{"id":"8a52b03f-1eeb-4a43-8a2b-f544a43920ae","arxiv_id":"2505.15942","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Using a 3D far-field potential-flow model, the authors show that passive hydrodynamic interactions can keep small swimmer groups together, while large groups break into smaller cohesive subgroups and the circular hydrodynamic milling state is always unstable.","lead":"This paper simulates groups of idealized swimmers whose only interactions come from the fluid flows they create. It finds that passive hydrodynamics alone can hold small groups together, but larger groups split into stable smaller clusters, including a newly identified circular chasing pattern.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central negative conclusion is conditioned on an inviscid, far-field dipole model; a wake- or near-field-corrected check is needed before generalizing to real swimmers.","rationale":"The reader's CONDITIONAL verdict already identifies the model idealization as the weakest assumption; I agree. The paper's equations are clear, the symmetry arguments are plausible, and the reported qualitative behaviors are internally coherent. But the central claim is a universal negative about passive hydrodynamics, and the evidence is entirely within a potential-flow dipole model that neglects viscous wakes and finite-body rotational coupling. This is not a matter of consensus but of external validity: if the fluid model changes the sign of the rotational feedback, the phase diagrams and instability conclusions change. The concrete test above would distinguish model-specific behavior from robust phenomenology. I also note that the '%indefinitely%' language for subgroups is stronger than the finite-time simulations support, but that is secondary to the model-scope issue. Thus the reader's CONDITIONAL verdict stands unchanged.","tokens_in":13102,"tokens_out":9078,"duration_ms":92806,"concrete_test":"Re-run the N=7 diamond and N=10 milling cases replacing the potential-dipole bead velocities (Section II) with an Oseen/wake-corrected pairwise interaction, or with a resolved 3D viscous self-propelled swimmer simulation, for the same initial conditions. If the same emergent subgroups appear and hydrodynamic milling remains unstable, the central claim survives; if large-scale cohesion persists or milling is stable, the conclusion must be restricted to the inviscid far-field limit.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The claim that passive hydrodynamics cannot sustain large-scale cohesion is stated generally, but every phase diagram and stability result is computed from one reduced model: each swimmer is a source-sink potential dipole and each bead is a material point (Section II, Eqs. 2-4). Under this model, orientation changes only through velocity differences sampled at two points. In real inertial swimmers, rotational dynamics are governed by the full velocity-gradient tensor, body geometry, and vorticity/wake fields; earlier wake-based models show stable schooling at scale (e.g., Filella et al. 2018). Therefore the subgrouping and milling instability may be an artifact of the dipole truncation rather than a general property of passive hydrodynamics. The paper is internally consistent, but the abstract and conclusion overstate external validity. A secondary concern is that '%maintain their structure indefinitely%' (Section VI) is inferred from finite-time trajectories without a boundedness argument, so even within the model the positive subgroup-cohesion clause is extrapolated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies whether passive hydrodynamic interactions alone can maintain cohesion in groups of inertial swimmers. It uses a three-dimensional, inviscid, far-field model in which each swimmer is a source–sink dipole pair with two material-point beads, following the authors' earlier pairwise model. The authors first map the dynamics of triangular configurations, finding finite-time cohesive (oscillatory) states for one leader and two followers and a stable relative equilibrium with a large basin of attraction for two leaders and one follower. They then simulate diamond lattices of 7, 13, and 27 swimmers and observe that the overall group loses cohesion while smaller peripheral subgroups, such as braided trios or quintets, remain cohesive. For circular configurations, they report discovering 'hydrodynamic milling' states for every N from 4 to 100, which are relative periodic orbits, and they claim these states are always unstable to exactly three modes: two asymmetric conjugate modes and one symmetric mode. Nonlinear evolution splits the milling group into smaller subgroups. The paper concludes that passive hydrodynamics alone cannot sustain large-scale cohesion