{"id":"8d393c4b-826e-4c54-93a1-31109d877bdf","arxiv_id":"2608.02906","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A categorical Gromov-Hausdorff distance for systems with families of pseudometrics preserves Lyapunov and asymptotic stability under convergence.","lead":"This paper introduces a new distance between dynamical systems, based on comparing how all points in a space move relative to each other over time. It shows that stability and asymptotic stability survive limit transitions under this distance, and that stable systems are an exceptional subset of all systems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Definition 2.1 quantifies stability over all g∈G; for flows with G=R this makes the systems in Example 2.9 (and any contracting flow) unstable, so the example's stability claim is false.","rationale":"The reader's conditional verdict is justified by real peripheral flaws: the factor-of-two error in Corollary 4.5 (the theorem's own bound gives (π√n)/2, not π√n), the self-map issue in Theorem 2.19 for ε<1, and the unproved convergence in Example 2.9. My stress-test focuses on a more fundamental issue that the reader did not flag: Definition 2.1's quantifier over all g∈G makes two-sided stability the operative notion for flows. This is internally inconsistent with Example 2.9, which asserts asymptotic stability of systems whose backward trajectories escape to a positive distance. The central proofs of Theorems 2.5 and 2.8 are logically sound relative to the stated definition, so I do not recommend rejecting the paper, but the presentation must reconcile the stability definition with standard one-sided Lyapunov stability or the examples and motivation lose their force. Agreement is partial because I share the reader's sense that corrections are needed, but the load-bearing issue I identify is different from the common-radius assumption.","tokens_in":24021,"tokens_out":31723,"duration_ms":292378,"concrete_test":"Verify the stability claim of Example 2.9 against Definition 2.1 for a fixed N with G=R: set ε=A_N/2 and consider x=A_N/2^k for arbitrarily large k. Compute the backward flow of ẋ=v_N(x) and show that sup_{t<0} |φ_N(t,x)| ≥ A_N (it reaches x=A_N at finite negative time), so Definition 2.1 fails because |φ_N(t,x)-0| > ε for some t. Also test the simpler system ẋ=-x on R with the same definition; the check should show instability for any δ by taking x=δ/2 and t→∞ in the negative direction. If the authors intend one-sided stability, the definition should quantify over g∈[0,∞) or over a semigroup, and the examples and theorems should be restated accordingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition 2.1 requires d_X(gx,gx_0)<ε for all g∈G. For a continuous flow with G=R this is a two-sided stability requirement: the inequality must hold for arbitrarily negative times as well as positive ones. The linear system ẋ=-Ax (A>0) fails this definition because its backward trajectories escape to infinity, so the 'classical' stable systems that motivate the paper are outside the stated class. Example 2.9 explicitly invokes G=R and claims that each X_N, with v_N=-A_N x on [-A_N,A_N], is asymptotically stable. Under Definition 2.1 this is false: fix N and take ε=A_N/2. For any δ>0 choose x∈(0,min{δ,A_N}); the backward trajectory of ẋ=v_N(x) reaches distance A_N from 0 in finite time, so sup_{g∈G} d(gx,0) ≥ A_N > ε, violating stability regardless of δ. Thus Example 2.9 does not demonstrate what it claims under the paper's own definition. The theorem proofs use the stated definition and remain internally valid, but the advertised connection to classical Lyapunov stability of ODE flows is broken unless G is restricted to a semigroup or the stability quantifier is restricted to forward times. This is the most load-bearing concern because it affects the interpretation and examples of the central stability theory, not merely a peripheral calculation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a categorical Gromov–Hausdorff distance for F-spaces, i.e. sets equipped with a family of generalized pseudometrics and marked points, and applies it to pointed dynamical systems by associating to each system the family of pseudometrics d_g(x,x') = d(gx,gx'). The main results are: preservation of Lyapunov stability under pointed continuous Gromov–Hausdorff convergence (Theorem 2.5); preservation of quasi-asymptotic stability under a common radius-of-attraction assumption (Theorems 2.8 and 2.14, generalized in Theorem 2.25); closedness and nowhere density of the stable and asymptotically stable classes (Theorems 2.19–2.20); a 1-Lipschitz property of the Hausdorff map for F-spaces and for