{"id":"937b12e5-565b-4948-b1ee-6a71398c537f","arxiv_id":"2610.01152","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For any static, spherically symmetric black hole, the logarithmic running of dynamical Love numbers is encoded in one traceless 2x2 monodromy matrix with three entries A, B, C.","lead":"Tidal Love numbers measure how a black hole deforms under a passing companion's pull. This paper shows that all the logarithmic running of these numbers at non-zero frequency is fixed by a single mathematical object, the monodromy around a singularity, and derives a universal equation for that running.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"The monodromy theorem is proved as a formal power series in ω², yet the exact claims ν_MST = ℓ − δ and the resummed RG flow require convergence (or Borel summability) that is neither stated nor established; the paper acknowledges the expansion may be asymptotic but does not address this gap.","rationale":"The reader correctly identified the analyticity/singularity-placement assumption as load-bearing, and the authors state it honestly. My concern is distinct and located in the step from the formal power series to exact statements, which the reader flagged only as 'the near-zone expansion possibly asymptotic' without elevating it to the central risk. The monodromy proof itself is sound as a formal statement and the low-order checks are good evidence, so I keep CONDITIONAL rather than escalating to REJECT. A concrete numerical comparison of ν_MST at finite small ω against the paper's prediction would settle whether the exact claims hold. This is a correctness-risk concern, not a circularity or authorship issue.","tokens_in":32838,"tokens_out":1893,"duration_ms":13866,"concrete_test":"For Schwarzschild monopole scalar (ℓ=0, s=0), where ν_MST is numerically known to high precision from the MST continued-fraction method (published, not to be re-derived), compare the prediction from Sec. 2.3, ν_MST = 1 − δ with δ² = C² − AB using the paper's Table 2 coefficients, against the exact numerical ν_MST at several small frequencies, e.g. ω = 10⁻¹, 10⁻², 10⁻³. If the difference scales as ω⁸ (the first omitted order), the series is asymptotic and the exact identification fails; if the difference is instead non-perturbative (e.g. exponentially small, ∼ e^{−1/ω} or omitted higher-order terms that vanish faster than any power), then the asymptotic series may be resummable and the exact claim could survive.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim (Sec. 2.2 Theorem, and its use in Sec. 2.3 ν_MST = ℓ − δ, and Sec. 3.1 exact Möbius/RG flow) rests on identifying, for finite small ω, the exact monodromy N(ω²) of the full ODE (2.1) with the formal power series N = ∑ ω²ᵏ Nₖ computed from the near-zone expansion. The paper itself states the near-zone expansion is '(possibly asymptotic)' (Sec. 2.1, just before eq. 2.4). The inductive proof (step i) is a correct formal power-series statement: it shows the coefficient tower has the monodromy structure order by order. But the transition from 'formal power series in ω²' to 'the exact monodromy, valid everywhere up to the nearest singularity' (Sec. 2.3) requires that the ω² series converge or be resummable in a domain containing finite ω, or a separate analytic-continuation argument. None is given. The cross-checks all verify coefficients through O(ω⁶) (Sec. 4.2, Ref. [44], ν_MST through O(ω⁴)), which is evidence for the low-order coefficients but cannot distinguish a convergent series from an asymptotic one. In particular, the claimed exactness of ν_MST = ℓ − δ as a function of ω, and the claim that the quadratic RG equation (3.4) is 'valid to all orders', are stronger than what the formal proof establishes. This does not invalidate the coefficient-level results — which are genuine, computable, and cross-checked — but it is the load-bearing assumption for the exact/global statements. The paper's own language ('possibly asymptotic', Sec. 2.1) flags the gap explicitly, so this is an acknowledged limitation, but the consequence for the exact claims is not addressed anywhere.