{"id":"346e7916-df85-4b02-9cc1-2ea2340c3a1a","arxiv_id":"2505.08415","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For the coherent quantum black hole geometry, scalar, electromagnetic, and gravitational quasinormal mode frequencies deviate from Schwarzschild by about 0.5% for a small core and up to roughly 5% in damping for a core at 70% of the Schwarzschild radius.","lead":"This paper computes the ringdown vibration frequencies of a model black hole called a coherent quantum black hole, where a fuzzy quantum core replaces the classical singularity. The frequencies are close to Einstein's prediction, but the damping times can differ by up to about five percent, which future gravitational wave detectors might measure.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The '~5%' ringdown deviation claim rests on WKB entries that are flagged non-convergent; the dominant even-parity ℓ=2 n=0 mode in Table 4 shows a ~15% shift in Imω, so a fully numerical QNM check is needed before the observational forecast can be accepted.","rationale":"I focused on the reliability of the WKB calculation rather than on the imported metric Vq from Refs [1,2]. The paper is explicitly a phenomenological study of a previously proposed geometry; the observational forecast is conditional on that model, and the authors do not claim to derive it within this work. Within the paper's own argument, the load-bearing step for the claimed observable magnitude is the QNM calculation. Here the paper's own tables signal trouble: a substantial fraction of entries at Rs=0.7 R_Sch are flagged non-convergent, including the dominant even-parity ℓ=2 n=0 mode. The text's '~5%' statement is not merely unverified; it appears inconsistent with the even-parity table, which shows ~15% for that mode. Since the even-parity ℓ=2 n=0 mode is the primary ringdown mode in gravitational-wave astronomy, this is the single most relevant quantity. A fully numerical QNM calculation on the same potential would settle whether the WKB result is trustworthy. The paper is appropriately cautious in calling the results preliminary, so I do not recommend rejection; the conditional verdict stands.","tokens_in":11460,"tokens_out":8887,"duration_ms":86063,"concrete_test":"Compute the ℓ=2, n=0 even-parity gravitational QNM frequency for the metric (1) with f=1−2(M/r)erf(r/Rs) at Rs=0.7 R_Sch using a fully numerical method (e.g., Leaver's continued fraction or time-domain integration) applied to the potential V_2^(e) in Eq. (34). Compare Imω to Table 4's −0.152i and to the text's '~5%' claim; if the numerically converged value differs by more than ~10%, or if the mode is not cleanly defined, the WKB-based quantitative claim is unreliable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claim, stated in Section 4, is that the imaginary part of the QNM frequencies becomes 'of the order of 5% smaller' for Rs = 0.7 R_Sch, leading to longer ringdown decay times and a possible observational signature. This claim is not robustly supported by the paper's own data. The frequencies are computed with the WKB formula (36) using a [6/7] Padé approximant, and many entries in Tables 1–4 are explicitly marked '?' for non-convergence. No uncertainty estimates are given, and no independent numerical check is provided. More seriously, the most astrophysically relevant mode, the even-parity gravitational ℓ=2, n=0 mode in Table 4, is marked '?' at Rs=0.7 R_Sch and gives Imω=−0.152, compared with −0.178 for Schwarzschild: a ~15% decrease, not 5%. Other even-parity entries shift by up to ~25%. Thus the '~5%' statement is inconsistent with the even-parity table, and the dominant ringdown mode is exactly the one for which the WKB result is unconverged. The even-parity potential (34) is moreover only sketched in Appendix A, so a verification of the WKB numbers with a fully numerical method is required before the observational forecast can be accepted.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies a static, spherically symmetric spacetime with the quantum-corrected potential Vq=-(M/r) erf(r/Rs), inherited from earlier coherent-state quantisation work. After reviewing the horizon, photon-ring, and critical-impact-parameter properties of this geometry, it computes quasinormal-mode frequencies for scalar, electromagnetic, and odd/even gravitational perturbations using a [6/7] Padé-improved WKB method. The central claim is that, for a quantum core of size Rs=0.7 R_Sch, the imaginary parts of the frequencies are about 5% smaller than in Schwarzschild, implying longer ringdown decay times that might be observable with next-generation gravitational-wave detectors.","tokens_in":11739,"tokens_out":13952,"duration_ms":141852,"significance":"If