{"id":"183afa5c-5d7d-42b7-9499-ab6812a5cb3a","arxiv_id":"2411.16012","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Kalkayotl 2.0 provides a flexible Bayesian code for inferring the 6D phase-space parameters of nearby stellar systems, finding a 2-sigma rotation in the Hyades and showing that Praesepe's reported rotation comes from its outer parts.","lead":"This paper introduces Kalkayotl 2.0, an open-source computer code that reconstructs the 3D positions and 3D velocities of stars in nearby clusters and associations using Gaia satellite data. It can measure whether a stellar group is expanding, contracting, or rotating, and the authors use it to remeasure the Hyades, Praesepe, and the beta Pictoris moving group.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The linear velocity field is never validated at T=0, so the HDI-based rotation/expansion detection criteria are uncalibrated; the Hyades 2σ signal sits in the poorly constrained weak-signal regime.","rationale":"The reader's conditional verdict is well aligned with the evidence: the code is open source, the synthetic validation is broad, and the real-data comparisons with Oh & Evans, Lodieu et al., and the β Pic literature mostly agree. I would not move to reject. However, I locate the single most load-bearing weakness differently. The paper's detection rules are stated as objective but are never calibrated under the null: all linear-velocity simulations use C=10, 50, 100, and no T=0 case appears. The importance of this gap is amplified because the headline Hyades rotation (32±11 m s−1 pc−1) is in the regime the authors themselves caution about (≲50 m s−1 pc−1), where relative errors of 50–200% are reported. The 100% credibility values for the tensor entries are a consequence of very wide HDIs, not of accurate point estimates. The sky-uncertainty inflation is a real but secondary contributor: it is applied everywhere and never varied, so it cannot be separated from weak-signal behaviour. A null-signal simulation is the decisive, cheap check; if it shows false-positive rates near the nominal 5%, the detections gain support, and if not, the specific astrophysical claims should be downgraded. The conditional verdict stands, with the condition expanded to include this calibration test.","tokens_in":31187,"tokens_out":11840,"duration_ms":114749,"concrete_test":"Run the Gaussian linear velocity model on the Sect. 3 synthetic grid with T=0 (C=0), using the same distances, n_stars, seeds, and Gaia DR3 uncertainties, and apply the paper's exact detection rule: expansion if the 95% HDI of |κ| excludes zero, rotation if the 95% HDI of any ω component excludes zero. Count the false-positive fraction across the grid. If it exceeds the nominal 5% level (or is inconsistent with the stated 2σ criterion), the Hyades rotation and the Praesepe null conclusion are not calibrated detections. As a secondary check, rerun the Hyades Gaia DR3 analysis with sky-scaling factors 10^3 through 10^7 to confirm the 32±11 m s−1 pc−1 signal is stable under the Sect. 2.6 heuristic.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3 validates the linear velocity field only for tensors T = C·M with C ∈ {10, 50, 100} (Eq. 4); no C=0 null-signal simulation is reported. The detection rules in Sect. 2.3.2 — expand/contract if the α-level HDI of |κ| excludes zero, rotate if the α-level HDI of any ω component excludes zero — are therefore never checked for false-positive rate. This is load-bearing because the headline results are weak signals in the poorly calibrated regime: the Hyades rotation is 32±11 m s−1 pc−1, and Sect. 3.1.2 reports relative errors of ~50% for C=50 and ~200% for C=10, with an explicit caution for tensor entries ≲50 m s−1 pc−1. The 100% credibility values for tensor entries are driven by very wide HDIs, not by accuracy of the point estimates. The sky-uncertainty inflation heuristic (Sect. 2.6, default 10^6, and 10^7 for β Pic linear runs) is applied in every real-data run and is never varied in the validation, so it could interact with the weak-signal regime. Without null-signal calibration, the claimed 2σ Hyades rotation and the Praesepe 'periphery-only' conclusion are not protected against systematic false positives.