{"id":"6f4eb1b3-c682-4a08-a421-e6e346a32623","arxiv_id":"2602.15279","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Applying a minimally invasive pairwise Hamiltonian control to a higher-order Kuramoto model, the authors find that triadic coupling raises the control strength needed near synchrony and produces a non-monotonic effect away from it.","lead":"This paper tests whether a cheap, minimally invasive pairwise control can desynchronize networks of oscillators that interact in threes, not just pairs. It finds that three-body interactions make desynchronization harder when the system starts near synchrony, but can help or hinder depending on coupling strength when it starts further away.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The simulated controller drops the B_i term in Eq. (A5) without quantifying it; if B_i is not negligible, the paper's efficiency claims apply to a different control than the claimed Hamiltonian one.","rationale":"The reader's weakest assumption identifies the same load-bearing concern: the simplified p_i used throughout the simulations relies on dropping B_i in Eq. (A5) and on the clique approximation. This is indeed the most critical point because every quantitative result—mu_c, phase diagrams, non-monotonic curves—is generated using p_i. The paper's own Appendix A states that B_i 'can be removed without losing the control efficiency' but does not demonstrate this for the higher-order model or for the off-sync initial conditions considered in Section IV. Since the central claim is about the efficiency of the pairwise Hamiltonian control, if the implemented control differs materially from that control, the paper's conclusions would be about a different controller. The concrete test—running the full \\hat h_i and comparing with p_i—would settle whether this concern actually lands. I found no other issue as fundamental: the numerical results are internally consistent, the threshold choices are arbitrary but affect only the precise definition of mu_c, and the lack of error bars is a reproducibility concern rather than a logical flaw. Thus the reader's CONDITIONAL verdict remains appropriate, with the B_i approximation as the key condition to verify.","tokens_in":21563,"tokens_out":5411,"duration_ms":57460,"concrete_test":"Compute B_i = \\hat h_i - p_i along representative simulated trajectories using the definitions in Appendix A. Choose hyperring r=2, N=100, K1=1, σ=0.1, and at least (K2, mu, epsilon) = (0, 0.05, 0), (5, 0.05, 4), (10, 0.1, 5), (20, 0.2, 5). For each, run the dynamics with Eq. (3) and with the full \\hat h_i of Eq. (A4) (keeping B_i). If the time-averaged \\hat R changes by more than 10% at any parameter point, or if max_t |B_i| / max_t |p_i| > 0.1 during the transient, the dropped term is not negligible and Figs. 1–6 need to be recomputed with the full control.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative results (Figs. 1–6, mu_c in Figs. 2 and 4) are obtained from Eq. (1) with p_i given by Eq. (3). In Appendix A, this p_i is obtained from \\hat h_i, Eq. (A4), by writing \\hat h_i = (2 K1^2 M^2 <k>^2)^{-1} R_M \\tilde R_{M,i} cos(...) + B_i and then deleting B_i with the statement that it 'can be removed without losing the control efficiency' (Ref. 35). This step is load-bearing: if B_i is not negligible for the higher-order dynamics studied here, the simulated controller is not the minimally invasive Hamiltonian control whose efficiency the paper claims to characterize. The original justification in Ref. 35 was given for pairwise networks and no higher-order coupling; the present paper applies it in a new setting (K2 up to 20, phase configurations far from synchrony) without quantifying B_i. In addition, item (i) replaces the true adjacency by a clique, which may make the control more powerful than the real pairwise Hamiltonian control. Both approximations are unquantified. Because mu_c and the non-monotonic phase diagrams are defined entirely through dynamics with p_i, an unvalidated approximation in p_i directly threatens the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies how higher-order (three-body) interactions affect the efficiency of a minimally invasive pairwise Hamiltonian control in desynchronizing the higher-order Kuramoto model. For synchronized initial conditions, numerical simulations on hyperrings and random Erdős–Rényi hypergraphs show that the critical control strength μ_c increases with the higher-order coupling strength K_2, indicating that higher-order interactions hinder control. For perturbed initial conditions, the reported R̂ versus K_2 curves become non-monotonic: intermediate K_2 impedes desynchronization while larger K_2 facilitates it. The authors interpret this as a competition between increased linear stability and shrinking basin of attraction of the synchronized state. They conclude that, in all cases, sufficiently strong control or a sufficiently large number of controlled nodes can desynchronize the system.","tokens_in":21875,"tokens_out":4700,"duration_ms":52933,"significance":"If