continuous compact dynamical systems (Theorems 3.6 and 3.13); and a modified trajectory-aware distance for which the distance between ergodic torus translations is computed (Theorem 4.4 and Corollary 4.5). The authors argue that the ordinary Gromov–Hausdorff distance cannot distinguish qualitative dynamical differences such as ergodicity, and that the new distances address this limitation.","tokens_in":24329,"tokens_out":12315,"duration_ms":112640,"significance":"Assuming the repairs described below, this is a genuinely useful framework. The distance satisfies the triangle inequality, the main stability-preservation theorems are non-trivial and internally coherent, and the Hausdorff-map estimates extend known results to F-spaces. The torus calculation, once the factor error is corrected, provides a clean example in which the modified distance distinguishes ergodic translations, and the paper makes its hypotheses explicit, in particular the common radius of attraction. The self-contained categorical setup, the explicit treatment of morphism classes, and the presence of counterexamples to sharpness are notable strengths. The main limitation is the mismatch between Definition 2.1 and classical Lyapunov stability for flows, which currently undermines the motivating examples rather than the internal logic of the stability proofs.","major_comments":[{"comment":"Definition 2.1 quantifies stability over all g in G. When G=R and the action is a flow, this includes arbitrarily negative times. Consequently the linear system x' = -A_N x, which is classically asymptotically stable in forward time, is not stable in the sense of Definition 2.1: for fixed N, epsilon = A_N/2 and any delta>0, a point x in (0,min{delta,A_N}) satisfies d(x,0)<delta, but the backward trajectory reaches distance A_N from 0 in finite time, so sup_{g in R} d(gx,0) >= A_N > epsilon. Thus Example 2.9 does not exhibit a sequence of asymptotically stable systems under the paper's own definition, and its advertised conclusion that the common-radius condition in Theorem 2.8 cannot be dropped is not established. The proof of Theorem 2.8 itself is unaffected, but the interpretation of the stability theorems as statements about classical Lyapunov stability of ODE flows requires either restricting G to a forward semigroup in Definition 2.1 or replacing Example 2.9 with an example that satisfies the definition as written.","section":"Section 2.1, Definition 2.1 and Example 2.9"},{"comment":"The constructed pointed system I_epsilon = ([0,epsilon],0) is used for arbitrary epsilon>0, but its action sends (g,x), for g != e and x != 0, to the point 1, which belongs to [0,epsilon] only when epsilon >= 1. For epsilon < 1 the map is not a self-map of I_epsilon, so the construction is invalid on neighborhoods of radius smaller than 1. The proof can likely be repaired by sending those points to epsilon rather than 1, but as written the nowhere-density argument does not cover all neighborhoods.","section":"Section 2.3, Theorem 2.19"},{"comment":"Theorem 4.4 concludes bd^G_GH,K(X,Y) = (1/2) max{diam(X),diam(Y)}. Since the flat n-torus has diameter pi*sqrt(n), Corollary 4.5 must conclude pi*sqrt(n)/2. The displayed equality in the proof of the corollary omits the factor 1/2 and therefore states a value twice the one forced by Theorem 4.4.","section":"Section 4, Corollary 4.5"}],"minor_comments":[{"comment":"In the definition of Mor_F, the condition is written as f(x_j)=f(y_j); it should be f(x_j)=y_j, since y_j is the marked point in Y, not an element of X.","section":"Definition 1.2"},{"comment":"The assertion that dis_{d_{X,f},d_{Y,f}}(f)=0 appears false as written: the graph distortion is sup_{x,x'} ||x-x'| - |f(f(x))-f(f(x'))||, which is not generally zero for an arbitrary bijection f. If a different map or metric was intended, the notation should be clarified.","section":"Example 1.20"},{"comment":"The proof begins 'Since the dynamical system X is continuous', but Definition 3.9 only introduces uniformly continuous maps; spelling out that uniform continuity gives the closure identity phi(g, closure(A)) subset closure(phi(g,A)) would improve readability.","section":"Section 3.2, Theorem 3.13"},{"comment":"The notation cdis^G and [codis^G] is introduced with an odd bracket and is not consistently defined; the definitions should be displayed cleanly, and the bracket should be removed.","section":"Section 4, Theorem 4.4"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the scope of the journal and contains several internally sound theorems. The three load-bearing issues are all fixable in principle: the mismatch between Definition 2.1 and the ODE examples, the invalid