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"This paper develops a model-agnostic framework for the running of dynamical Love numbers on four-dimensional, static, spherically symmetric, asymptotically flat spacetimes. Casting the separated perturbation equation as L_0 u + ω^2 R u = 0 with p, q analytic at x = 0 and R admitting a Laurent expansion starting at x^{-2} (Eqs. 2.1–2.2), the paper solves the near-zone tower in powers of ω^2 via a master recurrence (Eq. 2.6). The central theorem (Eq. 2.19) states that, as a formal power series in ω^2, the growing and decaying solutions equal x^N Φ with Φ free of logarithms and N the traceless monodromy generator with entries A, B, C. From this structure the paper derives the identification ν_MST = ℓ − δ with δ^2 = C^2 − AB (Eq. 2.26), the quadratic renormalization group equation dK/d log μ = −A − 2CK − BK^2 (Eq. 3.4), and a scheme-dependence dictionary (Eq. 2.29). Applications include Schwarzschild coefficients (Table 1) with an MS-scheme dictionary against Ref. [44] and the invariant δ^2 matched through O(ω^8) (Table 2, Eq. 4.19); closed-form leading dynamical running for axial Reissner-Nordström perturbations, vanishing at extremality (Eqs. 4.22–4.23); residue formulae (Eqs. 4.6–4.8); a universal tail coefficient (Eq. 4.11); a theorem on which multipoles probe metric deformations (Eq. 4.28); and exact results for Hayward and Bardeen black holes (Table 3).","tokens_in":33443,"tokens_out":30600,"duration_ms":323736,"significance":"If established, the framework is a genuine unification: it reduces the infinite tower of near-zone logarithms to a single algebraically computable monodromy generator, connects the EFT beta-function picture with the MST renormalized angular momentum, and yields concrete new results (vanishing extremal-RN running; quadrupole sensitivity to N = 5 metric deformations only at the dynamical level). The paper's strengths are substantial: the inductive proof of the monodromy structure of the coefficient tower (Sec. 2.2) is explicit and reproducible; the master recurrence is a concrete algebraic algorithm; and the cross-checks are independent and non-trivial (MS-scheme coefficients of Ref. [44] matched through the scheme dictionary, the invariant δ^2 agreeing through O(ω^8), and ν_MST agreeing with the MST continued-fraction result through O(ω^4)). The main caveat is the gap between the formal power-series theorem and the exact finite-ω claims, detailed below; the coefficient-level results are not affected by that caveat.","major_comments":[{"comment":"The theorem is proved and stated as a formal power series in ω^2, but it is then used to make exact finite-ω statements: Eq. (2.24) gives the 'correct leading powers of the exact solution' as x^{−(ℓ−δ)−1} and x^{ℓ−δ}, and the text claims the monodromy 'is thus valid everywhere up to the location of the nearest singularity.' Since the paper itself flags the near-zone expansion as '(possibly asymptotic)' (Sec. 2.1, just before Eq. 2.4), the identification of the formal series N(ω^2) = Σ N_k ω^{2k} with the exact monodromy of Eq. (2.1) at finite small ω is not established. The exact monodromy exists and is analytic in ω near 0; what is missing is either a convergence or resummability argument showing that the coefficient tower computes its Taylor expansion, or an explicit scoping of Eqs. (2.24)–(2.26) as formal-asymptotic statements. This is load-bearing for the global claims of Secs. 2.3 and 3.1 but does not affect the coefficient-level results, which are independently cross-checked through O(ω^8). A concrete test would be to compare ν_MST = ℓ − δ truncated at O(ω^4) against the exact MST continued-fraction value at a few moderate frequencies (e.g., ω ~ 0.05–0.2 on Schwarzschild, ℓ = 2).","section":"Sec. 2.2–2.3, Eqs. (2.19), (2.24), (2.26)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"'We emphasize that this is exact' (Eq. 3.4) is ambiguous: as an identity among formal power series in ω^2, the RG equation is established by differentiating the Möbius transformation, and in that sense 'valid to all orders' is defensible; but if the statement is meant literally at finite ω it inherits the unresolved issue of the first comment. The tracelessness argument in Sec. 2.2 step (iv) similarly uses continuity of N(ω^2) in ω^2 to conclude tr N = 0 from the discreteness of ℤ, which presupposes that N is a genuine analytic function rather than merely a formal series; this step should be restated as an order-by-order formal argument (for instance, det M = 1 as a formal power series together with tr N(0) = 0) or placed under the same convergence hypothesis. The revision should also reconcile the 'exact' language