{"id":"a59fb042-c370-4b74-9521-a9ef3433cfe7","arxiv_id":"2506.20386","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"G3425's nearly circular orbit points to a 4 to 4.4 solar-mass black hole from a failed supernova rather than a neutron star from a standard explosion.","lead":"A newly discovered binary system with an invisible heavy companion has a nearly circular orbit, which is hard to explain if a normal supernova ejected mass from the system. The authors analyze how orbits change during supernovae and conclude the invisible object is most likely a low-mass black hole created by a failed explosion, with mass between 4 and 4.4 times the Sun.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The failed-SN preference rests on an unstated rate prior and a zero-kick assumption; the apocenter fine-tuning calculation is only a likelihood factor, so the 'most likely' ranking is not yet established.","rationale":"The reader's weakest assumption is right on target: the preference for a failed SN is not established by the dynamical calculation alone. The strength of this paper is the clean orbital mechanics: the analytical e1 expression (Eq. 2) is consistent with the conservation derivation in Appendix A, the numerical homologous-expansion models agree with the analytical treatment for high expansion velocities, and the tidal-damping check is reasonable. The argument that SN II progenitors would engulf the secondary at the required e0 is also physically plausible. What is missing is the model-comparison layer. Table 1 gives only the geometric probability of being near apocenter; comparing scenarios requires multiplying by formation-rate priors, and the failed-SN channel requires marginalizing over BH kicks. The paper's own caveats mention asymmetric ejecta and tidal effects, but not these Bayesian inputs. I also note Eq. (1)/A6 appears to contain a typo — direct algebra from (A3)–(A5) gives the factor (1+e0 cosν0), not e0(1+cosν0) — but since the failed-SN ranking and the e1 calculation (Eq. 2) do not depend on a1, this is not the load-bearing issue. The central claim should therefore be presented as a scenario preference rather than a near-certainty, consistent with a CONDITIONAL verdict; no change to the reader's verdict is needed.","tokens_in":18678,"tokens_out":14462,"duration_ms":145602,"concrete_test":"Perform a posterior comparison with explicit priors: take an SN Ib/c-to-failed-SN rate ratio from population synthesis (e.g., Sukhbold et al. 2016; Shariat et al. 2025), and a BH natal-kick distribution (e.g., exponential with σ≈20 km/s, or the empirically motivated distribution in Vigna-Gómez et al. 2024). Sum over orbital phase and kick realizations to compute P(channel | e=0.05±0.01, a=current) for SN Ib/c and failed SN. If the prior odds against failed SNe exceed the ≲10 likelihood ratio for SN Ib/c, or if a few km/s kicks frequently move the failed-SN orbit outside the observed eccentricity, the 'most likely' claim is not robust; otherwise it becomes quantitatively supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central result is a scenario ranking: failed SN > SN Ib/c, with SN II ruled out. This ranking is made in §3 on the basis that SN Ib/c requires the explosion to occur within ~5°–15° of apocenter (probability ≲10%, Appendix C), while a failed SN is unconstrained in orbital phase. But Appendix C computes P(ν0 in 180°±Δν) for a bound orbit: that is a geometric likelihood factor, not a posterior odds. No prior over the intrinsic rates of failed SNe versus stripped-envelope SNe is stated, and no natal-kick model for BHs is included. The cited no-kick BH systems (e.g., Vigna-Gómez et al. 2024) constrain individual objects, not the distribution of kicks, and a kick of even a few km/s at a random phase would perturb e and a enough to require additional fine-tuning in the failed-SN channel. Without a rate ratio and a kick distribution, the conclusion that a failed SN is 'highly probable' (§3) is an argument from absence of constraints, not a quantitative posterior.