not rely on the rational-to-polynomial truncation, and check whether (i) both numerical methods still agree, (ii) the real part crosses zero at approximately 0.433 for m=1 and 0.655 for m=2, and (iii) the three m=2 branches persist at higher resolution. A discrepancy in any of these three checks would falsify the bifurcation claim.","supporting_citations":[],"review_version":1}