with the paper's own caveat, stated in the same section, that 'everything is assumed to be perturbative in ω.'","section":"Sec. 3.1, Eqs. (3.3)–(3.4)","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The closed-form expression for γ_1 contains the factor 1/[ℓ(ℓ+1)] and is not defined at ℓ = 0; the domain of validity (ℓ ≥ 1) should be stated explicitly, with the ℓ = 0 scalar entry in Table 1 identified as a separate evaluation.","section":"Eq. (4.13)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The claim that the invariant δ^2 agrees between the two schemes through O(ω^8) is checkable from Table 2, but the intermediate combination is not shown; displaying one line with C^2 − AB computed from the MS column would allow readers to verify Eq. (4.19) directly.","section":"Table 2 and Eq. (4.19)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The sentence reporting agreement of the O(ω^4) term of ν_MST with 'the numerical result obtained from the MST continued fraction formula' does not give the comparison value; reporting it, or a short table, would make this cross-check legible.","section":"Sec. 4.2, Eq. (4.18)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The universal tail formula (4.11) is derived in Appendix A for ℓ ≥ 1, while the ℓ = 0 case β_1 = 2 is established separately in Sec. 2.1 (Eq. 2.17); Sec. 4.1.2 should state this split explicitly.","section":"Sec. 4.1.2 and Appendix A","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"Several entries are arXiv-only without journal or status data (e.g., Refs. [5], [48], [101]); for Ref. [5], which is cited as a review, the publication status should be given.","section":"References","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The reader's conditional verdict aligns with my reading: the formal machinery is sound and well cross-checked, and the paper is a strong contribution that belongs in the journal once the formal-to-exact scoping is resolved. The skeptic's convergence concern is the right issue to press; the fix is likely achievable either by proving that the coefficient tower computes the Taylor expansion of the exact monodromy (which is analytic in ω^2 near 0) or by rewriting the global claims in the language of formal power series and flagging the resummation as a conjecture supported by the O(ω^4)–O(ω^8) cross-checks. I would also encourage stating the assumption that all other singularities lie at |x| ≥ 1 more prominently in the abstract, since it delimits the 'any background' claim. No concerns about citation practice or novelty disclosure: the relation to Refs. [69], [71], [72] is clearly delineated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"This paper is worth a careful read, not a desk rejection. Garcia-Saenz shows that all near-zone logarithms of dynamical Love numbers are generated by a single traceless 2x2 monodromy generator N(w^2), whose three entries A, B, C — running, anomalous dimension, tail — are computed by one algebraic recurrence (2.6) on any static spherically symmetric background in the stated class. The quadratic RG equation is a clean corollary of the Mobius structure, the scheme-dependence dictionary via basis covariance is genuinely clarifying, and the applications deliver: closed-form RN axial results with vanishing in the extremal limit, a theorem that dynamical running probes lower multipoles than static response, and exact Hayward/Bardeen entries. The author is also candid about prior work ([71,72] for nu_MST = ell - delta, [69] for the monodromy/Mobius structure); the new content is the general recurrence plus the physical dictionary. The checks are the strongest part. The invariant delta^2 matches Ref. [44]'s MS-scheme numbers through O(w^8) with the scheme shift exactly as the covariance argument predicts, and nu_MST at O(w^4) agrees with the MST continued fraction. That is real evidence the machine produces the right coefficients. Now the soft spots, in proportion. The stress-test concern lands, but moderately. The theorem is proved as a formal power series in w^2 — the paper says so explicitly in step (i) — and the near-zone expansion is called 'possibly asymptotic'. Yet (2.24) is stated as a fact about the true exponents, and (3.4) is called 'exact'. The bridge from formal to exact — convergence, Borel summability, or an analytic-continuation argument for the monodromy — is never given. My reading is that the