the numerical claims are correct, the paper provides a practically relevant answer to a nontrivial question: whether a simple pairwise controller, originally designed for ordinary Kuramoto networks, remains effective when the uncontrolled dynamics include higher-order interactions. The reported phenomenology is consistent with the 'deeper but smaller' effect of higher-order interactions and extends it to a control setting. The study is systematic in its parameter coverage (K_2, μ, M, ε, two topologies, two values of r), uses two complementary order parameters, and averages over many frequency and initial-condition realizations. The main caveat is that the simulated controller is an approximation of the claimed Hamiltonian controller, and this approximation is not quantitatively validated. Because all central results (μ_c, phase diagrams, non-monotonic curves) are produced with this approximate control term, the significance of the paper is conditional on that validation. The paper would be strengthened by providing code or a data repository, as none is currently supplied.","major_comments":[{"comment":"The control term actually simulated is p_i in Eq. (3), obtained by dropping the term B_i in Eq. (A5) with the statement that B_i 'can be removed without losing the control efficiency.' The original justification for neglecting B_i (Ref. 35) was developed for pairwise Kuramoto dynamics without higher-order interactions, whereas here it is applied to dynamics with K_2 up to 20, far-from-synchrony initial conditions, and partial control. The paper does not quantify the magnitude of B_i, does not compare the dynamics with p_i against the dynamics with the full ̂h_i of Eq. (A4), and does not test whether the neglected term remains small in the regimes where the non-monotonic effects are reported. Since μ_c and the phase diagrams are defined entirely through dynamics with p_i, an unvalidated approximation in the controller directly threatens the central claim. I ask for a numerical check: for","section":"Appendix A, Eq. (A5)"},{"comment":"The derivation replaces the actual adjacency A_ij among the pinned nodes by a complete graph, i.e., a clique, and the authors explicitly state that this makes the control 'potentially more powerful.' This approximation is load-bearing: the paper's claim is about the efficiency of the minimally invasive pairwise Hamiltonian control, but the simulated controller assumes a clique topology rather than the actual hyperring or Erdős–Rényi graph among controlled nodes. It is therefore possible that the reported efficiency (e.g., the statement that a sufficient number M of controlled nodes always achieves desynchronization) overstates what the real pairwise Hamiltonian controller would achieve on the actual network. A comparison simulation using the actual A_ij in Eq. (A3) is needed before the central efficiency claims can be accepted.","section":"Appendix A, item (i)"},{"comment":"The main phase diagrams and R̂-versus-K_2 curves are reported only as means over realizations, without error bars or confidence intervals. The threshold R_thr = 0.4 is set by 'preliminary visual inspection,' and the critical control strength μ_c is then defined as the smallest μ at which R̂ < R_thr. This makes the quantitative statements in Figs. 2 and 4 depend on an ad hoc threshold. While the qualitative trends appear to be robust, the manuscript should provide variability information for the main figures and a sensitivity analysis of μ_c with respect to R_thr (e.g., for R_thr ∈ {0.3, 0.4, 0.5}). This is particularly important because the non-monotonic regime in Figs. 5 and 6 is one of the central results.","section":"Figs. 1, 3, 5, 6; Section III"}],"minor_comments":[{"comment":"The text says 'R^(1) = 1 ⇒ R^(2) = 1 and R^(1) = 1 ⇒ R^(2) = 1'; the second implication should presumably involve R^(3). Please correct.","section":"Appendix B, first paragraph"},{"comment":"There is a typographical artifact in the sentence introducing the clique approximation: 'as if A AAactually defined a M−clique' should read 'as if A actually defined a M-clique.' There are also minor typos such as 'bellow' and 'Extention' in the Fig. 7 caption.","section":"Appendix A"},{"comment":"The authors state that the control is 'slightly modified' from Ref. 35 to use only controlled-node quantities. This is an important practical point and could be highlighted more explicitly in the main text, not only in Appendix A, because it is a key difference from the original minimally invasive control.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses an interesting and timely question, and the numerical results are likely to be of interest to the community. However, the central claim is conditional on the validity of the approximations made in deriving the simulated control term. I would request the comparison simulations with the full ̂h_i and with the actual network adjacency before publication, as well as error bars on the main phase diagrams. This is fixable by additional numerical work within the manuscript's scope, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The one thing to know: this paper gives a clear numerical map of when a cheap pairwise control can desynchronize a higher-order Kuramoto system. Near full synchrony, triadic coupling makes desynchronization harder (monotonically); away from synchrony, intermediate K2 impedes control while larger K2 helps. That second part is the genuinely new observation, and it fits naturally with the ‘deeper but smaller’ picture of higher-order interactions.