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the formation of the wide binary G3425 (P ≈ 877 d, e = 0.05 ± 0.01), whose unseen component has a mass of 2.9–4.4 M☉, by modeling the orbital response to the supernova that formed it. Using an instantaneous mass-loss solution of the two-body problem (Appendix A) and a homologous-expansion numerical model (Appendix B), the authors show that any mass-losing SN channel can reproduce the near-circular remnant orbit only if the explosion occurred near apocenter. They argue that SN II channels are excluded because the required initial eccentricity (e0 ≳ 0.4) puts the secondary inside the 500–1500 R☉ progenitor at pericenter; that SN Ib/c is possible but requires a fine-tuned near-apocenter explosion, with a geometric probability ≲ 10% (Appendix C); and that a failed SN with no mass ejection requires no such fine-tuning. They conclude that the failed-SN channel is the most likely origin and that the unseen component is a mass-gap black hole of 4–4.4 M☉.","tokens_in":18908,"tokens_out":36211,"duration_ms":334215,"significance":"The dynamical core of the paper is worth publishing once corrected. The derivation in Appendix A is explicit and checkable; the eccentricity equation (2) and the circularization condition (3) are correct, and the numerical model is validated against the analytical solution at high ejecta velocity (Fig. 2). The tidal-circularization timescale estimate (Eq. 6, τe ≳ 26 Gyr) appropriately rules out post-explosion circularization, and the stated caveats (§2.3) show due care. The paper makes a falsifiable prediction — a low-mass BH in the 3.5–4.4 M☉ range on a wide, nearly circular orbit — and the method is genuinely applicable to other wide binaries with unseen companions. The circularity concern raised in review does not land: the observed eccentricity is an external input, not a fitted quantity. However, the scenario ranking in §3 rests on an implicit uniform prior over channel rates and an unmodeled kick distribution, and the printed semi-major-axis formula contains a load-bearing error (below); both require correction.","major_comments":[{"comment":"Equating the pre- and post-explosion specific energies (Appendix A) yields a1/a0 = (μ1/μ0)/[1 + 2(μ1/μ0 − 1)(1 + e0 cos ν0)/(1 − e0²)]. The printed Eqs. (1)/(A6) instead contain 2e0(μ1/μ0 − 1)(1 + cos ν0)/(1 − e0²). The printed form is algebraically inconsistent with the rest of the paper: at apocenter it gives a1/a0 = μ1/μ0 < 1 for any mass loss (orbital shrinkage), whereas the circularization condition e1 = 0 of Eq. (3) requires a1 = r_apo = a0(1 + e0); the two equations can only agree if μ1 = μ0. The error is load-bearing for the SN II exclusion argued in §2.2 and §3: with the printed formula the required progenitor pericenters are ~3–5 AU (≈ 640–1100 R☉), outside the 500 R☉ SN II progenitor, so SN II would not be excluded; with the correct formula the pericenters are ~0.25–0.95 AU (≈ 54–200 R☉), inside the progenitor, and the exclusion stands. Because the dashed curves in Figure 2 are described as the analytical solution, the authors should verify which formula generated them, correct Eqs. (1)/(A6), and rerun the analytical–numerical comparison.","section":"Eq. (1) and Eq. (A6), §2.1"},{"comment":"The statement that the failed-SN channel is 'the most likely scenario' and that the system 'highly probable[ly]' contains a mass-gap BH is a likelihood ranking, not a posterior. Appendix C computes a geometric factor P(ν0 ∈ 180° ± Δν | e0), and §3 multiplies this by nothing: the conclusion implicitly assumes a uniform prior over the intrinsic rates of failed SNe versus SNe Ib/c in the relevant mass range and assumes zero BH natal kick. The paper cites several low-kick BH systems (Vigna-Gómez et al. 2024, etc.), but those constrain individual objects, not the kick distribution; for this binary, the orbital speed is ~35–40 km s−1, so a kick of even a few km s−1 changes the eccentricity by Δe ~ 0.1–0.3, comparable to or larger than the observed 0.05, and would reintroduce a phase constraint into the failed-SN channel. A failed SN also loses some mass (neutrinos, residual ejecta), which the 'no mass loss' scenario ignores. To support the headline ranking, the authors should either combine published rate estimates (e.g., Adams et al. 2017; Gerke et al. 2015) with a kick sensitivity analysis, or state the conclusion as conditional on these assumptions.","section":"§3 and Appendix C"},{"comment":"The headline mass claim is internally inconsistent. The abstract states the BH mass is 'constrained between the theoretical minimum for failed supernova progenitors (4 M☉) and the observed upper limit (4.4 M☉)', while §2 defines the failed-SN scenario with M1 = M'1 = [3.48, 4.4] M☉ and §1 quotes 3.48 M☉ as the rotation-extended NS mass limit. The origin of the 4 M☉ lower bound is not derived or cited in the body. Either the failed-SN scenario allows remnant masses down to 3.48 M☉, in which case the abstract's interval [4, 4.4] is too narrow, or a specific theoretical minimum failed-SN progenitor mass must be introduced and justified.","section":"Abstract and §3"}],"minor_comments":[{"comment":"The