bridge is probably there: for fixed x away from the singularities, the ODE depends polynomially on w^2, so the fundamental matrix is analytic in w^2 and the monodromy is analytic, with the recurrence computing its Taylor coefficients. But the paper does not make that argument, and a referee should ask for it or for the claims to be softened. Either way, the coefficient-level results stand. Second, minor: Tables 1 and 3 are asserted without shown working, and the appendices are terse exactly where the algebra is heaviest. Some entries are anchored by closed forms (4.15), the universal tail (4.11), and the delta^2 check, but the rest is not independently reproducible from the text. Third, the scope is stated carefully — four-dimensional, static, spherically symmetric, asymptotically flat, w << 1, singularities at |x| >= 1 — and the universal claims are exactly as strong as those hypotheses. That is the paper being honest, not a hidden flaw. For the community working on dynamical tidal response, MST, and world-line EFT matching, this is an organizing framework plus usable numbers. I would send it to a serious referee and expect it to survive if the formal/exact question is pinned down. I would cite it.","headline":"A genuinely unifying treatment of dynamical Love running — the coefficient-level results are solid and cross-checked, but the 'exact' claims need a convergence argument a referee should pin down.","tokens_in":34090,"tokens_out":10079,"would_cite":true,"duration_ms":116913,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"This paper shows that the entire structure of logarithmic (running) Love numbers of black holes, at every order in frequency, is fixed by a single traceless 2×2 matrix — the monodromy of the perturbation equation at spatial infinity —…","keywords":["Love numbers","monodromy","renormalization group","black hole perturbations","dynamical tidal response","renormalized angular momentum","Fuchsian theory","Reissner-Nordström"],"falsifier":"Pick a background whose metric functions fail to be analytic at spatial infinity, or whose perturbation equation has a singularity inside the unit disk, and check whether the $x^N \\Phi$ structure and the quadratic renormalization-group equation still hold; failing that, numerically integrate the full frequency-dependent perturbation equations for a superposition of multipoles and verify that every logarithm coefficient is reproduced by the three series $A,B,C$, or independently evaluate $\\delta^2$ via the MST continued-fraction method at $O(\\omega^4)$ for a multipole where no closed form is known.","tokens_in":34199,"feed_emoji":"⚫","tokens_out":5120,"duration_ms":32793,"temperature":0.7,"pith_summary":"The paper claims that all the 'running' of black-hole tidal Love numbers — the logarithms that appear once perturbations are made dynamical — is governed by one object: the monodromy of the perturbation equation around the singular point at spatial infinity. Formed as a traceless 2×2 matrix $N$ whose three entries $A,B,C$ are power series in $\\omega^2$, this object resums every logarithmic term at every order without solving any differential equation. From it follows an exact quadratic renormalization-group equation for the Love number, here proved to all orders for any tidal field on any static spherically symmetric background. The same matrix reproduces the Mano-Suzuki-Takasugi 'renormalized angular momentum' and makes the scheme dependence of logarithmic coefficients transparent. Applications include closed-form results for Reissner-Nordström black holes, where the leading dynamical running vanishes at extremality, and exact results for the Hayward and Bardeen regular black holes.","feed_headline":"One 2×2 matrix governs black-hole Love running","feed_subtitle":"Three coefficients — running, anomalous dimension, tail — fix every logarithmic tidal term on any static background.","key_machinery":"The monodromy generator $N(\\omega^2)$ is the traceless $2\\times 2$ matrix that implements analytic continuation once around $x=0$ on the basis $(u_g,u_d)$ of growing and decaying solutions; equivalently, all logarithmic terms of the near-zone solution are generated by the matrix exponential $e^{N\\log x}$. It is computed through a master recurrence formula that solves, to all orders, a tower of sourced Fuchsian problems sharing the same resonant exponents $\\ell$ and $-\\ell-1$. The