{"id":"d4b265ea-dd3d-41e9-8d77-1304be4fe288","arxiv_id":"2412.02793","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A Keldysh spectral expansion built on dual pairings reproduces black hole quasinormal-mode time series, including Schwarzschild power-law tails, and supplies H^p pseudospectral and transient-growth tools.","lead":"The authors recast black hole quasinormal-mode expansions using a Keldysh resolvent scheme in a hyperboloidal setting, showing that the construction needs no scalar product for the null-infinity time series. They demonstrate that the method accurately reconstructs full time-domain signals, including unexpected recovery of Schwarzschild late-time tails, and provide new H^p pseudospectrum and transient-growth analyses.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Tail recovery rests on an unproven continuum limit: App. B.3 makes the Keldysh sum exact for the finite-rank matrix, but convergence of that matrix branch-cut dynamics to the true Price-law tail is not established; a direct N-convergence check against the continuum branch-cut integral is needed.","rationale":"The reader's CONDITIONAL verdict is the right one, and my analysis keeps it. The core Keldysh construction of QNM expansions is on solid ground: the derivation in Sec. 2 is careful, the scalar-product independence is clearly argued, and the four testbed demonstrations, including the exact finite-rank identity in App. B.3, provide strong numerical and analytic support for the QNM-dominated part of the signal. The soft spot is precisely the tail-recovery claim. I partially disagree with the reader's framing: Sec. 5.1 already supplements the Riemann-sum heuristic with the exact matrix-evolution argument in App. B.3, so the missing piece is not the Keldysh projection coefficients for the matrix, but the continuum limit of the finite-rank branch-cut dynamics. Fig. 12 shows qualitative extension of the tail window with N, yet no fixed-τ error against the true continuum tail is given. A direct numerical comparison with an independent branch-cut evaluation, or with the known Price-law asymptotic amplitude, for increasing N would settle whether the reported β_fit is a genuine spectral approximation of the branch cut or a serendipitous finite-rank window effect. Because this is a missing check rather than a demonstrated contradiction, and because the paper's other contributions—scalar-product independence, H^p pseudospectra and transient growths, and the Weyl-law numerics—stand independently, the verdict remains CONDITIONAL and my recommendation is to keep the reader's verdict unchanged.","tokens_in":58874,"tokens_out":4347,"duration_ms":47120,"concrete_test":"For Schwarzschild ℓ=2, evaluate the true continuum branch-cut contribution to the waveform at fixed late times (e.g., τ = 100, 200, 400) by an independent frequency-domain integration along the branch cut, or from the known Price-law asymptotic amplitude with prefactor. Compare this against the Keldysh sum over the non-convergent branch-cut eigenvalues for increasing grid sizes N = 100, 200, 400, 600. Compute the relative error at each fixed τ and the fitted exponent β_fit(N); if the relative error decays with N and β_fit(N) approaches -(ℓ+1) within the fitting error, the tail recovery is a genuine spectral approximation; if the error plateaus or the exponent does not converge, the reported Price-law recovery is a finite-rank window artifact.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The headline beyond-validity result in Sec. 5.1 is that applying the Keldysh scheme to the non-convergent eigenvalues discretizing the Schwarzschild branch cut accurately reproduces the Price-law tail. The paper's own justification is not the Riemann-sum heuristic quoted by the reader, but the exact finite-rank statement in App. B.3: for the discretized generator L_N, the Keldysh formula reproduces exactly the full matrix propagator e^{iτL_N}, so the branch-cut sum is exactly the branch-cut part of that finite-dimensional evolution. This shifts the load-bearing assumption from 'Riemann sum of the Bromwich integral' to 'the finite-rank matrix dynamics converges to the continuum hyperboloidal evolution on the tail-dominated time window'. That assumption is not established. The branch-cut eigenvalues themselves do not converge as N grows (Figs. 2c, 3c), and Fig. 12 shows only that the apparent power-law window extends to later times as N increases. There is no quantitative comparison of the Keldysh branch-cut sum against the true continuum branch-cut contribution, no error-versus-N curve at fixed τ, and no demonstration that the fitted exponent β_fit converges to -(ℓ+1) as N grows. Without that, the excellent-looking β_fit values in Fig. 11 could be a serendipitous finite-rank effect: a finite sum of exponentials with non-convergent imaginary parts can mimic a power law over a fitting window, as the paper itself concedes when it notes the eventual exponential cutoff due to the finite number of modes. Since the tail recovery is one of the paper's headline 'beyond its limits of validity' claims, this continuum-limit gap is the most load-bearing concern.