the reported deviations are quantitatively correct, the paper supplies a concrete, falsifiable prediction linking a quantum-core scale to ringdown observables, which is of genuine interest to strong-field gravity and gravitational-wave tests. The manuscript is transparent about the limitations of the WKB method, explicitly flags non-convergent entries, and presents the perturbation potentials explicitly. These strengths make the paper a useful starting point, but the numerical basis of the headline claim needs to be strengthened before the observational forecast can be accepted.","major_comments":[{"comment":"The central quantitative claim that the imaginary part of omega becomes 'of the order of 5% smaller' for Rs=0.7 R_Sch is not supported uniformly by the tabulated data and is contradicted by the most astrophysically relevant mode. In Table 4, the even-parity gravitational ell=2, n=0 mode changes from Im omega=-0.178 (Schwarzschild) to Im omega=-0.152 at Rs=0.7 R_Sch, a reduction of roughly 15%, and this entry is marked with a question mark for non-convergence. Several other even-parity entries shift by 10-25%, and even at Rs=0.5 R_Sch the n=2, ell=2 entry in Table 4 slightly violates the statement that the imaginary part is always smaller in magnitude than Schwarzschild. The text should quantify the variation by sector, explicitly exclude or justify non-converged entries, and avoid presenting the 5% figure as the mode-independent prediction.","section":"Section 4, Tables 1-4"},{"comment":"The WKB frequencies are the sole numerical evidence for the ringdown prediction, yet they carry no error estimates and are not checked against an independent method. Section 3 correctly notes that there is no proof of convergence for the WKB series and that including higher orders can worsen the estimate; nonetheless, the observational claim depends on the magnitude of the Im omega shifts. Many entries at Rs=0.7 R_Sch, including the dominant even-parity gravitational mode, are flagged as non-convergent. The authors should provide a fully numerical solution of Eq. (27) with boundary conditions (28)-(31), or at least a quantitative convergence study of the Padé-WKB results, before the abstract's ringdown statement can be regarded as supported.","section":"Section 3, Eq. (36); Tables 1-4"},{"comment":"The even-parity gravitational potential in Eq. (34) is the central input for the dominant ringdown mode, but its derivation in Appendix A is only sketched. After the gauge choices encoded in Eqs. (58)-(59), the text states that 'a short calculation' and a 'straightforward, even though tedious' reduction lead to Eq. (34), without showing the intermediate steps and without verifying that the expression reproduces the Zerilli potential in the Schwarzschild limit f=1-2M/r. Given the size and complexity of Eq. (34), this omission makes Table 4 impossible to check; the authors should provide the missing derivation or an independent verification.","section":"Appendix A, Eq. (34)"}],"minor_comments":[{"comment":"The displayed horizon-existence bound appears to read Rs < 4M sqrt(pi), but the quoted numerical value 1.13 R_Sch corresponds to 4M/sqrt(pi); please correct the typesetting, since the printed formula is otherwise wrong by a factor of pi.","section":"Section 2, Eq. (17)"},{"comment":"The sentence 'the horizon exists only for inner cores of size Rs <~ 0.84 R_Sch' conflicts with Eq. (17), which gives Rs < 1.13 R_Sch for the existence of a horizon; the 0.84 R_Sch bound is the stricter condition for the core to be hidden inside the horizon. Please rephrase this sentence to distinguish the two conditions.","section":"Section 4"},{"comment":"The caption labels both gravitational panels as 'bottom left'; the odd-parity panel should presumably be labeled 'bottom right'.","section":"Figure 2 caption"},{"comment":"The symbol m is used for the Misner-Sharp mass, for the scalar field mass, and for the azimuthal quantum number. Please rename at least one of these to avoid confusion for the reader.","section":"Eqs. (9)-(10), (21)-(26)"},{"comment":"The Mathematica code for the WKB calculations and Fig. 1 is said to be available upon request; for reproducibility, please deposit the code and data in a permanent public repository.","section":"Section 3"},{"comment":"The observational phrasing should explicitly state that Eq. (6) is an input inherited from Refs. [1,2] and that Rs is a free regulator of the model, so the ringdown prediction is conditional on that coherent-state construction.","section":"Abstract and Section 4"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of the journal and addresses a timely topic, but the headline numerical claim needs verification. I recommend that the editor seek a second opinion from an expert in black-hole perturbation theory, particularly concerning the even-parity gravitational potential in Appendix A. The paper's reliance on the authors' own coherent-state construction is not circular, but the physical conclusion is only as strong as the input metric in Eq. (6)."