{"id":"a78096aa-94a1-4e85-9cb4-fa424b08a50d","arxiv_id":"2608.08091","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For feature-lifted dynamical systems with unknown finite-group symmetry, the paper proves a representation-theoretic characterization of the minimal identification trajectory length and gives a random-generator algorithm that discovers the symmetry from the same trajectory.","lead":"This paper studies how to identify the equations of a dynamical system from one observed trajectory when the system has a hidden finite symmetry, such as permutation or rotation invariance. It proposes an algorithm that discovers the symmetry group from the data and analyzes how much trajectory data is actually needed.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.5 guarantees success only at the worst-case threshold max_H T_Phi(H), not at the true group's threshold T_Phi(G); the abstract's 'same optimal trajectory length as in the known-symmetry case' is therefore an overclaim.","rationale":"The reader's verdict identifies the core gap: the formal guarantee in Theorem 4.5 uses the worst-case identification threshold over the candidate family, while the abstract and concluding remarks claim the same trajectory length as the known-symmetry case, i.e., the threshold of the true group G. This discrepancy is load-bearing because it is exactly what makes the adaptive setting appear to have 'near-zero overhead.' My reading of the proof confirms that the worst-case threshold is not an artifact of a loose proof: the generic separation condition in Definition 4.4 is only asserted for T >= max_H T_Phi(H), so the proof cannot be re-run at T = T_Phi(G) for arbitrary candidate families. The paper's only fully justified positive result for the adaptive setting is the worst-case guarantee, which is strictly weaker than the advertised claim and degenerates to the generic threshold when the trivial group is included. The reader's weakest assumption is essentially the same: Definition 4.4 is assumed for general families rather than derived, and the advertised equality with the known-symmetry threshold requires an additional unstated condition. I therefore agree with the REJECT verdict: the paper contains a substantial and mostly correct toolkit, but the central advertised claim is not supported by the theorems as stated. The concrete test above would determine whether the claim is merely unproven or actually false; either way, the current manuscript overstates its main contribution.","tokens_in":26022,"tokens_out":8393,"duration_ms":89550,"concrete_test":"Run Algorithm 1 with a candidate family G = {C_2, C_3} inside Gamma = S_4, where C_2 and C_3 are cyclic subgroups generated by a transposition and a 3-cycle, and let the true group be G = C_2. Choose Phi = identity (linear features) with d = 4, and compute T_Phi(C_2) and T_Phi(C_3) from Theorem 4.1; for linear systems these are T_lin(H) = max_{pi:n_pi>0} ceil(n_pi/d_pi), so T_Phi(C_3) > T_Phi(C_2) is expected. Generate a generic C_2-equivariant trajectory of length T = T_Phi(C_2), then run Algorithm 1. If the C_3 candidate is feasible and has larger cardinality, the algorithm returns the wrong group, disproving the claimed optimal trajectory length; if it is infeasible, the test can be repeated with another incomparable pair (e.g., C_3 vs. C_4 in S_5) to determine whether the abstract claim fails or merely lacks a proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central advertised contribution is that adaptive symmetry discovery identifies a G-equivariant system from a single trajectory of length T_Phi(G), the same length needed when the symmetry group is known. The formal Theorem 4.5 only proves success for T >= max_{H in G} T_Phi(H). Definition 4.4 defines generic candidate separation at this worst-case horizon, so it cannot be invoked at T = T_Phi(G). If G contains the trivial group, max_H T_Phi(H) = T_Phi({e}) = m, the generic no-symmetry threshold, and the theorem's guarantee degenerates to the baseline. The proof of Theorem 4.5 itself only needs T >= T_Phi(G) together with separation; for nested families (Proposition B.5) separation is automatic, so the advertised claim is plausibly true there. But for general families neither the required separation at T_Phi(G) nor the equality max_H T_Phi(H) = T_Phi(G) is established. Remarks 4.8-4.10 and the abstract present the worst-case threshold as if it were the known-symmetry threshold, which is unsupported. The separation assumption for incomparable candidates is also nontrivial: a candidate H neither a subgroup nor a supergroup of G could be feasible with many degrees of freedom when T < T_Phi(H), and the paper gives no argument excluding this. Thus the load-bearing premise of the abstract is absent from the stated theorems.