text says the probability of being within 180°±5°, ±10°, and ±15° of apocenter is 'equally ≲10%', but the matching-solution entries in Table 1 are 10% (Δν=5°, e0=0.65), 12% (Δν=10°, e0=0.4), and 4% (Δν=15°, e0=0.2); these are not equal and the 12% entry exceeds the quoted cap.","section":"Table 1 and §3"},{"comment":"The figure captions contain garbled fragments (e.g., 'M1=8, M1=20, M1=20, M1=30, M1=5, M1=10, M1=5, M1=10, → → 1' and '6, 0000 and 30, 0000 km s−1'); these need to be cleaned before publication.","section":"Figure captions 1, 2, and 5"},{"comment":"The sentence 'This is highly unlikely for SN II progenitors but guaranteed for SN Ib/c progenitors, which range in size from 500−1500 R☉ and 1−10 R☉, respectively' is confusingly phrased, since the two size ranges are attached to the scenarios in the order listed but the clause reads as if they were attached to the preceding adjectives.","section":"§2.1"},{"comment":"The sentence beginning 'However, at lower ejecta velocity the final eccentricity is larger...' has an unmatched parenthesis and a garbled clause ('vmax = 6, 000 km s−1, square symbols), see panels'); rephrase.","section":"§2.2, fourth paragraph"},{"comment":"The caption does not identify the panels; since §3 refers to 'panel (b)' and 'panel (c)', the caption should explicitly map panels to progenitor masses or Δν values.","section":"Figure 3 caption"}],"recommendation":"major_revision","confidential_remarks":"The reader's and skeptic's prior concern is, in my reading, correct and needs a quantitative answer. Independently, I found a more technical problem: the printed Eqs. (1)/(A6) are inconsistent with Eq. (3) and with energy conservation, and the printed formula would invalidate the SN II exclusion in §2.2 that the authors themselves rely on. I view both as repairable in a revision — the corrected formula actually strengthens the SN II exclusion — and the paper is otherwise a reasonable ApJL contribution. Please send back for major revision rather than reject."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper does a solid job with the dynamics of G3425 and gets a real result: the near-circular orbit is hard to explain with a Type II supernova, because the required pre-SN eccentricity would put the secondary inside the red supergiant. The Type Ib/c channel requires the explosion to happen within a few degrees of apocenter, about a 10% geometric chance. The analytical derivation in Appendix A is clean and the numerical homologous-expansion model matches it at high ejecta velocity. That part deserves credit.\n\nThe soft spot is the jump from 'fine-tuned' to 'most likely failed SN.' The 10% number is a likelihood factor for orbital phase, not a posterior probability. To rank failed SNe above SNe Ib/c you need a prior on how often stars in the relevant mass range produce failed explosions versus stripped-envelope ones, and you need a natal-kick model for black holes. The paper cites evidence that some BHs get small kicks, but that is not the same as showing this one got none. A kick of even a few km/s at a random phase would perturb the orbit enough to require its own fine-tuning. The caveat section covers tides and envelope asymmetry, but not kicks, which seems like an omission.\n\nThe mass-gap claim itself also rests on a model-dependent lower bound: the 4 Msun minimum for failed-SN progenitors is theoretically motivated, not observed. The upper limit 4.4 comes from the observed mass range, so the 4–4.4 interval mixes theory and data. It would be cleaner to say the unseen companion is a low-mass-gap BH candidate and note that the lower bound depends on the assumed minimum failed-SN mass.\n\nOverall, the analysis is honest and the math is sound; the conclusion just needs to be framed as conditional. This is a useful paper for anyone working on low-mass-gap BHs or binary evolution. It deserves a serious referee, but the strongest claims should be toned down. I would accept it for review and ask for a discussion of formation-rate priors and natal kicks.","headline":"A clean dynamical case against a normal SN for G3425, but the 'most likely failed SN' conclusion skips the prior on rates and the kick distribution.","tokens_in":19447,"tokens_out":3574,"would_cite":true,"duration_ms":39485,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The nearly circular orbit of G3425 points to a failed supernova and a 4 to 4.4 solar-mass black hole.","keywords":["G3425","mass-gap black hole","failed supernova","variable-mass two-body problem","binary eccentricity","stripped-envelope supernova","supernova