three entries $A,B,C$ of $N$ encode the running of the Love number, the anomalous dimension of the tidal field, and a tail-like term, and the covariance of $N$ under basis changes organizes all scheme dependence.","core_discovery":"To all orders in $\\omega^2$, the growing and decaying near-zone solutions of a tidal perturbation take the form $x^N \\Phi$, where $\\Phi$ contains no logarithms and $N(\\omega^2)$ is a traceless $2\\times 2$ matrix whose three independent entries carry the three logarithmic coefficients: $A$ measures the running of the Love number, $C$ is the anomalous dimension of the tidal field, and $B$ is a tail-like term. The immediate consequences are an exact renormalization-group equation $\\partial K/\\partial\\log\\mu = -A - 2CK - BK^2$, valid to all orders for any type of Love number, and the identification $\\nu_{\\mathrm{MST}} = \\ell - \\delta$ with $\\delta^2 = C^2 - AB$ for the MST renormalized angular momentum, computable algebraically on any background.","pith_inferences":["Editor's inference: the same monodromy argument should extend to the Teukolsky equation for spinning black holes, where the tower is organized in powers of $\\omega$ rather than $\\omega^2$ and the angular eigenvalue acquires frequency dependence; the paper expects this structure to survive but does not prove it.","Editor's inference: because $\\delta^2 = C^2 - AB$ is the discriminant of the quadratic flow, the renormalization-group fixed points could merge and become complex when $\\delta^2$ changes sign, a mechanism the paper compares to conformality loss while noting it lies outside its perturbative domain.","Editor's inference: the scheme-invariant quantity $\\delta^2$ is a natural target for gravitational-wave tests, since it sidesteps the scheme ambiguities that afflict individual logarithmic coefficients.","Editor's inference: since the logarithms are fixed by the exterior geometry alone, applying the master recurrence to the exterior metric of a neutron star or exotic compact object should produce identical coefficients, in line with the paper's universality statement."],"forward_implications":["The quadratic form of the renormalization-group equation is exact for every multipole and tidal field, not an artifact of a Schwarzschild-specific calculation.","The MST renormalized angular momentum $\\nu_{\\mathrm{MST}}$ becomes computable by linear algebra on any static spherically symmetric background, without explicitly solving the perturbation equation.","The leading dynamical running of axial Reissner-Nordström Love numbers vanishes in the extremal limit for all multipoles $\\ell \\geq 1$.","The leading tail coefficient $\\beta_{\\ell+1}$ is universal — fixed entirely by the Newtonian potential — and identical for every black hole and tidal field.","Dynamical running probes deformations of the metric at lower multipoles than the static Love number, so the quadrupole becomes sensitive to curvature-cubed effective-field-theory corrections even where the static number vanishes."],"supporting_citations":[{"why":"Mano-Suzuki-Takasugi analytic solutions of the Teukolsky and Regge-Wheeler equations; supplies the renormalized angular momentum $\\nu_{\\mathrm{MST}}$ that the paper identifies with $\\ell - \\delta$ via monodromy.","marker":"[63]"},{"why":"Caron-Huot et al. Born-series scalar tidal matching; the MS-scheme coefficients whose scheme dependence the paper reconciles with its own unit pure-power scheme through the covariance of the monodromy generator.","marker":"[44]"},{"why":"Closed-form static logarithmic Love number formula that the dynamical framework generalizes, and whose structure the level-0 analysis mirrors.","marker":"[37]"},{"why":"Monodromy generator around infinity for the Regge-Wheeler equation and its Möbius structure, which the paper extends to arbitrary backgrounds, any tidal field, and a direct algebraic recursion.","marker":"[69]"},{"why":"Static logarithmic Love number formula whose form underlies the second line of Eq. (2.11) and the level-0 analysis.","marker":"[39]"},{"why":"Hypergeometric master equation for Reissner-Nordström perturbations used in Appendix B to prove the closed-form results for all multipoles.","marker":"[90]"},{"why":"Fuchs's theorem and asymptotic-expansion background that grounds the monodromy proof for the tower of sourced equations.","marker":"[77]"}],"fun_headline_variants":["Love