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies identification of discrete-time feature-lifted linear dynamical systems f(x)=WΦ(x) from a single noiseless trajectory, under finite group equivariance. It first gives a representation-theoretic characterization T_Φ(G) of the generic identification threshold for a known symmetry group, with several examples (linear, affine, finite Abelian, permutation-equivariant polynomial) showing large reductions relative to the generic threshold. It then proposes adaptive symmetry discovery: Algorithm 1 samples random generators for each candidate group, tests feasibility of the trajectory and equivariance constraints, and returns the largest feasible group. Theorem 4.5 guarantees success for T at least the worst-case threshold over the candidate family, under a 'generic candidate separation' assumption and a random-generation condition. Algorithm 2 handles bounded-index subgroups by rejection-sampling from an ambient group under a 'generic elementwise separation' assumption. The appendix gives detailed proofs and a small matched-symmetry experiment.","tokens_in":26316,"tokens_out":10084,"duration_ms":106744,"significance":"If the results were as advertised, the paper would be a useful contribution: a clean sample-complexity theory for equivariant system identification, an elegant use of random generating sets and Cayley graphs to avoid enumerating finite groups, and a striking example where permutation equivariance reduces the required trajectory length from Θ(d^2) to a constant. The known-symmetry part (Theorem 4.1 and its corollaries) appears coherent, and Proposition A.2 on random generation is elementary and correct. However, the central adaptive claim in the abstract is not supported by Theorem 4.5, and the paper's own Remark 4.10 concedes that general candidate families require a larger horizon. This overclaim materially reduces the significance of the paper in its current form.","major_comments":[{"comment":"The central claim that adaptive discovery \"achieves the same optimal trajectory length as in the known-symmetry case\" is not what Theorem 4.5 proves. Theorem 4.5 requires T ≥ max_{H∈G} T_Φ(H), whereas the known-symmetry length for the true group G is T_Φ(G). If G contains the trivial group, max_{H∈G} T_Φ(H) = m, the generic no-symmetry threshold, so the theorem's guarantee degenerates to the baseline. Definition 4.4 defines generic separation only for T ≥ max_{H∈G} T_Φ(H), so it cannot be invoked at T = T_Φ(G). Remark 4.10 explicitly concedes that for incomparable candidate families the worst-case threshold need not suffice and a larger horizon is required, which contradicts the abstract and Remark 4.9. The authors must either weaken the abstract and main text to a worst-case threshold or to nested families, or add a new theorem establishing success at T = T_Φ(G) under an explicit and verified separation condition.","section":"Abstract; §4.2, Theorem 4.5"},{"comment":"Generic candidate separation is the load-bearing premise of Algorithm 1, but it is proved only for families totally ordered by inclusion (Proposition B.5). For incomparable candidates H that are neither subgroups nor supergroups of the true group G, no argument rules out an H-equivariant system that fits the trajectory when T is below T_Φ(H); in that regime the feasible set can have many degrees of freedom. The manuscript offers no example of a non-nested family satisfying Definition 4.4 and no structural condition beyond inclusion that implies it. Since Theorem 4.5's conclusion is conditional on this assumption, the adaptive result is much narrower than the introduction and abstract suggest. The assumption should be either derived for a substantive class of examples or explicitly represented as a restrictive condition in the statement of the main theorem.","section":"Definition 4.4; Proposition B.5"},{"comment":"Theorem 4.11 relies on \"generic elementwise separation,\" which is simply assumed. The acceptance test in Algorithm 2 is used to decide membership in the unknown subgroup H, and the proof requires that a sampled g ∈ Γ passes the feasibility test iff g ∈ H for a generic system. This is a strong correctness condition: for trajectory lengths below the known-symmetry threshold of a supergroup or an incomparable subgroup, spurious feasible elements may pass. No conditions on Γ, Φ, or T are given that imply this equivalence. As with Definition 4.4, the theorem is a conditional statement rather than an unconditional near-zero-overhead guarantee. The main text should state the assumption clearly and provide at least one nontrivial family where it