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a legitimate new application of standard WKB QNM methods to an imported quantum-corrected black hole metric, and the even-parity gravitational potential in Eq. (34) is a genuine new result. But the paper's headline claim about '~5%' deviations is undercut by its own Table 4, where the dominant ℓ=2 n=0 even-parity mode shifts by about 15% at Rs=0.7 R_Sch, and that entry is one of the ones flagged as non-convergent.\n\nThe new content is real: Tables 1-4 for scalar, electromagnetic, and gravitational QNMs on the erf-corrected metric are not in the prior literature, and the even-parity potential is a nontrivial extension of Zerilli to a non-vacuum, spherically symmetric background. Appendix A is a serious attempt at the derivation, though it does skip steps. The paper is also honest that the WKB expansion has known convergence issues and that these numbers are preliminary.\n\nNow the soft spots. First, the 5% statement in Section 4 is not a fair summary of the tables. For scalar, vector, and odd-parity gravitational modes, the imaginary-part shifts are indeed around 5% or less at Rs=0.7. But the even-parity sector, which contains the ℓ=2 n=0 ringdown mode that actually matters for gravitational wave observations, shows a 15% shift, and higher overtones deviate by more. The paper's own data contradict its summary. Second, the WKB results come with no error bars, and a substantial fraction of entries are marked '?', including the most relevant mode. The authors say a fully numerical calculation is beyond the scope, which is fair, but then the observational forecast in the abstract is premature. Third, the even-parity potential derivation is sketched; the gauge choices and the 'short calculation' that yields Eq. (64) should be spelled out or referenced more fully for a referee to verify. Finally, the geometry is imported from Refs. [1,2], so the phenomenology is conditional on that model being physical; that is not a flaw in this paper, but it is worth saying explicitly that the QNM numbers are properties of that model, not independent predictions.\n\nMy own view on the stress-test note: it is right. The discrepancy between the '5%' summary and the even-parity table is real and should be fixed before publication.\n\nWho is this for? People working on quantum-corrected black holes and ringdown phenomenology. It deserves a serious referee: the calculation is new, the potential is useful, and the model is falsifiable in principle. I'd send it to review, but the referee should push for either a numerical check of the key modes or a much more careful statement of what the WKB results actually show.\n\nRecommendation: send to peer review, but expect revision.","headline":"Solid new QNM tables for a quantum-corrected metric, but the paper's '~5%' summary conflicts with its own even-parity table, where the dominant mode shifts ~15% and is flagged non-convergent.","tokens_in":12305,"tokens_out":3420,"would_cite":false,"duration_ms":32244,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.70.-s","04.30.-w"],"model":"deepseek-v4-flash","headline":"The ringdown quasinormal modes of a coherent quantum black hole deviate from Schwarzschild: for a core of size 0.7 Schwarzschild radii the damping time grows by about 5 percent, making the core size potentially visible in merger…","keywords":["quasinormal modes","coherent quantum black holes","ringdown damping","WKB approximation","quantum core size","error-function potential","gravitational wave observables"],"falsifier":"One decisive check is a fully numerical computation of the fundamental $l=2$, $n=0$ even-parity gravitational mode at $R_s=0.7R_{\\mathrm{Sch}}$; the WKB table marks this entry with a question mark, so if a converged calculation does not reproduce an imaginary part about 5 percent smaller than the Schwarzschild value $-0.178$, the claimed lengthening is a WKB artifact rather than a property of the coherent quantum black hole geometry.","tokens_in":11245,"feed_emoji":"🕳️","tokens_out":13546,"duration_ms":119330,"temperature":0.7,"pith_summary":"This paper argues that a specific quantum-corrected Schwarzschild geometry, built from a coherent-state mean-field treatment of gravity, produces a ringdown that is slightly but detectably different from general