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a Keldysh (modes/comodes) expansion of the resolvent for hyperboloidal, non-selfadjoint black-hole perturbation generators, and uses it to construct quasinormal-mode (QNM) expansions. The authors clarify that the dual-pairing form of the expansion does not require a scalar product, while constant excitation coefficients require one, and they apply the scheme to Pöschl-Teller, Schwarzschild, Schwarzschild-dS, and Schwarzschild-AdS testbeds with Gaussian data. They report accurate reconstruction of boundary time series from early times onward, recovery of Schwarzschild late-time power-law tails by applying the Keldysh prescription to discretized branch-cut eigenvalues, studies of early-time convergence and H^p transient growths, H^p pseudospectra consistent with Warnick's band structure, and numerical evidence for a universal Weyl law for QNM counting.","tokens_in":59216,"tokens_out":4078,"duration_ms":43359,"significance":"If the central claims hold, the paper provides an efficient and conceptually clean spectral route to QNM expansions and, strikingly, to late-time tails. The analytic Keldysh formulation in terms of transpose operators is a genuine clarification of the role of scalar products, and Appendix B.3's finite-rank identity is exact and useful. The numerical comparisons between spectral and time-domain signals use no fitted parameters, and the H^p pseudospectra and Weyl-law results are useful exploratory contributions. However, the headline tail-recovery result and the early-time convergence claim rest on heuristic or single-data evidence, and several supporting studies are deferred to future papers by the same authors, so the significance is conditional on those gaps being closed.","major_comments":[{"comment":"The claim that applying the Keldysh scheme to the non-convergent eigenvalues discretizing the Schwarzschild branch cut accurately recovers Price-law tails is load-bearing but not quantitatively supported. Appendix B.3 shows exactly that the finite-rank matrix L_N reconstructs e^{iτL_N}, but this does not establish convergence of the finite-rank branch-cut dynamics to the continuum hyperboloidal evolution. The branch-cut eigenvalues do not converge as N grows (Figs. 2c, 3c), and Fig. 12 only shows that the apparent power-law window extends to later times with increasing N. The paper lacks a fixed-τ error-versus-N comparison against the continuum branch-cut integral, a demonstration that the fitted exponent β_fit converges to −(ℓ+1) as N grows, or an estimate of the discretization error. Without such evidence, the excellent β_fit values in Fig. 11 could be a serendipitous finite-rank effect rather than a property of the true resolvent.","section":"Sec. 5.1 and App. B.3"},{"comment":"The convergence result for the bulk QNM series at fixed τ_o depends critically on the estimate C(N_QNM,L) ≲ C e^{N_QNM} in Eq. (89). This estimate is read off from a tangent at the origin in Fig. 16 for a single Gaussian initial data u_0 in Eq. (204), while Eq. (37) states that the error constant should be independent of the initial data. The manuscript does not provide a data-independent bound or even numerical evidence that the growth exponent is stable across initial data. Therefore the conclusion that τ_o > 1/κ implies convergence is, at present, a heuristic observation for a special test case, not a general result. This should be stated explicitly and, ideally, supported by additional data or by a proof of the exponential bound.","section":"Sec. 5.2.2, Eqs. (88)–(90), Fig. 16"},{"comment":"The conclusions state that Pöschl-Teller and dS asymptotics 'present good, indeed uniform, convergence properties' of the QNM time series, but the body of the paper only reports that Fig. 14 'suggests' pointwise convergence and that the Schwarzschild-dS case is 'more difficult to evaluate.' The contour-line visual inspection is performed for one initial data and is not a proof of uniform convergence. Please align the summary statements with the exploratory status of the numerical evidence, or provide a rigorous or more systematic numerical test of uniform convergence.","section":"Sec. 5.2.1 and Sec. 6, conclusions ii.3"},{"comment":"The abstract and conclusions state that the paper 'demonstrates the efficiency and accuracy' of the Keldysh approach, but all demonstrations use a single Gaussian initial data family (Appendix E.3) and are limited by the tolerance of the ODE/DAE