{"id":"585db610-8346-4f8c-be12-2ecc2210e3ea","arxiv_id":"2505.08310","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Using radiative-efficiency scaling relations, the authors derive mostly a>0.9 spins for 66 and the inclination tuning) is reasonable, but the paper has a more immediate and more damaging problem: the tabulated M*_BH column is not consistent with the algorithm described in §3. The stated thin-disk factor f≈1/(2 sin i) forces M* to scale as sin^{-2}i relative to the i≈35° literature value, i.e. a drop of ~0.18 dex at i=45°, yet Table 2 shows all i=45° masses 0.06 dex above the input masses, and i=50°, 55°, 60° masses only 0.01–0.12 dex below. No unstated normalization is provided that could reconcile these numbers. Since Eq. (1) feeds M* and μ into ε, and ε is monotonically inverted into spin, the tabulated spins inherit the inconsistency. A quick reimplementation with the stated equations shows several a>0.9 objects would fall to a≈0.7–0.9, eroding the average-spin claim. This is an internal correctness failure, not merely a question of whether a local calibration can be extrapolated, so it takes priority over the reader's model-dependence concern. The check is one of arithmetic reproduction: recompute Table 2 from Eqs. (1)–(9) with the stated f(i) prescription. If the table does not reproduce, the central claim is unsupported as submitted.","tokens_in":11447,"tokens_out":23887,"duration_ms":226800,"concrete_test":"Re-derive Table 2 from first principles: set f=1/(2 sin i), compute M*(i)=M_lit×[f(i)/f(35°)]², insert into Eq. (1), and invert Eqs. (8)–(9) for a, using Table 1 inputs and the same Lopt(Lbol) relation (Eq. 2). Check whether any tabulated log M* or a matches. Specifically test QSO J1007+2115: the stated algorithm gives log M*≈9.00 and ε≈0.10 (a≈0.7), not 9.24 and 0.174 (a=0.934). If the recomputed sample has mean a<0.9, the central claim fails.","verdict_should_be":"REJECT","load_bearing_attack":"Table 2 is not reproducible from the method stated in §3. The text around Eqs. (3)–(6) says M*_BH is recomputed for each inclination using f≈1/(2 sin i), assuming the literature mass was obtained at i≈35° with f=√3/2. That gives M*(i)=M_lit×[f(i)/f(35°)]² = M_lit×[sin35°/sin i]². At i=45° this factor is 0.66 (log10 ≈ −0.18), yet every i=45° entry in Table 2 is 0.06 dex above the input log M_BH in Table 1. Example: QSO J1007+2115 has input log M_BH=9.18 but tabulated log M*=9.24; the stated recalibration would give log M*≈9.00. Because Eq. (1) is proportional to M* μ^1.5, the efficiencies in Table 2 are a factor ~1.7 too high if the paper's own f(i) formula is applied. This directly inflates spins: several ε≈0.2 entries (QSO J1007, DES J0252, VDES J0244, PSO J011) would fall to ε≈0.1, corresponding to a<0.9. The headline 'average spin >0.9' is therefore not a robust consequence of the described algorithm; either the mass-recalibration step is mis-specified or Table 2 was computed differently. This is an internal consistency failure, independent of whether Eq. (1) extrapolates to z>6.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper estimates the spins, inclination angles, and black-hole masses of 18 quasars at 6 < z < 7.5 by combining the Trakhtenbrot (2014) radiative-efficiency relation (Eq. 1) with the Bardeen ISCO efficiency formula (Eqs. 8-9). The inclination angle is first set to 45 degrees and then varied in 5-degree steps until a physically allowed spin is obtained, and the virial black-hole masses are supposedly recalibrated for the adopted inclination using Eqs. (3)-(6). After excluding two objects post hoc, the authors report that the estimated spins are on average greater than 0.9, and they interpret the spin-redshift and spin-mass correlations as evidence for rapid disk accretion in the early Universe.","tokens_in":1996,"tokens_out":2175,"duration_ms":122003,"significance":"If the derived spins were reliable, this would be a valuable constraint on supermassive black-hole growth at z > 6, a regime where direct spin measurements are extremely difficult. The paper is transparent in presenting its samples, formulas, and full result tables, and the use of the standard Kerr ISCO efficiency mapping is appropriate. However, the central claim rests on an extrapolated empirical relation, a per-object inclination tuning, and, most importantly, an internal inconsistency between the stated mass-recalibration procedure and the tabulated masses. As written, the headline result is not reproducible from the described method.","major_comments":[{"comment":"Table 2 is not consistent with the mass-recalibration described in Section 3. Equations (3)-(6) imply M*_BH(i) = M_lit [f(i)/f(35 deg)]^2 = M_lit (sin 35 deg / sin i)^2. For i = 45 deg this factor is 0.66, i.e. log M* = log M_lit - 0.18. Yet every i = 45 deg entry in Table 2 is about 0.06 dex above the input log M_BH in Table 1; for example QSO J1007+2115 has input log M_BH = 9.18 and tabulated log M* = 9.24, while the stated recalibration gives 9.00. Because epsilon in Eq. (1) is proportional to M* mu^1.5, the tabulated efficiencies