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents Kalkayotl 2.0, a Bayesian hierarchical modeling code for inferring 3D positions and 6D phase-space parameters of low-number stellar systems from Gaia astrometry and radial velocities. The model includes Gaussian, Student-T, and Gaussian mixture distributions, plus a linear velocity field that yields expansion, contraction, rotation, and shear. The code is validated on synthetic Gaia DR3-like datasets and applied to beta Pictoris, the Hyades, and Praesepe, recovering literature parameters and reporting an expansion age for beta Pic of 19.1 +/- 1.0 Myr, a 2-sigma rotation signal in the Hyades, and evidence that Praesepe's rotation arises from its periphery.","tokens_in":31533,"tokens_out":7172,"duration_ms":61736,"significance":"If the detection criteria are properly calibrated, this is a valuable open-source contribution: it handles missing radial velocities, models angular correlations in Gaia astrometry, and provides posterior predictive checks and decontamination tools. The synthetic validation is extensive and transparent about the poor recovery of the linear velocity tensor at low signal amplitudes. The real-data benchmarks reproduce literature values in most cases. However, the paper's headline detections rely on weak signals whose false-positive rate is not established, and the sky-uncertainty inflation heuristic used in all real-data runs is never subjected to a sensitivity analysis.","major_comments":[{"comment":"The validation of the linear velocity field uses only tensors T = C.M with C in {10, 50, 100}; no null-signal (C = 0) simulation is reported. The detection criteria in Sect. 2.3.2, which declare expansion or contraction if the HDI of |kappa| excludes zero and rotation if the HDI of any omega component excludes zero, are therefore never tested for false-positive rate. This is load-bearing because the headline results, particularly the 2-sigma Hyades rotation (Table 4) and the Praesepe periphery conclusion (Sect. 4.3), are weak signals in the regime where Sect. 3.1.2 reports relative errors of about 50% for C=50 and about 200% for C=10. I request a null-signal validation: simulate clusters with T=0, run the same pipeline, and report the fraction of runs in which the HDI criteria falsely declare expansion or rotation, as a function of distance and number of stars.","section":"Sect. 3, Eq. (4)"},{"comment":"The heuristic of inflating Gaia sky-position uncertainties by factors of 10^5 to 10^6 (and 10^7 for the beta Pic linear runs) is applied to every real-data run, but its effect on the posterior is never tested. The claim that this has negligible impact on the recovered parameter values is not supported by any sensitivity analysis. Because the likelihood is altered for all astrometric features, this could bias the inferred positions, velocity gradients, expansion ages, and rotation detections. I request a sensitivity test on synthetic data in which the scaling factor is varied (for example, 1, 10^2, 10^4, 10^6, and 10^7) and the recovery of the linear tensor and population parameters is compared across these settings.","section":"Sect. 2.6"},{"comment":"The credibility metric, defined as the fraction of simulations in which the true value falls in the 95% HDI, reaches 100% for the tensor entries for all C values, including C=10 where the relative error is up to 200%. This shows that the credibility metric is insensitive in the weak-signal regime because the HDIs are extremely wide. The paper should report the bias and HDI width explicitly for the tensor entries, and it should use null-signal simulations to quantify the false-positive rate of the detection criteria. Without this, the 100% credibility values are potentially misleading as evidence of good performance in the regime relevant to the paper's main detections.","section":"Sect. 3.1.2"}],"minor_comments":[{"comment":"The default hyper-parameter for the Student-T degrees of freedom is given as nu_beta = 10 in Table 1 but as nu_beta = 0.1 in the text; please reconcile these values.","section":"Table 1 and Sect. 2.5"},{"comment":"The parameters w1 through w5 are reported in these tables but are never defined in the text or in Eq. (3); if they are the five components of the traceless symmetric shear tensor, please state this explicitly and give the mapping to the tensor entries.","section":"Tables 4 and 7"},{"comment":"Section 4.1 reports expansion detections at 12 sigma, 3 sigma, and 1 sigma in the X, Y, and Z directions, while the conclusion states that the model detects expansion at the 2 sigma level; please clarify which quantity, such as the combined |kappa|, the 2 sigma statement refers to.","section":"Sect. 4.1 and Sect. 5"},{"comment":"The sentence 'we exemplify the use of Kalakyotl by applying it to beta-Pictoris' contains a typo in the code name; it should read 'Kalkayotl'.","section":"Sect. 1"},{"comment":"The diagonal entries of T are labeled kappa_x, kappa_y, kappa_z in Eq. (3) but are subsequently referred to as kappa_0, kappa_1, and kappa_2; please unify the notation.","section":"Sect. 2.3.2"},{"comment":"The abstract states that the code is intended for systems with up to about 1000 stars, but the synthetic validation grid only goes up to N_stars = 400; please either extend the validation or add a statement about expected performance up to 1000 stars based on the real-data applications.","section":"Abstract and Sect. 3"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid methods contribution and the requested additions, namely a null-signal test for the detection criteria and a sensitivity analysis for the sky-uncertainty inflation, are well within the scope of a revision. I would not reject over these issues, but the detection claims in the abstract and conclusions should be tempered until the false-positive rate is quantified. The missing definitions of w1-w5 in Tables 4 and 7 should also be fixed, as they currently hinder reproducibility."