solver in several cases (Fig. 7). The paper itself acknowledges that a systematic study of generic initial data is left for future work [56]. For a proof-of-principle article this is acceptable, but the wording should be softened from 'demonstrate' to 'illustrate in proof-of-principle testbeds' so that the claims match the evidence presented.","section":"Sec. 3.3, Sec. 6, and abstract"}],"minor_comments":[{"comment":"Footnote 14 explicitly states 'We lack a proof of the later statement' regarding uniqueness of the QNM time series at null infinity, yet conclusion i.3 presents this uniqueness as a result. Please add a qualifier such as 'conjectured' or 'supported by the Lax-Phillips framework'.","section":"Footnote 14 and Sec. 6, item i.3"},{"comment":"The caption of Fig. 14b uses Λ = 0.11, whereas Section 3.2.1 and the surrounding text mainly use Λ = 0.07/M². Please unify the notation so the reader can reproduce the parameter choices.","section":"Fig. 14 caption and Sec. 5.2.1"},{"comment":"The Weyl-law evidence is based on a linear fit to 'a few dozens of points at the end of the series' for each spacetime. Please report the number of fitted points, the residuals, and the sensitivity of the fitted exponent to the fitting window, since the claim of universality across asymptotics rests on this numerical fit.","section":"Sec. 5.4, Fig. 24"},{"comment":"The statement |⟨v̂−_p, v̂+_p⟩_{H^p}| ∼ 1 − 1/p^4 is presented without derivation or numerical table; please indicate whether this is an analytic result, a fit, or a numerical observation, and provide the corresponding data or reference.","section":"Sec. 5.3, Eq. (102)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a broad proof-of-principle paper; several claims that appear in the abstract and conclusions are stronger than the evidence in the body, and important validation steps are deferred to the authors' future works [56], [58], [76]. In particular, the tail-recovery claim needs a direct continuum-limit check. I recommend major revision with the understanding that the central Keldysh formalism is sound and the remaining work is to either supply the missing quantitative evidence or substantially qualify the claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: this is a serious spectral-methods paper, and the Keldysh prescription—expansion coefficients from dual pairings of modes and comodes, no scalar product needed—is clearly presented and numerically vindicated on four testbed spacetimes. The headline surprise, that blindly applying the same projection to non-convergent discretized branch-cut eigenvalues recovers Schwarzschild Price tails, is real but rests on one unproven continuum limit, and that is the place to focus any referee effort.\n\nThe genuinely new content: the transpose/dual-pairing reformulation of the Keldysh expansion (over their earlier adjoint-based work) is clean; the H^p-Sobolev pseudospectra and transient growth analysis in Pöschl-Teller is a nice demonstration that Warnick's band structure controls the pseudospectrum; the Weyl-law numerics for dS/AdS is a useful first; the second-order QNM coefficient formula is a sketch but a reasonable one. The comparisons with time-domain evolutions show agreement below solver tolerance for the three non-flat cases, which is strong evidence the spectral algorithm is doing the right thing.\n\nThe soft spots. The tail recovery is the most load-bearing. The paper itself flags the Riemann-sum heuristic, and App. B.3 shows only that the Keldysh sum over all matrix eigenvalues exactly reproduces the finite-rank propagator—not that the finite-rank branch-cut dynamics converges to the true continuum tail. The branch-cut eigenvalues do not converge with N; Fig. 12 shows only that the apparent power-law window extends later; there is no error-vs-N at fixed tau and no evidence that beta_fit tends to -(ell+1). A finite sum of non-convergent exponentials can mimic a power law over a fitting window, and the paper concedes the eventual exponential cutoff. So the tail claim is plausible but not established. Minor: the C(N)~e^N convergence bound is inferred from one dataset, and every numerical demonstration uses the same Gaussian initial data. No code or data are provided; for a numerical-heavy paper that is a real reproducibility gap.