indefinitely, but that cohesive subgroups form robustly and that controlling only the periphery or inter-subgroup interactions might suffice to maintain a larger group.","tokens_in":13229,"tokens_out":4354,"duration_ms":42264,"significance":"If the results hold, the paper makes a useful contribution to the understanding of passive hydrodynamic mechanisms in collective swimming. The model is fully specified and parameter-free in the sense that no constants are fitted to produce the observed cohesion, subgrouping, or milling behavior; the governing equations and the numerical methods are stated clearly enough to reproduce the main simulations. The discovery of hydrodynamic milling states is novel and potentially relevant for bioinspired robotics, and the systematic treatment of triangular, diamond, and circular configurations provides a useful map of behaviors. The main limitation is that all conclusions are derived from an inviscid, far-field potential-flow model whose external validity for real swimmers is not established; in addition, two of the central claims—the universal three-mode instability of milling and the indefinite cohesion of subgroups—rest on numerical evidence that is not fully documented. These issues are fixable, but they currently leave the strongest claims less secure than the presentation suggests.","major_comments":[{"comment":"The claim that hydrodynamic milling is 'always' unstable to exactly three modes for every N in [4,100] is not supported by the reported numerical evidence. The Arnoldi iteration uses finite-difference Jacobian-vector products via Eq. (A1), but the manuscript does not state the perturbation size epsilon, the number of Arnoldi vectors, the number of iterations, or any convergence criterion. No eigenvalue spectra, growth rates, or residuals are shown for any N; only the radius and speed appear in Fig. 12, and the unstable modes are illustrated for N=10 only. Because the universal three-mode instability is a central quantitative claim, please report the eigenvalues (or at least the unstable growth rates) as functions of N, and validate the finite-difference/Arnoldi computation against a dense eigenproblem for a small case such as N=4, while demonstrating convergence in epsilon and in Krylov dimension.","section":"Section V A and Appendix A"},{"comment":"The statement that emergent subgroups 'maintain their structure indefinitely' and 'remain cohesive for the rest of time' is an extrapolation from finite-time simulations. The paper provides no boundedness argument, no invariant set, and no quantitative measure of subgroup cohesion over time; Figs. 9, 14, and 15 show trajectories only until divergence or collision. Either soften these statements to 'for the duration of the simulations' or provide quantitative evidence such as bounded subgroup separations over long integration times, a norm that remains bounded, or an explicit statement of the simulated time horizon. This is load-bearing because the positive claim that small subgroups are indefinitely cohesive is a key part of the paper's message.","section":"Section VI (and Sections IV, V B)"},{"comment":"The abstract and conclusions state a general negative answer: 'passive hydrodynamics alone cannot sustain large-scale cohesion indefinitely.' All results, however, are computed with a single reduced model in which each swimmer is an inviscid, far-field source–sink dipole and each bead is a material point (Section II, Eqs. 2–4). The paper itself cites wake-based models, such as Ref. [24], that produce stable schooling at scale, and the model neglects viscous wakes, near-field forces, and finite body geometry. The general claim should be explicitly restricted to the inviscid far-field model in the abstract and conclusions, or supplemented by a robustness check that adds a near-field repulsion or a simple wake term and shows whether the qualitative conclusions persist. As written, the paper overstates the scope of its conclusions.","section":"Abstract and Section VI"},{"comment":"The loss of mirror symmetry in the diamond configurations is attributed to 'round-off error in our simulations,' and this symmetry breaking is used to infer that 'certain symmetries are not robust to perturbations.' No precision study or quantitative perturbation analysis is reported. Please provide the magnitude of the asymmetry at which the symmetry breaks, and ideally verify the claim by comparing double-precision results with higher-precision arithmetic or by applying controlled asymmetric perturbations of known amplitude. This would strengthen the otherwise qualitative robustness claim.","section":"Section IV"}],"minor_comments":[{"comment":"There is a typo in the text after Eq. (1): 'in terms of of xc,i, ni' should read 'in terms of