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's what you should know. This is a genuinely useful methods paper: it upgrades Kalkayotl from 1D distance inference to full 3D and 6D phase-space modelling with a linear velocity field, missing radial velocity handling, and corrections for Gaia angular correlations. The code is public, GPU-friendly, and the validation is honest about where it works. For common parameters (location, dispersions) it recovers true values with ~100% credibility up to 1.5 kpc (800 pc for GMM), and the real-data benchmarks on β-Pic, Hyades, and Praesepe reproduce literature values at the 2σ level. That part is solid.\n\nThe combination it offers—joint position-velocity inference with flexible distributions, missing data, and spatial correlations—is not in Lindegren et al. (2000) or Oh & Evans (2020). The extension is incremental but real, and shipping code and data is credit-earning.\n\nThe soft spot is real, and the stress-test note gets it right. Section 3 validates the linear velocity tensor only for C = 10, 50, 100 m s−1 pc−1. There is no null-signal (C=0) simulation, so the HDI-based detection rules in §2.3.2 (exclude zero from the HDI of |κ| or ω) have no measured false-positive rate. The paper itself warns that tensor entries ≲50 m s−1 pc−1 are unreliable, with relative errors ~50% for C=50 and ~200% for C=10, yet the headline Hyades rotation is 32±11 m s−1 pc−1, right in that regime. The 100% credibility values for tensor entries are driven by wide HDIs, not accuracy of point estimates.\n\nThe sky-uncertainty inflation heuristic (§2.6) is a second concern. Increasing sky errors by 10^5–10^6, and 10^7 for some β-Pic runs, is applied in every real-data run and never varied in validation. It may be benign, but it is a load-bearing choice for the exact weak-signal regime where the rotation claims live.\n\nThe Praesepe conclusion (rotation from the periphery) depends on post-hoc decontamination and tidal-radius cuts. The paper is transparent about these choices, and the cleaning does remove the signal, so the narrative is coherent, but a sensitivity analysis would strengthen it.\n\nWho is this for? Anyone doing Bayesian phase-space inference of nearby clusters and associations with Gaia. It is a practical, well-documented tool with validation above the norm for this literature. The Hyades and Praesepe science claims are suggestive, not settled. This deserves a serious referee. I would send it out, with a request for a C=0 calibration and a sensitivity test on the sky-uncertainty scaling before publication.","headline":"A solid, genuinely useful methods paper with real code and honest validation, but the rotation/expansion detection rules are never calibrated at null signal and the Hyades 2σ claim sits in the weak-signal regime the authors themselves caution about.","tokens_in":32030,"tokens_out":3080,"would_cite":true,"duration_ms":27762,"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":"A single Bayesian hierarchical model can infer the 3D positions, velocities, expansion, and rotation of stellar systems with up to about 1000 stars from Gaia data, as demonstrated on beta Pictoris, the Hyades, and Praesepe.","keywords":["Bayesian hierarchical model","Gaia astrometry","phase-space inference","linear velocity field","open clusters","stellar associations","expansion age","rotation"],"falsifier":"Run the same Hyades and beta Pictoris analyses with the default sky-uncertainty inflation and with a much smaller inflation factor, using longer chains so both runs converge; if the inferred rotation component or the expansion age shifts by more than the quoted uncertainties, the inflation assumption is refuted.","tokens_in":30996,"feed_emoji":"🌌","tokens_out":11486,"duration_ms":98952,"temperature":0.7,"pith_summary":"This paper claims that the 3D structure and internal motions of star-forming regions, stellar associations, and open clusters can be inferred jointly from Gaia astrometry and radial velocities with one flexible Bayesian hierarchical model. The model, the