\n\nBottom line: the Keldysh machinery is worth engaging with and the H^p tools are a plus. The Price-law recovery needs a proper N-convergence check against the continuum branch-cut integral before it becomes a headline result. This deserves peer review, but with the expectation that the tail section is reworked. I'd cite the formalism and pseudospectra; I'd wait on the tails.","headline":"Solid Keldysh formalism and impressive numerics, but the Schwarzschild tail recovery rests on an unproven continuum limit.","tokens_in":59784,"tokens_out":2277,"would_cite":true,"duration_ms":24622,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C57","83C35","35P05","47A10","65M70"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that Keldysh resolvent expansions, built from modes and comodes of a hyperboloidal non-selfadjoint generator, provide a scalar-product-free spectral construction of quasinormal-mode expansions that accurately reproduces…","keywords":["quasinormal modes","black hole perturbations","Keldysh expansion","non-selfadjoint operators","hyperboloidal foliation","Price law tails","pseudospectrum","Weyl law"],"falsifier":"Take a fixed Schwarzschild case (say ell=2), evaluate the truncated Keldysh sum over the branch-cut eigenvalues at a late time tau, and compare with a direct numerical evaluation of the Bromwich integral around the branch cut for increasing grid sizes N. If the difference does not decrease toward zero as N grows, the recovered Price tail is a finite-rank artifact rather than a property of the true resolvent.","tokens_in":2073,"feed_emoji":"🕳️","tokens_out":2471,"duration_ms":82432,"temperature":0.7,"pith_summary":"The paper argues that quasinormal-mode (QNM) expansions of black hole perturbations are best built not from orthogonality or completeness of modes, but from a Keldysh expansion of the resolvent of the non-selfadjoint time generator in a hyperboloidal slicing. In this setting QNMs are proper eigenvalues with normalisable eigenfunctions, and the expansion coefficients come from dual pairing with the left eigenfunctions, the comodes, so no scalar product is needed. For Gaussian test data the truncated Keldysh series reproduces the boundary time-domain signal from early times dominated by high overtones through late ringdown, with remarkable accuracy in all studied asymptotics. The most striking claim is that applying the same Keldysh prescription to the non-convergent eigenvalues that discretise the Schwarzschild branch cut recovers the power-law late-time tails, including the Price law $tau^{{-(ell+1)}}$. The paper also extracts second-order QNMs, constructs H^p-pseudospectra, identifies H^p transient growths, and confirms the QNM Weyl law N(omega) ~ $omega^{3}$ in flat, de Sitter and anti-de Sitter asymptotics.","feed_headline":"Keldysh resolvent expansion rebuilds black hole ringdown and tails","feed_subtitle":"Modes and comodes of a non-selfadjoint generator reproduce waveforms and recover the Price power-law tail.","key_machinery":"The load-bearing object is the Keldysh expansion of the resolvent of the infinitesimal time generator L in a hyperboloidal compactification. The hyperboloidal foliation and coordinate compactification turn outgoing boundary conditions into regularity requirements, making QNMs proper eigenvalues of a non-selfadjoint operator with normalisable eigenfunctions in a Hilbert or Banach space. The Keldysh identity expresses R_L(omega) as a sum over poles omega_n with residues given by rank-one operators <alpha_n, .> v_n, where alpha_n are comodes solving L^t alpha_n = omega_n alpha_n, plus a holomorphic remainder; this remainder provides the explicit error bound for truncated QNM series. The discrete Chebyshev pseudospectral approximation then replaces L by a matrix, and the same identity, applied to all eigenvalues of the finite-rank approximant, reconstructs the full evolution, which is what makes the branch-cut tail recovery possible.","core_discovery":"The central discovery is that the Keldysh expansion of the resolvent, R_L(omega) = sum_n [<alpha_n, .>/(omega_n - omega)] v_n + H(omega), built from modes v_n and comodes alpha_n of L and its transpose L^t, gives a spectral, scalar-product-free construction of Lax-Phillips resonant expansions. The paper demonstrates numerically, in the Poschl-Teller model and in Schwarzschild, Schwarzschild-de Sitter and Schwarzschild-anti-de Sitter spacetimes, that the resulting QNM series u(tau,x) ~ sum_n $e^{{i omega_n tau}}$ a_n v_n(x), with a_n = <alpha_n,u_0>/<alpha_n,v_n>, matches the hyperboloidal time evolution of Gaussian data at boundaries throughout the signal. Beyond its strict domain of validity, applying the same formula to the discretised branch-cut eigenvalues reproduces the Schwarzschild late-time power-law tails with the correct Price exponent. If this claim survives scrutiny, the Keldysh scheme is a single spectral algorithm that accounts for both the discrete QNM content and the continuous branch-cut content of black hole perturbation dynamics.","pith_inferences":["If the Riemann-sum interpretation of the branch-cut reconstruction is made rigorous with an error estimate, the Keldysh scheme would provide a unified spectral algorithm for ringdown and tails that could be applied to Kerr or higher-dimensional spacetimes.","The tau > 1/kappa