is verified.","section":"Theorem 4.11"}],"minor_comments":[{"comment":"The symbol T_Φ(G) is used both for the known-symmetry threshold of a group G and for the worst-case threshold max_{G∈G} T_Φ(G) of a candidate family. This notation collision makes the gap between the theorem and the abstract easy to miss; use a distinct symbol such as T_Φ(𝒢) for the family threshold.","section":"§3.3 and Definition 4.4"},{"comment":"In the paragraph on representation stability, the phrase \"stabilize stabilize\" contains a duplicated word.","section":"§4.1"},{"comment":"The examples paragraph contains grammatical slips: \"Here is a few examples\" and \"permutation group (also known as symmetric group) of group of all permutations\" should be rewritten.","section":"Appendix A.1"},{"comment":"The experiments only illustrate the known-symmetry thresholds for matched symmetry classes; they do not run Algorithm 1 or Algorithm 2. A small experiment executing the adaptive discovery procedure, even on the linear systems in Table 1, would directly test the paper's main algorithmic claim.","section":"Appendix C"},{"comment":"In the proof, the line \"Since T ≥ T_Φ(G) ≥ T_Φ(G)\" prints the same symbol for the family threshold and the group threshold; this is a consequence of the notation collision noted above and should be corrected for readability.","section":"Proof of Theorem 4.5"}],"recommendation":"reject","confidential_remarks":"The rejection is driven by the mismatch between the abstract's central claim and the formal content of Theorem 4.5, which the authors themselves acknowledge in Remark 4.10. The known-symmetry characterization and the nested-family separation result appear sound and could form the basis of a revised submission that either restricts the adaptive claim to nested families or proves separation at the true group's threshold for a nontrivial class of families. No concerns about attribution or novelty disclosure beyond the overclaim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new result here is Theorem 4.1: an exact characterization of the minimal trajectory length for identifying an equivariant feature-lifted linear system, in terms of isotypic multiplicities and generic excitation ranks. That is a real contribution and it is proved carefully. The follow-up bounds for permutation-equivariant polynomial systems, where the trajectory length stays constant as the state dimension grows, are also convincing and nicely illustrate the theory. The random-generator idea, using logarithmically many sampled group elements to test candidate symmetries, is clever, and the appendix gives honest proofs.\n\nThe problem is the gap between the theorems and the advertised message. The abstract and introduction say adaptive discovery achieves the same optimal trajectory length as the known-symmetry case. Theorem 4.5 does not prove that. It proves success at T >= max_{H in G} T_Phi(H), the worst-case threshold over the candidate family. If the candidate family contains the trivial group, that maximum reverts to the generic no-symmetry threshold m, so the guarantee collapses to the baseline. The paper tries to patch this with Definition 4.4, generic candidate separation, but that is assumed for general families and only proved automatic for nested ones. So the central claim in the abstract is unsupported; Remark 4.10 effectively concedes that by saying a larger horizon may be needed.\n\nThe experiments are also oddly placed: they only verify the known-symmetry thresholds from Theorem 4.1, not the adaptive algorithm at all. That is a missed opportunity, because a simple experiment on nested groups could have shown the advertised claim working in a concrete case.\n\nThe underlying work is serious and the theoretical core is worth engaging. The paper deserves a serious referee, but not in its current form. A revision should either prove the known-symmetry claim under stated assumptions (for nested families it may be true) or rewrite the abstract and conclusions to match the worst-case guarantee. The bounded-index material is a nice complement but rests on a similarly strong elementwise separation assumption that needs scrutiny. I would not desk-reject, but I would send it back with the expectation that the claims be brought in line with the theorems.","headline":"A strong representation-theoretic core with a load-bearing overclaim: the formal theorem only guarantees the worst-case candidate threshold, not the advertised known-symmetry trajectory length.","tokens_in":611,"tokens_out":711,"would_cite":true,"duration_ms":22425,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37M10","68Q32","20C15","05C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Unknown