multidimensional successor of the 1D Kalkayotl, treats each star's position and velocity as latent variables and lets the whole system be described by Gaussian, Student-t, or Gaussian-mixture populations, optionally with a linear velocity field that separates expansion, contraction, rotation, and velocity dispersion. Validated on synthetic clusters that mimic Gaia DR3, the method recovers population-level and per-star parameters with small errors and high credibility out to about 1.5 kpc. Applied to real benchmark systems, it reproduces literature values and yields an expansion age of $19.1\\pm 1.0$ Myr for $\\beta$ Pictoris, a $2\\sigma$ rotation signal of $32\\pm 11\\ \\mathrm{m\\,s^{-1}\\,pc^{-1}}$ in the Hyades, and the conclusion that Praesepe's reported rotation comes from its periphery. If the method is right, these kinematic diagnostics can be compared directly with star-formation theory predictions.","feed_headline":"Bayesian model finds expansion and rotation in nearby star clusters","feed_subtitle":"Open-source Kalkayotl 2.0 recovers expansion ages and rotation signals from Gaia data for associations up to 1.5 kpc.","key_machinery":"The load-bearing object is a Bayesian hierarchical model with a linear velocity field. The system is described by a population-level location and covariance in 3D or 6D, while each star's position and velocity are latent variables; the velocity-field model sets $v_i = T(x_i - x_0) + D(v_0, \\Sigma_v)$, where $T$ is a $3\\times 3$ gradient tensor whose trace gives the expansion or contraction rate $|\\kappa| = \\tfrac{1}{3}\\mathrm{tr}(T)$ and whose antisymmetric part gives the rotation vector $\\omega$. The likelihood is a single high-dimensional multivariate Gaussian that keeps inter-source spatial correlations among Gaia parallaxes and proper motions, rather than assuming independent observations, and it naturally accommodates missing radial velocities. The prior structure uses flexible distributions on locations, scales, correlations, mixture weights, and degrees of freedom, together with a generalised-Gamma distance prior. The argument succeeds because this architecture lets expansion, rotation, and dispersion be separated from perspective effects and tested against zero with explicit posterior-interval criteria.","core_discovery":"The paper's central discovery is that a single hierarchical forward model can replace the usual two-step translation of Gaia measurements into physical space. In this model, individual stars' 3D positions and 3D velocities are source-level parameters drawn from a population-level distribution, and the likelihood is evaluated in the observed space by transforming proposed phase-space coordinates into astrometry and radial velocities. The new ingredient relative to earlier versions is a general linear velocity field: each star's velocity is the system's bulk velocity plus a 3x3 gradient tensor acting on the offset from the system centre, so expansion or contraction and rotation appear as distinct tensor components with objective posterior-based detectability criteria. The authors show, on realistic Gaia DR3-like simulations, that population parameters are recovered with relative errors typically below 10 to 20 percent and median credibility near 100 percent out to 1.5 kpc for the Gaussian, Student-t, and linear-field families, and out to 800 pc for Gaussian mixtures. On real data the model reproduces literature phase-space parameters for beta Pictoris, the Hyades, and Praesepe, and produces the new kinematic results listed in the abstract.","pith_inferences":["Editorial inference: the heuristic sky-uncertainty inflation implies that individual stellar positions near the cluster centre are effectively down-weighted relative to the population-level prior, so the recovered source-level coordinates should be read as shrinkage estimates rather than as independent measurements.","Editorial inference: the 2-sigma Hyades rotation appears in different components in different reference frames, namely $\\omega_z$ in ICRS and $\\omega_x$ in Galactic coordinates, so a rotation with a fixed spatial orientation should have been consistent across frames; this suggests the signal is marginal until confirmed with higher-precision radial velocities.","Editorial inference: the same linear velocity tensor could be used to quantify differential rotation or shear in young associations and to place empirical upper limits on dissolution rates of sparse groups, at the cost of assuming the velocity field is truly linear over the system's size.","Editorial inference: a direct way to stress-test the method beyond the paper's simulations is to feed it a synthetic cluster with a known rotation of $30\\ \\mathrm{m\\,s^{-1}\\,pc^{-1}}$ at 50 pc and