\n\nThe paper does well in several respects. The simulations are straightforward and cover two topologies (hyperrings and random Erdős–Rényi hypergraphs), with partial control (varying M pinned nodes). The results are consistent with the uncontrolled dynamics of Ref. 91, which is a good sanity check. There are no tuned constants or fitted curves — the control term is taken from prior theory. The authors are also transparent about the modifications they introduce: they assume the pinned nodes form a clique and they drop the B_i term in Eq. (A5). That transparency is a credit.\n\nNow the soft spots, in order of importance. The B_i term is dropped with the statement that it ‘can be removed without losing control efficiency,’ but the justification in Ref. 35 was for a purely pairwise network. Here K2 goes up to 20, and the system can be far from synchrony, so it is genuinely unclear whether B_i stays negligible. If it does not, Eq. (3) is not the claimed minimally invasive Hamiltonian control. The clique assumption is a second approximation that may make the control more powerful than the real one. Both are acknowledged but never quantified. This is not fatal — the qualitative K2-dependence could survive — but a referee should ask for a numerical estimate of B_i’s magnitude and some sensitivity runs with the full h_i.\n\nThe second issue is statistical: the main phase diagrams (Figs. 1, 3, 5, 6) show no error bars even though quantities are averaged over 50 realizations. The thresholds R_thr=0.4 and R_dot_thr=0.5 are chosen by visual inspection. The paper would be much stronger with sensitivity analysis, confidence bands, and with the code and data made available — the current ‘data available within the article’ is not sufficient for reproducibility.\n\nIf you work on control of higher-order oscillator networks, this paper is worth a read. It is a legitimate contribution that deserves peer review; I would not desk-reject it. The revisions should focus on the control-term approximation and on statistical rigor.","headline":"A useful numerical map of when a cheap pairwise control can desynchronize higher-order Kuramoto, but the unquantified approximation in the control term and missing error bars keep the evidence softer than the prose.","tokens_in":22363,"tokens_out":3920,"would_cite":true,"duration_ms":38943,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34C15","37N35","34D06"],"pacs":["05.45.Xt","89.75.-k"],"model":"deepseek-v4-flash","headline":"This paper claims that a minimally invasive pairwise Hamiltonian control can desynchronize a higher-order Kuramoto network, but three-body interactions systematically raise the control effort needed when starting near synchrony, while for o","keywords":["higher-order Kuramoto model","Hamiltonian control","desynchronization","pinning control","three-body interactions","attraction basin","linear stability","order parameter"],"falsifier":"Rerun the phase diagram with the full control expression including the dropped term B_i instead of the simplified version. If including B_i appreciably changes the monotonic increase of the critical control strength with triadic coupling, or if the control no longer desynchronizes for the same parameter values, then the simplification is load-bearing. A minimal check: measure the magnitude of B_i along trajectories at the desynchronization threshold; if it is not small compared with the simplified control, the results are not guaranteed.","tokens_in":21435,"feed_emoji":"🔄","tokens_out":3657,"duration_ms":35127,"temperature":0.7,"pith_summary":"The paper asks whether a cheap, pairwise control action built from Hamiltonian control theory can still break synchronization when oscillators also interact through three-body couplings. Its answer: yes, if enough nodes are pinned or the control is strong enough, but the efficiency depends on where the system starts. Starting at the synchronized state, stronger three-body coupling makes desynchronization strictly harder: the minimum control strength grows with the triadic coupling. Starting away from synchrony, the effect is non-monotonic: moderate three-body coupling makes the control less effective, while strong coupling makes it easier, because the attraction basin of synchrony shrinks. The two behaviors mirror the two competing effects of higher-order interactions on the uncontrolled system.","feed_headline":"Near synchrony, three-body coupling resists desync control","feed_subtitle":"Starting off-sync, moderate coupling impedes the control while strong coupling helps it","key_machinery":"The