target point in Theorem 2.19's construction, and the factor error in Corollary 4.5. I recommend major revision rather than rejection, but the examples and the final torus value must be corrected before the advertised claims are reliable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know upfront. The F-space construction and the categorical Gromov–Hausdorff distance are a real extension of Bogaty–Tuzhilin, and the stability-preservation theorems (2.5, 2.8, 2.14, 2.25) are internally consistent and look provable as stated. But the paper's advertised link to classical Lyapunov stability for flows is broken: Definition 2.1 quantifies over all g ∈ G, so for G = R it demands stability under backward time as well as forward time. A contracting flow ẋ = -Ax is unstable by that definition. Example 2.9 claims exactly such systems are asymptotically stable, so it is false under the paper's own definition. The theorems survive—they are about this two-sided notion—but the motivation and the main worked example are not measuring what the introduction says they measure. This needs to be fixed by restricting G to a semigroup or by changing the stability quantifier, and all examples must be checked against that.\n\nWhat is genuinely new: F-spaces with families of pseudometrics and marked points; the categorical GH distance; transfer of Lyapunov stability and asymptotic stability under pointed continuous GH convergence with a common radius of attraction; nowhere-density of stable systems; the Hausdorff map 1-Lipschitz extension; and the new trajectory-aware distance bdG that distinguishes torus translations. The Hausdorff map results look clean, and Lemma 2.18's l1-product argument is neat.\n\nBeyond the definition mismatch, three concrete errors. Corollary 4.5 states π√n, but Theorem 4.4 gives half the maximum diameter, which is π√n/2. In Theorem 2.19, the action on I_ε sends nonzero points to 1, which lies outside [0,ε] when ε < 1, so the nowhere-density construction needs a fix (sending to ε instead). Example 2.9's convergence via identity maps is asserted rather than proved, and with the stability definition issue the example cannot stand as written. None of these kill the central theorems, but they need correcting before acceptance.\n\nThe citation pattern looks appropriate; the novelty claims against earlier GH-for-dynamical-systems work are plausible. Who is this for? People working on GH distances for dynamical systems will find a useful language and some transfer theorems. As it stands I would not cite it without caveats, but it deserves a serious referee: the core ideas are sound enough to warrant a revision cycle, and the errors are local. Recommendation: send to peer review, asking for correction of the stability-definition/example issue, the factor of two, and the I_epsilon construction.","headline":"Useful F-space machinery and mostly sound stability-transfer theorems, but the paper's two-sided stability definition disqualifies its own contracting-flow examples, and there are a few concrete numerical/construction errors to fix.","tokens_in":24811,"tokens_out":5816,"would_cite":false,"duration_ms":56371,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C23","37C75"],"pacs":[],"model":"deepseek-v4-flash","headline":"Lyapunov stability passes to Gromov–Hausdorff limits under a new pointed continuous distance.","keywords":["Gromov-Hausdorff distance","Lyapunov stability","asymptotic stability","F-space","pointed dynamical systems","continuous Gromov-Hausdorff distance","Hausdorff map","torus translations"],"falsifier":"A concrete counterexample would be a sequence of pointed systems, each Lyapunov stable, that converges to a non-stable system in the pointed continuous Gromov–Hausdorff distance $d^G_{\\mathrm{GH},p,c}$; Theorem 2.5 says this is impossible, so exhibiting one would disprove the paper's main claim. The paper's Example 2.6 shows only that the non-continuous pointed distance allows this failure, which is why the continuity condition is part of the theorem.","tokens_in":23836,"feed_emoji":"📐","tokens_out":6414,"duration_ms":57266,"temperature":0.7,"pith_summary":"The paper introduces a new way to measure how close two dynamical systems are: it views each system as an F-space, a set carrying a family of time-shifted pseudometrics and, for pointed systems, a marked point, and defines a Gromov–Hausdorff distance on these objects. The central result is that Lyapunov stability survives limit transitions: if a sequence of Lyapunov stable pointed systems converges to a system under the pointed continuous version of this distance, then the limit is Lyapunov stable. Asymptotic stability is also