check that the recovered posterior interval contains the input; the paper's own Figure 3 suggests such a detection would be only marginal, so real claims at that level should be treated cautiously."],"forward_implications":["Kinematic diagnostics that are currently spread across different methods, namely expansion rate, contraction, rotation vector, velocity dispersion, and bulk motion, become outputs of one reproducible inference with stated posterior-interval detection criteria.","The method handles partial radial-velocity coverage, so the full astrometric sample is used instead of cropping to stars with radial velocities, avoiding bias toward bright members.","Users can fix any population parameter to a literature value and infer only the rest, making the code suitable for membership classification or single-star updates without rerunning the full system.","The validation limits give a practical applicability envelope: Gaussian, Student-t, and linear-field families work out to about 1.5 kpc and Gaussian mixtures out to about 800 pc for systems with up to about 1000 stars.","For real systems, the model yields outputs directly comparable to star-formation theory, such as the beta Pictoris expansion age of $19.1\\pm 1.0$ Myr and the Hyades rotation of $32\\pm 11\\ \\mathrm{m\\,s^{-1}\\,pc^{-1}}$."],"supporting_citations":[{"why":"Defines the linear velocity tensor model, astrometric radial velocities, and kinematically improved parallaxes that Kalkayotl's 6D model builds on.","marker":"Lindegren et al. (2000)"},{"why":"Provides the Kinesis model and the Hyades DR2 kinematics results used as the main comparison benchmark.","marker":"Oh & Evans (2020)"},{"why":"The original 1D Bayesian hierarchical model whose parametrisation and prior choices this code extends to 3D and 6D.","marker":"Olivares et al. (2020, Paper I)"},{"why":"Establishes that parallax-to-distance transforms are prior-dominated, motivating the forward-modelling likelihood used here.","marker":"Bailer-Jones (2015)"},{"why":"Lays out the forward-modelling strategy for Gaia data that the paper adopts to avoid transforming observations before inference.","marker":"Luri et al. (2018)"},{"why":"Provides the conversion constant and the prescription for turning expansion rates into expansion ages used for beta Pictoris.","marker":"Mamajek & Bell (2014)"},{"why":"Supplies beta Pictoris membership lists and literature trace-back ages that the inferred expansion age is compared with.","marker":"Miret-Roig et al. (2020)"},{"why":"Supplies the spatial correlation functions for Gaia parallaxes and proper motions used to construct the likelihood covariance matrix.","marker":"Lindegren et al. (2021)"},{"why":"Defines the Gaia DR3 catalogue whose precision, missing radial velocities, and correlations shape the realistic simulations and real-data applications.","marker":"Gaia Collaboration et al. (2023)"}],"fun_headline_variants":["Kalkayotl 2.0 Bayesian code uncovers cluster expansion and rotation","Open-source Bayesian model decodes stellar kinematics from Gaia","Bayesian phase-space model maps cluster expansion and spin","New Bayesian code reveals beta Pic expansion, Hyades rotation","Kalkayotl 2.0: stellar phase-space modeling with Gaia"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method depends on assuming that inflating the tiny Gaia sky-position uncertainties by a factor of 100,000 to 1,000,000 leaves the inferred parameters unchanged; if that heuristic is wrong, the reported positions, gradients, expansion ages, and rotation detections would all be biased.","fun_headline_variants_meta":{"raw":{"variants":["Kalkayotl 2.0 Bayesian code uncovers cluster expansion and rotation","Open-source Bayesian model decodes stellar kinematics from Gaia","Bayesian phase-space model maps cluster expansion and spin","New Bayesian code reveals beta Pic expansion, Hyades rotation","Kalkayotl 2.0: stellar phase-space modeling with Gaia"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001448,"raw_usage":{"total_tokens":5918,"prompt_tokens":1114,"completion_tokens":4804,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":730,"completion_tokens_details":{"reasoning_tokens":4714}},"tokens_in":730,"tokens_out":4804,"duration_ms":29495,"temperature":1.0,"reasoning_tokens":4714,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:37:54.816616+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same Hyades and beta Pictoris analyses with the default sky-uncertainty inflation and with a much smaller inflation factor, using longer chains so both runs converge; if the inferred rotation component or the expansion age shifts by more than the quoted uncertainties, the inflation assumption is refuted.","supporting_citations":[],"review_version":1}