relativity. The input is the potential $V_q=-(M/r)\\,\\mathrm{erf}(r/R_s)$, in which a Gaussian regulator of size $R_s$ replaces the classical singularity with a milder integrable one. Computing scalar, electromagnetic, and gravitational quasinormal modes with a WKB method, the paper finds real frequencies close to Schwarzschild but imaginary parts that are smaller, implying longer damping times; for $R_s=0.7R_{\\mathrm{Sch}}$ the damping time grows by about 5 percent. If this holds, the duration of the ringdown phase in black hole merger observations would carry a measurable imprint of the quantum core. The paper is careful to frame the numbers as preliminary estimates, noting that the WKB expansion fails to converge in part of the interesting parameter range.","feed_headline":"Black-hole ringdown would last 5 percent longer with a quantum core","feed_subtitle":"A quantum core at 0.7 Schwarzschild radii would stretch ringdown damping, letting merger observations detect it.","key_machinery":"The load-bearing object is the quantum-corrected potential $V_q(r)=-(M/r)\\,\\mathrm{erf}(r/R_s)$: the error function is the Gaussian regulator that turns the classical divergence into an integrable singularity at $r=0$ and defines the core size $R_s$. This potential enters the metric function $f_q=1+2V_q$, from which the paper builds the effective potentials for scalar, electromagnetic, odd-parity and even-parity gravitational perturbations. Each perturbation obeys a Schrödinger-like equation in the tortoise coordinate $r_*$ defined by $dr_*/dr=1/f(r)$, and the frequencies are extracted with the WKB condition of Ref. [6] truncated with Padé approximants of order $[6/7]$. The mechanism of the argument is that the core changes the height and width of the effective potential barrier, which shifts the imaginary part $\\omega_I$ more than the real part $\\omega_R$.","core_discovery":"The central claim is that the coherent quantum black hole metric of Eq. (6) is not spectroscopically identical to Schwarzschild outside the horizon. Using the WKB approximation with Padé improvement, the paper computes the fundamental and first overtones for scalar ($j=0$), electromagnetic ($j=1$), and gravitational ($j=2$) perturbations. For a quantum core $R_s=0.5 R_{\\mathrm{Sch}}$ the quasinormal frequencies agree with Schwarzschild at the level of about 0.5 percent, but for $R_s=0.7 R_{\\mathrm{Sch}}$ the imaginary part is systematically about 5 percent smaller in magnitude while the real part shifts little. A smaller imaginary part means longer-lived modes, so the ringdown signal would persist noticeably longer than in general relativity. The paper also establishes that the event horizon exists only for $R_s \\lesssim 0.84 R_{\\mathrm{Sch}}$ and that $R_s=R_{\\mathrm{Sch}}$ corresponds to a horizonless black hole mimicker, so the quoted deviations apply to the black hole regime.","pith_inferences":["I infer that if the geometry is real, existing GR-based ringdown templates would fit the inspiral and merger well but leave a small systematic residual in the damping tail; rescaling the template damping time by $|\\omega_I|^{-1}$ would test the effect on archival merger events.","A natural next calculation is to replace the WKB estimate with a fully numerical time-domain or continued-fraction computation in the regime $R_s\\gtrsim0.7R_{\\mathrm{Sch}}$, where several entries in the tables fail to converge; the 5 percent figure should be treated as provisional until confirmed there.","Because the paper's metric is static and spherically symmetric while real merger remnants are rotating, whether the core imprint survives in the ringdown of a rotating black hole is an open extension the paper does not address."],"forward_implications":["The damping time of the fundamental ringdown modes becomes up to about 5 percent longer for a core size $R_s=0.7R_{\\mathrm{Sch}}$, so the duration of a merger's ringdown carries information about the quantum core.","The real part of the quasinormal frequencies stays nearly unchanged, so the oscillation pitch of the ringdown is a poor probe of the core while the decay envelope is the sensitive observable.","For a smaller core $R_s=0.5R_{\\mathrm{Sch}}$, deviations are only about 0.5 percent, so detecting the effect requires either large cores or high-precision next-generation detectors.","The horizon exists only for $R_s\\lesssim0.84R_{\\mathrm{Sch}}$; larger cores are horizonless mimickers whose oscillation spectrum depends on unknown surface boundary conditions, so the quoted deviations do not extend beyond this range."],"supporting_citations":[{"why":"Defines the coherent-state mean-field picture and shows the exact Schwarzschild