controlled dynamics augment the pairwise-plus-triadic Kuramoto model with a pinning term that is derived from embedding the pairwise Kuramoto model into a Hamiltonian system and then applying Hamiltonian control theory, truncated to a minimally invasive form. The term is built only from the phases and natural frequencies of the pinned nodes, treating them as if they formed a clique, and it scales with the squared pairwise coupling and with the order parameter restricted to the pinned set, so it acts strongly when phases are clustered and fades when they are incoherent. This feedback term is what pushes trajectories out of the synchronized basin; its vanishing at low order parameter leave","core_discovery":"The central result is that the efficiency of the minimally invasive pairwise Hamiltonian control on the higher-order Kuramoto model is governed by the competition between increased linear stability and shrinking basin of the synchronized state. For synchronized initial conditions, the increased local stability dominates: the critical control strength needed to push the time-averaged order parameter below a threshold increases monotonically with the triadic coupling strength, on both hyperring and random hypergraph topologies. For initial conditions perturbed away from synchrony, an opposite effect appears for large triadic coupling: the shrunk basin lets the control push trajectories out mor","pith_inferences":["A natural testable extension is to choose the pinned nodes near the basin boundary rather than uniformly at random; the basin-shrinking effect suggests that boundary-targeted pinning could desynchronize with fewer controlled nodes.","The paper leaves open whether a minimally invasive control that itself includes higher-order terms would shift the critical control strength differently from the pairwise version used here.","The non-monotonic effect is likely generic to any perturbation that pushes trajectories out of a shrinking basin, so similar dependence on coupling strength may appear in other control schemes beyond Hamiltonian pinning."],"forward_implications":["Near synchrony, the minimum control strength needed for phase desynchronization grows monotonically with the strength of three-body coupling, so stronger higher-order interactions demand stronger control.","For off-sync initial conditions, intermediate three-body coupling impedes the control while large coupling helps it, so desynchronization cost is not a monotone function of higher-order coupling.","The pairwise minimally invasive control suffices to desynchronize the higher-order system when enough nodes are pinned, even without using higher-order control terms.","The control can leave frequency-synchronized cluster or twisted states in between, meaning phase desynchronization is easier to achieve than frequency desynchronization.","The qualitative results hold on both hyperring and random hypergraph topologies, indicating that the mechanism is robust to network structure and only the thresholds shift."],"fun_headline_variants":["Triadic coupling resists desync control near synchrony","Near-sync triadic effects hinder, off-sync they help desync","Non-monotonic triadic effect on desync control efficiency","Pairwise desync control faces triadic barrier near sync","Initial phase flips triadic coupling's desync impact"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The simulated controller is the simplified version of the Hamiltonian control in which the pinned nodes are treated as a clique and the term B_i is dropped from the full expression; if that term actually matters for the three-body dynamics, the reported critical control strengths could shift.","fun_headline_variants_meta":{"raw":{"variants":["Triadic coupling resists desync control near synchrony","Near-sync triadic effects hinder, off-sync they help desync","Non-monotonic triadic effect on desync control efficiency","Pairwise desync control faces triadic barrier near sync","Initial phase flips triadic coupling's desync impact"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000166,"raw_usage":{"total_tokens":1070,"prompt_tokens":702,"completion_tokens":368,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":446,"completion_tokens_details":{"reasoning_tokens":282}},"tokens_in":446,"tokens_out":368,"duration_ms":4082,"temperature":1.0,"reasoning_tokens":282,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T22:53:21.810649+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Rerun the phase diagram with the full control expression including the dropped term B_i instead of the simplified version. If including B_i appreciably changes the monotonic increase of the critical control strength with triadic coupling, or if the control no longer desynchronizes for the same parameter values, then the simplification is load-bearing. A minimal check: measure the magnitude of B_i along trajectories at the desynchronization threshold; if it is not small compared with the simplified control, the results are not guaranteed.","supporting_citations":[],"review_version":1}