natal kick","wide binary evolution"],"falsifier":"Measure the unseen component's mass directly by combining radial velocities and astrometry; if it comes out below about 4 solar masses or above 4.4, or if the system's space velocity shows a natal kick of more than a few km/s relative to the local stellar population, the failed-supernova no-kick explanation is ruled out. A statistically robust census showing that failed supernovae make up only a few percent of core collapses would also overturn the likelihood ranking.","tokens_in":18472,"feed_emoji":"🕳️","tokens_out":11098,"duration_ms":97113,"temperature":0.7,"pith_summary":"The paper asks how the wide binary G3425—a red giant with a massive unseen companion on a nearly circular orbit—came to be, since a supernova that removes most of the primary's mass should normally leave the companion on a wider, more eccentric orbit or unbind it entirely. Applying the variable-mass two-body problem, it finds that ordinary type II supernovae cannot produce the observed eccentricity of $0.05 \\pm 0.01$, and that a stripped-envelope SN Ib/c can do so only if the explosion happened within a few degrees of the companion's apocenter, a fine-tuning with probability at most about 10 percent. A failed supernova, which ejects essentially no mass and leaves a black hole, requires no such fine-tuning because the binary simply inherits its pre-explosion orbit. The paper therefore concludes that the unseen component is most probably a mass-gap black hole with mass between about 4 and 4.4 solar masses, and that the same reasoning can be applied to any wide, nearly circular binary with a dark companion.","feed_headline":"G3425's unseen partner is likely a failed-supernova black hole","feed_subtitle":"A nearly circular orbit rules out ordinary supernovae and points to a quiet collapse into the mass gap.","key_machinery":"The carrying object is the analytical solution of the instantaneous variable-mass two-body problem: closed-form expressions for the post-explosion semi-major axis $a_1$ and eccentricity $e_1$ in terms of the initial total mass $\\mu_0$, the post-explosion total mass $\\mu_1$, the initial eccentricity $e_0$, and the true anomaly $\\nu_0$ (the companion's angular position along its ellipse). The critical identity is the apocenter circularization condition, $e_0=(\\mu_0-\\mu_1)/\\mu_0$, which converts a measured near-zero eccentricity into a constraint on how much mass the primary lost and where the companion was when it exploded. A numerical homologous-envelope expansion model, with ejecta speeds of $6{,}000$-$30{,}000$ km s$^{-1}$, checks the instantaneous approximation and also tracks the growth of the semi-major axis and possible mass loss from the secondary. Together these tools map each supernova type onto an allowed region of progenitor mass and orbital phase, which is what separates the fine-tuned SN Ib/c channel from the unconstrained failed-SN channel.","core_discovery":"On the paper's own terms, the central discovery is that the nearly circular orbit of G3425 selects a failed supernova as the formation channel. For instantaneous mass loss from the primary, the post-explosion eccentricity $e_1$ is a function of the initial eccentricity $e_0$, the pre- and post-explosion total masses $\\mu_0$ and $\\mu_1$, and the companion's true anomaly $\\nu_0$ at the moment of explosion. Circularization ($e_1 = 0$) at apocenter requires $e_0 = (\\mu_0-\\mu_1)/\\mu_0$. Type II progenitors (8-30 $M_\\odot$) would need high initial eccentricities whose pericenters fall inside the $\\sim 500\\,R_\\odot$ progenitor envelope, so the companion would be engulfed; SN Ib/c progenitors (5-10 $M_\\odot$) match the observed $e = 0.05\\pm0.01$ only when $\\nu_0 \\approx 180^\\circ \\pm 5^\\circ$ to $\\pm15^\\circ$, which carries at most a $\\sim 10\\%$ probability. Failed supernovae, by contrast, eject no mass, so the remnant inherits the pre-explosion orbit unchanged; the only requirements are a progenitor mass in the range $3.48$-$4.4\\,M_\\odot$ and an initial eccentricity already equal to the observed one. This is why the paper proposes a mass-gap black hole of $4$-$4.4\\,M_\\odot$ as the most probable identity of the unseen component.","pith_inferences":["If failed supernovae turn out to be intrinsically rare, the likelihood ranking could invert: a roughly 10 percent apocenter coincidence in an SN Ib/c might be more probable than a rare failed event, so the mass-gap conclusion should be re-tested with measured failed-SN statistics.","A natal kick of even a few km/s would break the no-mass-loss