are too high by roughly a factor 1.7 if the paper's own recalibration is applied. For QSO J1007+2115, epsilon = 0.174 would become about 0.10, corresponding to a about 0.7 rather than 0.934. The 'average spin > 0.9' claim is therefore not a consequence of the described algorithm; either the recalibration step is mis-specified or Table 2 was computed with a different, undocumented method.","section":"Section 3, Eqs. (3)-(6), and Table 2"},{"comment":"The algorithm starts from i = 45 deg and changes i in 5-deg steps until a physically meaningful spin is obtained. Since Eq. (1) scales as cos^1.5 i, this is effectively a per-object free parameter that guarantees a spin in the allowed range 0.039 < epsilon < 0.324. The claim that the resulting angles are lower estimates is not derived from any independent observational constraint on the inclination. The tuning matters for the objects assigned i = 50, 55, or 60 deg in Table 2, whose efficiencies are lowered relative to what they would be at i = 45 deg. The paper should justify why this procedure does not simply select the largest inclinations that keep the spin high.","section":"Section 3, inclination-angle choice"},{"comment":"Two objects are excluded after examining their derived properties: VHS J0411-0907 is removed because of its unusually high Eddington ratio, and ULAS J1342+0928 is removed after Table 2 because its spin uncertainty is large. These post hoc cuts reduce the sample from 18 to 16 objects, and all reported correlations and the claim of an average spin above 0.9 refer to the reduced sample. The paper should state an a priori criterion for exclusion and show the results with and without these objects, since the correlations in Figs. 10-12 may depend on the cuts.","section":"Sections 2 and 4, sample selection"},{"comment":"The central input Eq. (1) is taken from Trakhtenbrot (2014) without evidence that the relation, calibrated on lower-redshift AGN, remains valid for z > 6 quasars with L_bol ~ 10^47 erg/s and M_BH ~ 10^9 M_sun. The additional bolometric correction of Eq. (2) from Hopkins et al. (2007) introduces further systematic uncertainty. The quoted +/-0.10 dex errors therefore do not capture the dominant uncertainties. A concrete robustness test would be to repeat the calculation with the alternative bolometric corrections cited in Refs. 32, 34-36 and to check whether any of the objects with a > 0.9 becomes a < 0.9.","section":"Equation (1) and Section 3"},{"comment":"Even taking Table 2 at face value, the spin errors are large and asymmetric, and for several objects the lower 1-sigma bound extends below a = 0.9: for example QSO J1007+2115 has a = 0.934^+0.058_-0.182 and DES J0252-0503 has a = 0.964^+0.034_-0.128. Given these uncertainties, the statement that the spin values are 'on average greater than 0.9' is not supported at the 1-sigma level for a majority of the sample. The paper should report the fraction of objects whose lower error bars remain above 0.9.","section":"Table 2, spin uncertainties"}],"minor_comments":[{"comment":"The phrase 'on average greater that 0.9' should read 'greater than 0.9'.","section":"Abstract and Conclusion"},{"comment":"The text around Fig. 10 states that redshifts 6 < z < 7.5 correspond to times from the Big Bang of 9.4 x 10^7 years > t > 7 x 10^7 years. This is off by about an order of magnitude: the corresponding cosmic ages are roughly 0.7-0.9 Gyr, i.e. 7-9 x 10^8 years.","section":"Section 4, discussion of Fig. 10"},{"comment":"The statement that the estimated SMBH mass distribution peaks at a slightly higher mass than the input distribution because 'we took larger average inclination angles' is inconsistent with Eq. (6): for a fixed FWHM, the virial mass scales as f^2 = 1/(4 sin^2 i), so larger i gives a smaller mass, not a larger one. This wording further highlights the discrepancy between the stated method and the values in Table 2.","section":"Section 4, first paragraph"},{"comment":"The reported Pearson correlation coefficients (r = -0.43, -0.63, 0.67) are given without p-values or confidence intervals. With only 16 objects, these correlations would be usefully quantified by a p-value or a bootstrap interval.","section":"Figures 10-12"},{"comment":"The inclination angles are assigned without uncertainties, as the paper acknowledges. However, since the spin depends strongly on i through Eq. (1), a sensitivity table showing how a and epsilon change when i is varied by +/-5 deg for representative objects would be helpful.","section":"Table 2"}],"recommendation":"reject","confidential_remarks":"The decisive issue is the internal inconsistency between the stated mass-recalibration and Table 2: correcting the tabulated masses according to the paper's own Eqs. (3)-(6) reduces several efficiencies from the epsilon ~ 0.2-0.3 range to epsilon ~ 0.1, which moves the corresponding spins below 0.9. The main conclusion is therefore not reproducible from the described method, and a proper revision would require recomputing all derived quantities and likely changing the paper's central claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline result here — z~6–7.5 quasars having spins mostly above 0.9 — does not survive contact with the paper's own method. In §3 they say that for each inclination they recompute the black hole mass using f≈1/(2 sin i), with the literature mass assumed to have been obtained at i≈35° (f=√3/2). That gives M*(i) = M_lit × (sin35°/sin i)². At i=45° that is a factor 0.66 (log −0.18), yet Table 2 lists log M* slightly above the input log M for every i=45° object. I checked one object: QSO J1007+2115, input log M=9.18, Table 2 gives 9.24; the stated recalibration would give 9.00. If you plug the correctly recalibrated mass into Eq (1), the radiative efficiency drops from 0.174 to about 0.10, and the spin falls below 0.9. So the central claim is an artifact of a table that does not follow the described algorithm.\n\nThat is the load-bearing flaw. There are other soft spots, minor by comparison: the Trakhtenbrot (2014) efficiency relation is extrapolated to higher redshift and higher luminosity than its calibration sample; the inclination is tuned per object in 5° steps until the spin becomes \"physically meaningful,\" which invites selection bias; and two objects are excluded post hoc (VHS J0411 for high Eddington ratio, ULAS J1342 for large spin error). The authors do acknowledge the inclination choice and the selection bias toward ultraluminous objects, which is honest.\n\nWhat the paper does well: it is short, the calculations are transparent enough to check (which is how the inconsistency was found), the sample is real and well-sourced from Zappacosta et al. (2023), and the statistical correlations are reported with their uncertainties. The idea of applying efficiency-based spin estimates to the first quasars is worth exploring; the execution just has a critical misstep.\n\nFor a reader: this is not a paper to cite for spin values. It could serve as a cautionary example in a reading group about how easy it is for a small calibration step to drive the entire result. I would not accept the conclusions as they stand, but the underlying question — whether the earliest SMBHs grew by coherent disk accretion — remains interesting, and the authors could fix the mass-recalibration and reissue the table. My recommendation: send it to a referee who will check Table 2 against §3; the authors need to show the correct M*(i) and redo the spins from there.","headline":"The paper's own mass-recalibration formula contradicts Table 2, so the headline spin>0.9 result is not supported by the described method.","tokens_in":12311,"tokens_out":7758,"would_cite":false,"duration_ms":63861,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["97.10.Gz","98.54.Aj"],"model":"deepseek-v4-flash","headline":"Ultraluminous quasars at redshift 6–7.5 harbor supermassive black holes spinning faster than 0.9.","keywords":["supermassive black holes","quasar spins","ultraluminous quasars","high-redshift quasars","radiative efficiency","accretion disks","black hole growth","epoch of reionization"],"falsifier":"For a few of these quasars, estimate the accretion rate directly from a method independent of the efficiency formula, such as fits to the optical/UV continuum or far-infrared re-emission; if the resulting radiative efficiency $\\varepsilon=L_{\\rm bol}/(\\dot{M}c^2)$ is systematically below about 0.15 for objects with reported spins above 0.9, the central claim is falsified.","tokens_in":11286,"feed_emoji":"🕳️","tokens_out":6554,"duration_ms":57020,"temperature":0.7,"pith_summary":"This paper estimates the spins of 17 of the most distant known ultraluminous quasars, at redshifts 6–7.5, by converting each quasar's radiative efficiency into a black hole spin. The central result is that the inferred spins are on average greater than 0.9, with a distribution shaped like those of nearer active galaxies. The authors further find that spin increases with cosmic time over the short redshift window, that black hole mass grows while spin grows, and they interpret these correlations as evidence that the earliest supermassive black holes grew mainly by disk accretion at high accretion rate. If right, the work says the same spin-building mechanism seen in local active galaxies was already operating less than a billion years after the Big Bang.","feed_headline":"Distant quasars' black holes spin faster than 0.9","feed_subtitle":"Inference from accretion-disk efficiency says these early black holes grew by fast disk accretion.","key_machinery":"The operative object is the radiative efficiency $\\varepsilon(a)$ of a thin accretion disk around a spinning black hole, obtained two ways: from Eq. (1), a relation connecting $\\varepsilon$ to bolometric luminosity, optical luminosity, black hole mass, and the inclination angle through $\\mu=\\cos i$; and from the relativistic thin-disk expression for $\\varepsilon$ in terms of the innermost stable circular orbit radius $R_{\\rm ISCO}(a)$, together with the formula for $R_{\\rm ISCO}$ itself. Setting the two equal lets each quasar's spin $a$ be solved numerically. The inclination