preserved, provided all approximating systems share the same radius of attraction; the paper gives an example showing this radius condition cannot be dropped. A second, modified distance takes into account not only how distances between points change over time but also the geometry of whole trajectories, and it yields the exact distance between non-resonant torus translations. This matters because local stability properties could previously be lost under coarse Gromov–Hausdorff approximations; the new distance keeps the topology of orbits intact.","feed_headline":"Lyapunov stability survives Gromov-Hausdorff limits","feed_subtitle":"A new pointed distance preserves local stability of dynamical systems; asymptotic stability needs one shared radius.","key_machinery":"The central object is the F-space: a set equipped with a family of generalized pseudometrics indexed by some set $I$ and, in the pointed case, a distinguished marked point. Each dynamical system is turned into an F-space by adding the time-shifted pseudometrics $d_{X,g}(x_1,x_2)=d_X(gx_1,gx_2)$ for every $g\\in G$, so the distance sees how the whole arrangement of points evolves at every moment of time. The Gromov–Hausdorff distance between two such F-spaces is half the infimum, over pairs of structure-preserving maps, of the largest distortion of distances and codistortion; the pointed continuous variant restricts both maps to be continuous and to send the marked point to the marked point. The proofs work through distortion inequalities: if an approximating system has small distortion relative to the limit, then a ball around the marked point in the limit maps into a ball in the approximating system, and the stability estimate transfers back with an error controlled by the distortion. The common radius of attraction $\\delta_0$ enters as the uniform scale on which this transfer works.","core_discovery":"On the paper's own terms, the discovery is a pair of preservation theorems for local stability under a carefully chosen pointed continuous Gromov–Hausdorff distance $d^G_{\\mathrm{GH},p,c}$ between dynamical systems. Theorem 2.5 states that if a sequence of Lyapunov stable pointed dynamical systems converges in this distance to a pointed dynamical system, then the limit system is Lyapunov stable. Theorem 2.8 states that if a sequence of quasi-asymptotically stable pointed systems shares a common radius of attraction $\\delta_0>0$ and converges in the (not necessarily continuous) pointed Gromov–Hausdorff distance, then the limit is quasi-asymptotically stable with the same radius $\\delta_0$; combining the two gives preservation of asymptotic stability under the pointed continuous distance. The paper also proves that the stable systems form a closed, nowhere dense subset of the space of compact dynamical systems, while asymptotically stable systems form a nowhere dense $F_\\sigma$-set, and it shows that the Hausdorff map sending a space to its space of closed bounded subsets is 1-Lipschitz for F-spaces and for uniformly continuous dynamical systems with bounded action.","pith_inferences":["By the same distortion-transfer mechanism, one might expect Theorem 2.5 to extend to parametrized families of systems, such as random or time-dependent systems, because the proof only uses continuity of the maps and a Lipschitz transfer of balls; the paper does not state this.","The modified distance $\\hat{d}^G_{\\mathrm{GH},K}$ records whole-trajectory geometry, so it should be able to separate ergodic rotations with different rotation vectors beyond the non-resonant case; the paper only computes the non-resonant pair.","The nowhere-density result suggests that Lyapunov stability is fragile in a topological sense: in the Baire-category sense, most compact dynamical systems are unstable; quantifying how often stability appears in concrete parameter families would be a natural next step.","Example 2.6 indicates that continuity of the approximating maps is essential: the non-continuous pointed distance allows a sequence of stable systems to converge to an unstable limit, so any numerical or applied use of the distance must check that the identified correspondences are continuous."],"forward_implications":["If the paper is right, local stability properties can be studied through pointed continuous Gromov–Hausdorff limits: a converging sequence of stable systems cannot suddenly become unstable in the limit.","Asymptotic stability is stable under limits only when the approximating systems attract a common-size neighborhood of the base point; Example 2.9 shows that radii shrinking to zero can destroy asymptotic stability even when the limit is stable.","The class of Lyapunov