numbers run by a traceless 2x2 matrix","Three logs, one matrix: universal Love running","All-order Love running from monodromy at infinity","Renormalized Love numbers: a 2x2 matrix governs all","Black-hole Love running: a 2x2 matrix at all orders"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The framework rests on the near-zone perturbation equation having the form $\\hat{L}_0 u + \\omega^2 R u = 0$ with $p,q$ analytic at $x=0$, $R$ a Laurent series starting at $x^{-2}$, and all other singularities on the complex plane located at $|x| \\gtrsim 1$, conditions enforced for black holes by the analyticity of the metric functions at spatial infinity.","fun_headline_variants_meta":{"raw":{"variants":["Love numbers run by a traceless 2x2 matrix","Three logs, one matrix: universal Love running","All-order Love running from monodromy at infinity","Renormalized Love numbers: a 2x2 matrix governs all","Black-hole Love running: a 2x2 matrix at all orders"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000237,"raw_usage":{"cost":0.000237048,"is_byok":false,"cost_details":{"upstream_inference_cost":0.000237048,"upstream_inference_prompt_cost":0.000117488,"upstream_inference_completions_cost":0.00011956},"total_tokens":1471,"prompt_tokens":1044,"completion_tokens":427,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":256,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":337}},"tokens_in":1044,"tokens_out":427,"cost_source":"openrouter_usage","duration_ms":3226,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":337,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-02T18:40:31.569047+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Pick a background whose metric functions fail to be analytic at spatial infinity, or whose perturbation equation has a singularity inside the unit disk, and check whether the $x^N \\Phi$ structure and the quadratic renormalization-group equation still hold; failing that, numerically integrate the full frequency-dependent perturbation equations for a superposition of multipoles and verify that every logarithm coefficient is reproduced by the three series $A,B,C$, or independently evaluate $\\delta^2$ via the MST continued-fraction method at $O(\\omega^4)$ for a multipole where no closed form is known.","supporting_citations":[{"cited_title":"Analytic Solutions of the Teukolsky Equation and Their Low Frequency Expansions","cited_arxiv_id":"gr-qc/9603020","evidence_quote":"Mano-Suzuki-Takasugi analytic solutions of the Teukolsky and Regge-Wheeler equations; supplies the renormalized angular momentum $\\nu_{\\mathrm{MST}}$ that the paper identifies with $\\ell - \\delta$ via monodromy."},{"cited_title":"Gravitational Wave Scattering via the Born Series: Scalar Tidal Matching to $\\mathcal{O}(G^7)$ and Beyond","cited_arxiv_id":"2503.13593","evidence_quote":"Caron-Huot et al. Born-series scalar tidal matching; the MS-scheme coefficients whose scheme dependence the paper reconciles with its own unit pure-power scheme through the covariance of the monodromy generator."},{"cited_title":"On the logarithmic Love number of black holes beyond general relativity","cited_arxiv_id":"2512.19111","evidence_quote":"Closed-form static logarithmic Love number formula that the dynamical framework generalizes, and whose structure the level-0 analysis mirrors."},{"cited_title":"Black hole scattering amplitudes via analytic small-frequency expansion and monodromy","cited_arxiv_id":"2409.06681","evidence_quote":"Monodromy generator around infinity for the Regge-Wheeler equation and its Möbius structure, which the paper extends to arbitrary backgrounds, any tidal field, and a direct algebraic recursion."},{"cited_title":"Schwarzschild quasi-normal modes of non-minimally coupled vector fields","cited_arxiv_id":"2202.07131","evidence_quote":"Static logarithmic Love number formula whose form underlies the second line of Eq. (2.11) and the level-0 analysis."},{"cited_title":"Ladder Symmetries and Love Numbers of Reissner--Nordstr\\\"om Black Holes","cited_arxiv_id":"2404.06544","evidence_quote":"Hypergeometric master equation for Reissner-Nordström perturbations used in Appendix B to prove the closed-form results for all multipoles."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Fuchs's theorem and asymptotic-expansion background that grounds the monodromy proof for the tower of sourced equations."}],"review_version":1}