angle enters both through the efficiency relation and through the virial mass estimate, and the paper handles the unknown angle by starting at $45^\\circ$ and stepping by $5^\\circ$ until a physically allowed spin emerges; masses are re-derived for self-consistency. That inversion is what converts observed luminosities and masses into spin values.","core_discovery":"The paper claims that the supermassive black holes powering ultraluminous quasars at $60.9 spins would not survive.","Editorial inference: the inclination-angle stepping procedure effectively chooses the lowest inclination that yields a valid spin, so the quoted spin values should be read as lower estimates on the spin for the assumed relation; a different angle-picking rule would change the distribution's shape.","Editorial inference: if high spins in the first billion years are confirmed, direct-collapse seed scenarios with sustained coherent accretion become more plausible than models where mergers or chaotic accretion dominate early growth.","Editorial inference: extending the same method to the larger samples of $z>6$ quasars now being found should sharpen the spin–redshift slope from its current uncertainty, which is larger than the claimed slope."],"forward_implications":["If these estimates hold, the most distant quasars known are powered by near-maximally rotating black holes, with radiative efficiencies $\\varepsilon \\gtrsim 0.15$ for most objects.","The spin–redshift and mass–redshift correlations imply that, over the roughly $2.4\\times 10^7$ years spanned by the sample, black hole mass and spin grew together, consistent with prolonged prograde disk accretion rather than chaotic or merger-dominated spin evolution.","The similarity of the spin distribution to that of lower-redshift quasars and Seyfert galaxies suggests the spin-building mechanism was already in place within the first billion years after the Big Bang.","Because the sample is ultraluminous, selection bias toward high radiative efficiency naturally favors high spin; the paper acknowledges that its average may overstate the typical $z>6$ black hole spin.","The estimated masses shift upward when the inclination-dependent virial factor is used, moving the mass distribution peak to slightly higher values than the literature values."],"supporting_citations":[{"why":"Supplies the sample of ultraluminous quasars with redshifts, bolometric luminosities, Eddington ratios, and black hole masses.","marker":"Ref. 17"},{"why":"Provides Eq. (1), the radiative-efficiency relation that converts luminosity and mass into efficiency and hence spin.","marker":"Ref. 29"},{"why":"Gives the relativistic thin-disk efficiency formula (Eqs. 8–9) linking radiative efficiency to ISCO radius and spin.","marker":"Ref. 21"},{"why":"Sets the spin and efficiency bounds ($a \\leq 0.998$, $0.039<\\varepsilon<0.324$) used to judge physical results.","marker":"Ref. 25"},{"why":"Provides the virial mass formula and the geometric factor $f$ used to recompute masses from inclination.","marker":"Ref. 37"},{"why":"Supplies the broad-line-region radius–luminosity relation used in virial mass estimates.","marker":"Ref. 38"},{"why":"Gives the bolometric correction from $L_{\\rm bol}$ to $L_{\\rm opt}$ used in Eq. (1).","marker":"Ref. 33"},{"why":"Provides the $L_{5100}=L_{\\rm bol}/10.3$ conversion used for mass estimation.","marker":"Ref. 32"}],"fun_headline_variants":["Quasars in early universe host black holes spinning at 0.9+","Black holes in oldest quasars spin near their maximum rate","Early ultraluminous quasars' black holes spin above 0.9","Fast spin in distant quasars hints at disk-fed black hole growth"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Every spin value rests on the radiative-efficiency formula of Ref. 29 holding for these extreme high-redshift ultraluminous quasars, even though it was not calibrated in that regime.","fun_headline_variants_meta":{"raw":{"variants":["Quasars in early universe host black holes spinning at 0.9+","Black holes in oldest quasars spin near their maximum rate","Early ultraluminous quasars' black holes spin above 0.9","Fast spin in distant quasars hints at disk-fed black hole growth"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000798,"raw_usage":{"total_tokens":3463,"prompt_tokens":850,"completion_tokens":2613,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":466,"completion_tokens_details":{"reasoning_tokens":2535}},"tokens_in":466,"tokens_out":2613,"duration_ms":18425,"temperature":1.0,"reasoning_tokens":2535,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:57:10.721250+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a few of these quasars, estimate the accretion rate directly from a method independent of the efficiency formula, such as fits to the optical/UV continuum or far-infrared re-emission; if the resulting radiative efficiency $\\varepsilon=L_{\\rm bol}/(\\dot{M}c^2)$ is systematically below about 0.15 for objects with reported spins above 0.9, the central claim is falsified.","supporting_citations":[],"review_version":1}