inheritance; a precise space-velocity measurement of G3425 relative to its local stellar population could therefore rule the failed-SN scenario in or out.","A systematic search for wide, nearly circular binaries with dark companions could test the paper's prediction that most such systems host low-mass black holes rather than neutron stars."],"forward_implications":["If the identification holds, G3425 becomes a directly measured example of a roughly 4 solar-mass black hole produced by a failed supernova, anchoring the low-mass end of black-hole birth masses.","Because the apocenter timing needed for an SN Ib/c is a roughly 10 percent geometric fine-tune while a failed SN needs none, wide nearly circular binaries with unseen companions can be used to estimate how often core collapses fail.","Type II supernova progenitors are excluded as formation channels for such systems even without observing the explosion, since their required initial eccentricities would place the companion inside the progenitor envelope.","The same analytic condition, $e_0=(\\mu_0-\\mu_1)/\\mu_0$, can be applied to any other wide, nearly circular post-supernova binary to recover its pre-explosion eccentricity and progenitor mass."],"supporting_citations":[{"why":"discovered G3425 and measured the unseen component's mass range, the red giant's mass, the 877-day period, and eccentricity 0.05.","marker":"Wang et al. (2024)"},{"why":"showed that extreme mass loss in a binary can drive eccentricity above unity and unbind the secondary, founding the variable-mass treatment.","marker":"Hadjidemetriou (1966b)"},{"why":"provided analytical and numerical solutions for supernova-like mass-loss regimes that this paper extends to the G3425 parameters.","marker":"Veras et al. (2011)"},{"why":"introduced the homologous envelope expansion model for post-supernova binaries used here as the numerical check.","marker":"Regály et al. (2022)"},{"why":"extended the homologous expansion binary model and is used for the numerical eccentricity calculations.","marker":"Fröhlich et al. (2023)"},{"why":"supplies the theoretical basis for black-hole formation in failed supernovae via fallback or direct collapse.","marker":"O'Connor & Ott (2011)"},{"why":"maps supernova outcomes to progenitor mass, supporting the adopted SN II and SN Ib/c progenitor ranges.","marker":"Sukhbold et al. (2016)"},{"why":"provides observational evidence that some black holes receive little or no natal kick, supporting the failed-SN assumption.","marker":"Shenar et al. (2022)"},{"why":"standard derivation of the probability density over true anomaly used to compute the at-most-10 percent apocenter fine-tuning probability.","marker":"Murray & Dermott (1999)"}],"fun_headline_variants":["Failed supernova best explains G3425's circular orbit","G3425's unseen partner: a quiet collapse, not a supernova","Quiet black hole birth: G3425's odd orbit explained","G3425's circular orbit points to a failed-supernova black hole"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The scenario ranking assumes, without an explicit prior, that failed supernovae are not intrinsically much rarer than normal core-collapse supernovae and that the black hole received no natal kick; if either assumption fails, the fine-tuned stripped-envelope supernova channel becomes competitively probable.","fun_headline_variants_meta":{"raw":{"variants":["Failed supernova best explains G3425's circular orbit","G3425's unseen partner: a quiet collapse, not a supernova","Quiet black hole birth: G3425's odd orbit explained","G3425's circular orbit points to a failed-supernova black hole"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001008,"raw_usage":{"total_tokens":4352,"prompt_tokens":1125,"completion_tokens":3227,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":741,"completion_tokens_details":{"reasoning_tokens":3150}},"tokens_in":741,"tokens_out":3227,"duration_ms":23930,"temperature":1.0,"reasoning_tokens":3150,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:49:17.866398+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the unseen component's mass directly by combining radial velocities and astrometry; if it comes out below about 4 solar masses or above 4.4, or if the system's space velocity shows a natal kick of more than a few km/s relative to the local stellar population, the failed-supernova no-kick explanation is ruled out. A statistically robust census showing that failed supernovae make up only a few percent of core collapses would also overturn the likelihood ranking.","supporting_citations":[],"review_version":1}