xc,i, ni.'","section":"Section II"},{"comment":"The sentence 'whereas the model used in the present is appropriate for more commonly encountered long and narrow swimmers' is missing a noun; it should read 'the model used in the present work.'","section":"Section IV"},{"comment":"The phrase 'rotated π/48 from the vertical' appears without a space before the angle; it should be 'rotated by π/48 from the vertical' for clarity.","section":"Section V A"},{"comment":"Figure 12 plots the radius and speed as functions of N, but the claim that the radius is a linear function of N is not quantified. Adding a linear fit or reporting the slope and intercept would make the scaling claim more precise.","section":"Figure 12"},{"comment":"The definition of the Poincare map is somewhat informal: the dimension of the section, the choice of the sectioning condition, and the method used to locate the fixed point p are not described. A few sentences specifying these details would improve reproducibility.","section":"Appendix A"}],"recommendation":"major_revision","confidential_remarks":"The paper is a well-written modeling study with a plausible and interesting central mechanism, but the strongest claims—universal three-mode instability of milling and indefinite subgroup cohesion—are currently supported by insufficiently documented numerics and finite-time simulations. The external-validity caveat for the inviscid far-field model should appear in the abstract and conclusions. I would be comfortable seeing the paper published after these points are addressed; I do not see a need for rejection, provided the authors either supply the requested numerical evidence or explicitly qualify the claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version. This is a carefully done numerical study of a specific idealized model, and the main result—passive far-field dipole interactions alone don't keep large groups cohesive, while small emergent subgroups do—is convincing for that model. The genuinely new piece is hydrodynamic milling: for every N between 4 and 100, uniformly spaced circular arrangements are fixed points of a Poincare map, and they are always unstable to three modes, one symmetric and a conjugate pair of asymmetric modes. That is a concrete, falsifiable statement about the model, and the paper supports it with both linear stability and nonlinear trajectories.\n\nCredit where due: the model is parameter-free, built on the authors' prior pairwise work [27], with no fitted constants. The phase diagrams for triangular groups come with physical explanations that make sense. The diamond simulations show a clear periphery effect—subgroups form from the outside in, and the core stays cohesive longer. The comparison with Gazzola et al. [11] is honest and identifies the difference in how rotational dynamics are sampled. For a paper whose main currency is numerical exploration, the analysis is unusually clear.\n\nNow the soft spots, in rough order of severity.\n\nFirst, the stability claim. The three-unstable-modes result comes from a finite-difference Arnoldi iteration on a Poincare map. The paper reports no convergence tests: no epsilon study for the Jacobian-vector product, no check that the number of Arnoldi vectors is adequate, no residuals. Since the 'always three modes' claim is a headline result, this needs a paragraph showing the eigenvalues have converged. It is a fixable omission, not a fatal flaw.\n\nSecond, the conclusion says subgroups 'maintain their structure indefinitely,' but the evidence is finite-time simulation. The paper elsewhere uses 'seems to' and 'much longer,' so the indefinite claim is an extrapolation. Either add a boundedness argument or weaken the wording.\n\nThird, and this is a real but moderate concern: the abstract and conclusion generalize from 'this far-field inviscid dipole model' to 'passive hydrodynamics.' The model treats swimmers as source-sink dipoles with beads moving as material points, which is a legitimate far-field approximation but a specific one. Wake-based models, e.g. Filella et al., show stable schooling at scale with different hydrodynamic physics. So the negative answer is a property of this model class, not of all passive hydrodynamics. The paper's framing mostly acknowledges this, but the abstract overstates it.\n\nMinor: the symmetry-breaking in the largest diamond lattice is attributed to round-off error, but no quantification is given. That's a small thing.