convergence threshold hints at a first-principles selection criterion for the start time of ringdown fits in gravitational-wave data analysis; testing it on realistic inspiral-merger-ringdown waveforms would be a natural next step.","The delta-like H^p transient growth in the p to infinity limit suggests that high-derivative norms can expose an initial loss of regularity; connecting this with Aretakis-type instability at extremal horizons is a testable extension.","Since the Keldysh scheme extends to quadratic pencils, it could provide a practical spectral route to second-order QNM excitation coefficients for mode coupling with different angular quantum numbers."],"forward_implications":["Keldysh QNM expansions are independent of the scalar product: only the product a_n v_n(x) is defined, while constant excitation coefficients require an additional norm.","The scheme generalises Ansorg-Macedo expansions to arbitrary dimensions and to non-diagonalisable cases, going beyond effective one-dimensional problems.","The truncated Keldysh series can be used as a spectral time-domain reconstruction tool: given an acceptable error and a number of modes, the contour plots fix the earliest valid time for the expansion.","For Gaussian data in the Poschl-Teller case the bulk QNM series converges for tau > 1/kappa, providing a natural timescale for ringdown fitting.","The QNM Weyl law N(omega) ~ omega^3 holds in flat, de Sitter and anti-de Sitter asymptotics."],"supporting_citations":[{"why":"Introduces the Keldysh approach to BH QNM expansions and the scalar-product discussion that this paper revisits and refines.","marker":"[1]"},{"why":"Provides the hyperboloidal non-selfadjoint formulation and pseudospectrum tools on which the numerical implementation relies.","marker":"[10]"},{"why":"Establishes H^p Sobolev regularity as the setting in which QNMs are proper eigenvalues, used for pseudospectra and convergence bands.","marker":"[21]"},{"why":"Introduces the discrete spectral QNM expansion that Keldysh generalises, and the branch-cut Riemann-sum tail interpretation used in Section 5.1.","marker":"[22]"},{"why":"Supplies the Lax-Phillips resonance expansion framework of which the Keldysh series is presented as a spectral version.","marker":"[7]"},{"why":"Gives the Keldysh expansion of the resolvent for non-selfadjoint operator pencils, the core identity of the scheme.","marker":"[8]"},{"why":"Provides the Keldysh expansion theorem for holomorphic Fredholm operator pencils underlying the construction.","marker":"[35]"},{"why":"States the QNM Weyl-law conjecture that the paper tests numerically in de Sitter and anti-de Sitter asymptotics.","marker":"[71]"}],"fun_headline_variants":["Keldysh resolvent expansion unifies black hole ringdown and tails","Single spectral formula recovers black hole quasinormal modes and tails","Hyperboloidal Keldysh scheme rebuilds black hole ringdown and Price tail","Non-selfadjoint modes and comodes predict black hole waveforms and tails"],"cache_read_input_tokens":61824,"weakest_assumption_plain":"The tail recovery rests on the assumption that the non-convergent eigenvalues discretising the Schwarzschild branch cut, each weighted with its Keldysh coefficient, form a valid discrete approximation (like a Riemann sum) of the continuous branch-cut contribution to the inversion integral; the paper does not prove this and notes it only 'could be understood' that way.","fun_headline_variants_meta":{"raw":{"variants":["Keldysh resolvent expansion unifies black hole ringdown and tails","Single spectral formula recovers black hole quasinormal modes and tails","Hyperboloidal Keldysh scheme rebuilds black hole ringdown and Price tail","Non-selfadjoint modes and comodes predict black hole waveforms and tails"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000667,"raw_usage":{"total_tokens":3126,"prompt_tokens":1114,"completion_tokens":2012,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":730,"completion_tokens_details":{"reasoning_tokens":1930}},"tokens_in":730,"tokens_out":2012,"duration_ms":16003,"temperature":1.0,"reasoning_tokens":1930,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T23:06:54.815755+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a fixed Schwarzschild case (say ell=2), evaluate the truncated Keldysh sum over the branch-cut eigenvalues at a late time tau, and compare with a direct numerical evaluation of the Bromwich integral around the branch cut for increasing grid sizes N. If the difference does not decrease toward zero as N grows, the recovered Price tail is a finite-rank artifact rather than a property of the true resolvent.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the QNM Weyl-law conjecture that the paper tests numerically in de Sitter and anti-de Sitter asymptotics."}],"review_version":1}