potential cannot be realised without regularisation; it is the conceptual basis for the quantum geometry.","marker":"[1]"},{"why":"Supplies the Gaussian regulator and the error-function quantum-corrected potential used as the input metric for every calculation in the paper.","marker":"[2]"},{"why":"Provides the higher-order WKB method used to compute the quasinormal frequencies and their overtone expansion.","marker":"[6]"},{"why":"Gives the photon-ring and critical-impact-parameter observables used to characterise deviations from Schwarzschild.","marker":"[11]"},{"why":"Supplies the electromagnetic perturbation potential on a static spherically symmetric background.","marker":"[12]"},{"why":"Provides the Padé-improved semianalytic approach used to stabilise the WKB frequency estimates.","marker":"[14]"},{"why":"Delivers the higher-order WKB recipe and the code the paper uses to evaluate the frequencies.","marker":"[15]"}],"fun_headline_variants":["Quantum core delays black-hole ringdown by 5%","Black-hole ringdown lasts 5% longer with quantum core","Quantum black holes ring 5% longer than GR predicts","5% longer ringdown: quantum core vs classical black hole","Coherent quantum black holes slow ringdown by 5%"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the coherent-state mean-field construction leading to $V_q=-(M/r)\\,\\mathrm{erf}(r/R_s)$ describes the actual spacetime of a physical black hole; if that metric is only an ad hoc phenomenological regulator, the computed frequencies describe a toy model, not an observational prediction.","fun_headline_variants_meta":{"raw":{"variants":["Quantum core delays black-hole ringdown by 5%","Black-hole ringdown lasts 5% longer with quantum core","Quantum black holes ring 5% longer than GR predicts","5% longer ringdown: quantum core vs classical black hole","Coherent quantum black holes slow ringdown by 5%"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001214,"raw_usage":{"total_tokens":4962,"prompt_tokens":876,"completion_tokens":4086,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":492,"completion_tokens_details":{"reasoning_tokens":4002}},"tokens_in":492,"tokens_out":4086,"duration_ms":30513,"temperature":1.0,"reasoning_tokens":4002,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:55:09.109219+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One decisive check is a fully numerical computation of the fundamental $l=2$, $n=0$ even-parity gravitational mode at $R_s=0.7R_{\\mathrm{Sch}}$; the WKB table marks this entry with a question mark, so if a converged calculation does not reproduce an imaginary part about 5 percent smaller than the Schwarzschild value $-0.178$, the claimed lengthening is a WKB artifact rather than a property of the coherent quantum black hole geometry.","supporting_citations":[{"cited_title":"Geometry and thermodynamics of coherent quantum black holes,","cited_arxiv_id":null,"evidence_quote":"Defines the coherent-state mean-field picture and shows the exact Schwarzschild potential cannot be realised without regularisation; it is the conceptual basis for the quantum geometry."},{"cited_title":"Horizon quantum mechanics for coherent quantum black holes,","cited_arxiv_id":null,"evidence_quote":"Supplies the Gaussian regulator and the error-function quantum-corrected potential used as the input metric for every calculation in the paper."},{"cited_title":"Black Hole Normal Modes: A WKB Approach. 1. Foundations and Application of a Higher Order WKB Analysis of Potential Barrier Scattering,","cited_arxiv_id":null,"evidence_quote":"Provides the higher-order WKB method used to compute the quasinormal frequencies and their overtone expansion."},{"cited_title":"Observational properties of coherent quantum black holes,","cited_arxiv_id":null,"evidence_quote":"Gives the photon-ring and critical-impact-parameter observables used to characterise deviations from Schwarzschild."},{"cited_title":"Quantization of the electromagnetic field outside static black holes and its application to low-energy phenomena,","cited_arxiv_id":null,"evidence_quote":"Supplies the electromagnetic perturbation potential on a static spherically symmetric background."},{"cited_title":"Quasinormal modes of black holes. The improved semianalytic approach,","cited_arxiv_id":null,"evidence_quote":"Provides the Padé-improved semianalytic approach used to stabilise the WKB frequency estimates."},{"cited_title":"Higher order WKB formula for quasinormal modes and grey-body factors: recipes for quick and accurate calculations,","cited_arxiv_id":null,"evidence_quote":"Delivers the higher-order WKB recipe and the code the paper uses to evaluate the frequencies."}],"review_version":1}