\n\nWho is this for? People working on hydrodynamic models of collective behavior, especially fish schooling and model reduction. It deserves a serious referee. I'd recommend sending it out, with the request that the authors add convergence checks, ship code and data, and temper the external-validity language.","headline":"A solid, internally consistent numerical study of an idealized dipole-swimmer model; the new milling instabilities are real for this model, but the general claim that passive hydrodynamics cannot keep groups cohesive outruns the model.","tokens_in":13775,"tokens_out":2634,"would_cite":true,"duration_ms":23669,"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":"Passive hydrodynamic interactions alone cannot sustain large-scale cohesion in groups of swimmers, which instead break into smaller cohesive subgroups.","keywords":["collective motion","swimmer cohesion","hydrodynamic interactions","source-sink dipole","milling","subgroup formation","linear stability","fish schools"],"falsifier":"An experiment with a school of self-propelled underwater robots, or a simulation with resolved viscous vortical wakes, in which a group of more than roughly ten identical swimmers maintains a single cohesive milling or schooling configuration without any active control, would contradict the claim that passive inviscid far-field interactions alone cannot sustain large-scale cohesion.","tokens_in":12862,"feed_emoji":"🐟","tokens_out":3677,"duration_ms":30630,"temperature":0.7,"pith_summary":"The paper asks whether a group of swimmers can stay together purely through the flow fields they create, with no social attraction or active steering. Using a three-dimensional far-field model in which each swimmer acts as a source-sink dipole, it finds that small symmetric groups can remain cohesive, but larger groups cannot. Instead, large schools break apart into smaller self-organized subgroups that keep their own structure. For circular arrangements, the model produces 'hydrodynamic milling' states in which swimmers chase each other around a circle, but these states are always unstable. The authors conclude that passive hydrodynamics alone cannot hold a large group together, and suggest that controlling only the periphery or the interactions between subgroups could maintain cohesion with minimal active input.","feed_headline":"Passive flows break big swimmer schools into stable small groups","feed_subtitle":"A 3-D dipole model finds cohesion fails at scale, while pairs and trios stay together and milling is unstable.","key_machinery":"The load-bearing object is the three-dimensional source-sink dipole swimmer: each swimmer is a dumbbell with a source bead at the front and a sink bead at the rear, separated by a fixed length, so that the far-field flow is that of a dipole. Beads move as material points in the inviscid flow induced by all other beads, with a Lagrange multiplier enforcing constant body length, giving a 6N-dimensional dynamical system. For milling, the central analytic device is the Poincaré map on a rotating frame: hydrodynamic milling appears as a fixed point of this map, and its linear stability is computed by Arnoldi iteration on the Jacobian-vector products.","core_discovery":"The central discovery is that passive hydrodynamic interactions are enough to create small cohesive clusters but not to sustain cohesion of a large group. In the model, every group of three or more swimmers evolves in characteristic ways: symmetric trios with one leader and two followers show only finite-time locking before divergence, while two leaders and one follower admit a stable relative equilibrium with a large basin of attraction. Diamond lattices and circular configurations are more striking: the outer layers destabilize first and the group splits into braided trios, pairs, or quintets that remain internally cohesive even as they separate from each other. For all N between 4 and 100, the model admits 'hydrodynamic milling' states—relative periodic orbits in which swimmers chase one another around a circle—but these states are linearly unstable to two asymmetric modes (a conjugate pair) and one symmetric mode. The milling state therefore cannot persist without active input.","pith_inferences":["A testable extension would be to add a weak social alignment term to the model; the prediction would be that a critical social strength exists below which large-scale cohesion still fails, quantifying how much 'active input' is minimally needed.","The subgrouping mechanism resembles a wave of instability propagating inward; one could measure this propagation speed and compare it with the time scale of flow-induced rotation to see if it is controlled by the dipole interaction geometry.","The model suggests that robotic swarm controllers should focus on edge agents; this could be tested in a water-tank experiment with propeller-driven robots programmed to hold only the boundary robots on a circle."],"forward_implications":["If the model is right, no passive far-field hydrodynamic interaction can hold a large school together; some