stable compact dynamical systems is closed and nowhere dense, so stable systems form a topologically small subset of the space of all compact systems.","The Hausdorff map is 1-Lipschitz in this theory, meaning that passing from a system to its system of closed bounded subsets does not increase the Gromov–Hausdorff distance between systems.","The modified trajectory-based distance distinguishes torus translations: for a non-resonant pair $\\omega,\\omega'$, the distance equals $\\pi\\sqrt{n}$, which the primary distance cannot detect because it is blind to the geometry of whole orbits."],"supporting_citations":[{"why":"Introduces the Hausdorff distance between sets, the root concept that the paper generalizes to F-spaces and dynamical systems.","marker":"[1]"},{"why":"Supplies standard facts about the Gromov–Hausdorff distance and the Hausdorff metric used throughout the paper.","marker":"[4]"},{"why":"Provides the continuous Gromov–Hausdorff distance whose pointed version the paper adapts to local dynamics.","marker":"[8]"},{"why":"Surveys existing Gromov–Hausdorff theory for dynamical systems, setting the context for the new distance and its emphasis on local behavior.","marker":"[9]"},{"why":"Gives the 1-Lipschitz property of the Hausdorff map for compact metric spaces that Theorem 3.6 extends to F-spaces and dynamical systems.","marker":"[15]"},{"why":"Supplies the theorem that a torus translation is ergodic if and only if its frequency vector is non-resonant, used in Corollary 4.5 for the exact distance computation.","marker":"[16]"}],"fun_headline_variants":["Stability preserved in pointed Gromov-Hausdorff limits","Lyapunov and asymptotic stability survive GH limits","Stable systems are closed and rare under GH distance","Gromov-Hausdorff distance keeps local stability intact","Asymptotic stability persists under GH limits with shared radius"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The asymptotic-stability result depends on all approximating systems pulling an entire neighborhood of the marked point toward it with one common positive radius; without that shared radius, a limit can end up stable but not asymptotically stable, as Example 2.9 shows.","fun_headline_variants_meta":{"raw":{"variants":["Stability preserved in pointed Gromov-Hausdorff limits","Lyapunov and asymptotic stability survive GH limits","Stable systems are closed and rare under GH distance","Gromov-Hausdorff distance keeps local stability intact","Asymptotic stability persists under GH limits with shared radius"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000769,"raw_usage":{"total_tokens":3413,"prompt_tokens":960,"completion_tokens":2453,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":576,"completion_tokens_details":{"reasoning_tokens":2372}},"tokens_in":576,"tokens_out":2453,"duration_ms":14769,"temperature":1.0,"reasoning_tokens":2372,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:57:13.650253+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete counterexample would be a sequence of pointed systems, each Lyapunov stable, that converges to a non-stable system in the pointed continuous Gromov–Hausdorff distance $d^G_{\\mathrm{GH},p,c}$; Theorem 2.5 says this is impossible, so exhibiting one would disprove the paper's main claim. The paper's Example 2.6 shows only that the non-continuous pointed distance allows this failure, which is why the continuity condition is part of the theorem.","supporting_citations":[{"cited_title":"Hausdorff.Grundz¨ uge der Mengenlehre, Leipzig, Veit, 1914.[reprinted by Chelsea],1949","cited_arxiv_id":null,"evidence_quote":"Introduces the Hausdorff distance between sets, the root concept that the paper generalizes to F-spaces and dynamical systems."},{"cited_title":"Burago, Yu","cited_arxiv_id":null,"evidence_quote":"Supplies standard facts about the Gromov–Hausdorff distance and the Hausdorff metric used throughout the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the continuous Gromov–Hausdorff distance whose pointed version the paper adapts to local dynamics."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Surveys existing Gromov–Hausdorff theory for dynamical systems, setting the context for the new distance and its emphasis on local behavior."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the 1-Lipschitz property of the Hausdorff map for compact metric spaces that Theorem 3.6 extends to F-spaces and dynamical systems."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the theorem that a torus translation is ergodic if and only if its frequency vector is non-resonant, used in Corollary 4.5 for the exact distance computation."}],"review_version":1}