finite symmetries can be discovered from a single trajectory at no additional observation cost, and the dynamics can be identified within the same trajectory length as if the symmetry were known, provided the family of candidate…","keywords":["system identification","equivariance","symmetry discovery","finite groups","sample complexity","representation theory","Cayley graphs","single trajectory"],"falsifier":"For a fixed feature map such as $\\Phi_{\\le 2}$, enumerate all pairs of incomparable subgroups $H_1,H_2$ of $S_4$ or $S_5$; for a generic system with full symmetry group $H_1$, solve the Algorithm 1 feasibility program for $H_2$ at trajectory length $T=\\max(T_\\Phi(H_1),T_\\Phi(H_2))$. If any $H_2$ that is not a subgroup of $H_1$ is feasible, generic candidate separation fails for that family, and the no-extra-trajectory guarantee does not hold unconditionally.","tokens_in":25697,"feed_emoji":"🎯","tokens_out":12164,"duration_ms":113936,"temperature":0.7,"pith_summary":"The paper addresses a practical challenge in scientific machine learning: when a dynamical system's only symmetry is an unknown finite group, can one still identify the system from a short single trajectory? For the class of feature-lifted linear systems $f(x)=W\\Phi(x)$, it first gives a representation-theoretic formula for the minimal trajectory length $T_\\Phi(G)$ needed when $G$ is known, showing that symmetries shorten the trajectory from the generic length $m$ to a quantity determined by multiplicities of irreducible representations. Its central claim is that the same length $T_\\Phi(\\mathcal G)=\\max_{H\\in\\mathcal G}T_\\Phi(H)$ suffices when the symmetry is unknown, under the assumption of generic candidate separation: no candidate group outside the true one's subgroups can fit the data. The paper proposes a concrete algorithm that tests random generating sets of candidate groups by linear feasibility and selects the largest feasible group, so the saving in data is real while the extra cost is computational. If right, symmetry discovery is observationally free for such systems, and the theory quantifies exactly when and how.","feed_headline":"No extra trajectory data needed when symmetry is unknown","feed_subtitle":"One short trajectory reveals both the dynamics and its symmetry group, provided candidates are distinguishable.","key_machinery":"Three pieces carry the argument. The first is the isotypic decomposition of the state and feature representations of a finite group: after a change of basis, every equivariant map takes the block form $\\bigoplus_\\pi C_\\pi\\otimes I_{V_\\pi}$, so identification reduces to checking whether the per-block design matrices $\\Phi_{\\pi,T}$ have full row rank. The second is the random-generation property of finite groups: $O(\\log|G|+\\log(1/\\delta))$ i.i.d. uniform samples generate $G$ with probability $1-\\delta$, equivalently making the Cayley graph $\\mathrm{Cay}(G,S)$ connected. The third is the feasibility formulation that combines the trajectory equation $W\\Phi(x_t)=x_{t+1}$ with the sampled intertwining constraints $\\rho(g)W=W\\rho_\\Phi(g)$, which are linear in the entries of $W$ and, by the generating-set property, enforce full equivariance. Generic candidate separation is the condition that lets the algorithm pick the true group as the unique largest feasible candidate.","core_discovery":"The central claim is that, under a separation condition on the candidate family, discovering the symmetry group is observationally free: Algorithm 1 recovers both the parameter matrix $W$ and a generating set of the full symmetry group $G$ from a single generic trajectory of length $T_\\Phi(\\mathcal G)$, the same worst-case trajectory length required when $G$ is known. The proof builds on a precise characterization of the known-symmetry threshold: $T_\\Phi(G)$ is the smallest $T$ such that, in every isotypic component $\\pi$ that appears in the state representation, the generic rank $h_{\\pi,\\Phi}(T)$ of the stacked feature matrix $\\Phi_{\\pi,T}$ reaches the feature multiplicity $m_\\pi$. Because equivariance on a generating set implies full equivariance, logarithmically many random samples per candidate suffice; the largest feasible candidate is then the true group, since every feasible candidate must be a subgroup of it. The paper also proves a bounded-index variant in which samples are drawn from a known ambient group and rejected until they generate the unknown subgroup, avoiding enumeration of candidate subgroups.","pith_inferences":["The 'same length as known-symmetry' promise is conditional on the threshold not being dominated by the least symmetric candidate: if the trivial group belongs to $\\mathcal G$, then $T_\\Phi(\\mathcal G)=m$ and the adaptive guarantee collapses to the generic no-symmetry length, so the meaningful regime is a family that excludes near-trivial candidates or a guarantee at $T_\\Phi(G)$ itself.","Generic candidate separation is proved automatically only for nested families; for incomparable candidates it is an extra hypothesis that should be checked. A concrete computational test on incomparable subgroups of a symmetric group, such as $C_4$ and $V_4$ inside $S_4$, would reveal how often separation holds for natural feature maps.","The feasibility tests assume exact data; under noise the true system would generically fail the equality constraints, so a practical variant would need tolerant feasibility, likely least-squares residual thresholds, and the question of how the trajectory-length gain degrades with noise remains open.","The bounded-index sampler requires elementwise separation, a stronger per-element condition than group-level separation; if spurious elements pass the feasibility test, rejection sampling no longer yields uniform samples from the hidden subgroup and the generating-set guarantee fails."],"forward_implications":["For candidate families totally ordered by inclusion, the separation condition is automatic, so adaptive discovery provably matches the known-symmetry trajectory length without any extra observations.","For fixed-degree permutation-equivariant polynomial systems, the known-symmetry threshold is $O_k(1)$, independent of the state dimension $d$, so the adaptive result would make unknown permutation symmetries equally cheap when separation holds.","When the true symmetry is a bounded-index subgroup of a known ambient group, the algorithm discovers and generates it without enumerating the candidate subgroups, with expected sampling overhead at most a factor $B$.","The computational cost scales polynomially in $d,m,T,|\\mathcal G|,\\log|G|_{\\max},\\log(1/\\delta)$, so groups of size exponential in the state dimension are handled with polylogarithmic random samples.","If a generating set for each candidate is supplied, randomization is unnecessary and the algorithm becomes a deterministic feasibility search."],"supporting_citations":[{"why":"Supplies the representation theory background, including decompositions into irreducibles and Schur's lemma, used to derive the per-block identification criterion in Theorem 4.1.","marker":"Serre et al., 1977"},{"why":"Standard reference for isotypic decompositions and equivariant maps, used for the block structure of equivariant parameter matrices.","marker":"Fulton and Harris, 2013"},{"why":"Sets the random-Cayley-graph expansion framework that motivates sampling logarithmically many generators; the paper uses the weaker connectivity consequence in its adaptive algorithm.","marker":"Alon and Roichman, 1994"}],"fun_headline_variants":["Unknown symmetry costs zero extra trajectory data","Discover symmetry and dynamics from a single trajectory","Symmetry discovery comes free with single-trajectory identification","One trajectory yields dynamics and hidden symmetry"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the candidate family is generically separating: for the true group $G$, no group $H$ outside the subgroups of $G$ can admit an equivariant system consistent with the observed trajectory; the paper proves this automatically only when the candidates are totally ordered by inclusion, and otherwise it is assumed rather than derived.","fun_headline_variants_meta":{"raw":{"variants":["Unknown symmetry costs zero extra trajectory data","Discover symmetry and dynamics from a single trajectory","Symmetry discovery comes free with single-trajectory identification","One trajectory yields dynamics and hidden symmetry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000704,"raw_usage":{"total_tokens":3177,"prompt_tokens":950,"completion_tokens":2227,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":566,"completion_tokens_details":{"reasoning_tokens":2183}},"tokens_in":566,"tokens_out":2227,"duration_ms":17550,"temperature":1.0,"reasoning_tokens":2183,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T00:27:41.964752+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed feature map such as $\\Phi_{\\le 2}$, enumerate all pairs of incomparable subgroups $H_1,H_2$ of $S_4$ or $S_5$; for a generic system with full symmetry group $H_1$, solve the Algorithm 1 feasibility program for $H_2$ at trajectory length $T=\\max(T_\\Phi(H_1),T_\\Phi(H_2))$. If any $H_2$ that is not a subgroup of $H_1$ is feasible, generic candidate separation fails for that family, and the no-extra-trajectory guarantee does not hold unconditionally.","supporting_citations":[],"review_version":1}