active or social input is required for large-scale cohesion.","The stable cohesive unit predicted is a small subgroup (pairs, trios, quintets), so engineered swarms could aim to control interactions between subgroups rather than every individual.","Hydrodynamic milling, though visually similar to fish milling, is intrinsically unstable, so observed milling in nature likely relies on active behavior or additional physical effects.","Because destabilization starts at the periphery and diffuses inward, controlling only the edge of a group may keep the core passively coherent.","The stability of diamond formations depends on the swimmer model, since the vortex-pair model used earlier gives different rotational dynamics."],"supporting_citations":[{"why":"Provides the pairwise interaction model and the result that beads move as material points, forming the basis for the group dynamics.","marker":"[27]"},{"why":"Supplies the force law for a sphere in unsteady inviscid flow that justifies treating the source and sink beads as material points.","marker":"[36]"},{"why":"Provides the analogous finite-dipole system and the stable equilateral-triangle equilibrium that the present trio equilibrium extends.","marker":"[28]"},{"why":"Gives the prior result that diamond formations are not passively stable under a planar vortex-pair model, the baseline this work compares against.","marker":"[11]"},{"why":"Supplies the Jacobian-free Newton-Krylov method used to find the hydrodynamic milling fixed points of the Poincaré map.","marker":"[38]"},{"why":"Provides the Arnoldi iteration technique used to estimate the eigenvalues and eigenmodes for the linear stability analysis of milling.","marker":"[39]"}],"fun_headline_variants":["Big swimmer groups break apart; small clusters stay cohesive","Hydrodynamic milling is unstable; only small groups stay cohesive","Passive flows can't keep large swimmer schools together","Symmetry fails to save big groups from splitting apart","Small schools stay cohesive; large ones break into subgroups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole analysis rests on treating swimmers as far-field source-sink dipoles in inviscid flow, with each bead moving as a material point—if viscous wakes, near-field forces, or social responses change the induced rotations, the predicted subgrouping and milling instability need not hold for real swimmers.","fun_headline_variants_meta":{"raw":{"variants":["Big swimmer groups break apart; small clusters stay cohesive","Hydrodynamic milling is unstable; only small groups stay cohesive","Passive flows can't keep large swimmer schools together","Symmetry fails to save big groups from splitting apart","Small schools stay cohesive; large ones break into subgroups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000469,"raw_usage":{"total_tokens":2325,"prompt_tokens":924,"completion_tokens":1401,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":540,"completion_tokens_details":{"reasoning_tokens":1322}},"tokens_in":540,"tokens_out":1401,"duration_ms":9095,"temperature":1.0,"reasoning_tokens":1322,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:09:06.458028+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"An experiment with a school of self-propelled underwater robots, or a simulation with resolved viscous vortical wakes, in which a group of more than roughly ten identical swimmers maintains a single cohesive milling or schooling configuration without any active control, would contradict the claim that passive inviscid far-field interactions alone cannot sustain large-scale cohesion.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the pairwise interaction model and the result that beads move as material points, forming the basis for the group dynamics."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the force law for a sphere in unsteady inviscid flow that justifies treating the source and sink beads as material points."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the analogous finite-dipole system and the stable equilateral-triangle equilibrium that the present trio equilibrium extends."},{"cited_title":"Gazzola, A","cited_arxiv_id":null,"evidence_quote":"Gives the prior result that diamond formations are not passively stable under a planar vortex-pair model, the baseline this work compares against."},{"cited_title":"Equilibria, periodic orbits and computing them","cited_arxiv_id":"1908.06730","evidence_quote":"Supplies the Jacobian-free Newton-Krylov method used to find the hydrodynamic milling fixed points of the Poincaré map."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Arnoldi iteration technique used to estimate the eigenvalues and eigenmodes for the linear stability analysis of milling."}],"review_version":1}