{"id":"3aa7012b-bf1c-469b-865f-07c25936741b","arxiv_id":"1908.03562","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Given a tautological control system and a submersion, the paper gives sufficient conditions under which a lifted control system exists and reachability transfers from the base to the lift.","lead":"Control systems describe how points move on a space. This paper shows when a control system on one space can be lifted to a larger space through a smooth map, and when reachability of the original system carries over to the lifted one.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3 inherits global-in-time from Grasse's C^1 lifting result without proof in the tautological/etale setting; if that transfer fails, Proposition 5 cannot lift reachability.","rationale":"The reader's weakest assumption identifies the same load-bearing step: Theorem 3 applies Proposition 5 only after asserting, without proof, that the morphism constructed in Theorem 1 is global in time because Phi is proper, citing Corollary 1 of [7]. This is the point where the proof crosses from ordinary C^1 control systems to tautological systems with etale sheaf trajectories, and the paper does not justify that crossing. I considered other potential objections and found them less damaging: the construction in Theorem 1 can likely be repaired by using a smooth horizontal distribution to obtain the required continuous linear operator; the claim that a piecewise smooth path in a connected fiber is a trajectory of the pwc vertical subsystem is plausible via extension of the tangent field along the path; and Proposition 5, once global-in-time is available, can be proved by transporting the base trajectory's flow backward from the target fiber to the initial fiber and using the fiber reachability set. Thus the unresolved transfer of global-in-time is the single most load-bearing concern. It is a missing proof rather than a demonstrated falsehood, and the compactness inherent in proper submersions makes the intended argument likely to succeed, so the appropriate verdict remains CONDITIONAL rather than REJECT.","tokens_in":16441,"tokens_out":26755,"duration_ms":333498,"concrete_test":"Take a globally generated tautological system H with a generator Y whose flow is not complete, a proper submersion Phi with compact connected fibers, and the horizontal lift Phi#Y constructed in Theorem 1. Verify analytically that for every x, the maximal interval J_{Phi#Y}(s,x) equals J_Y(s,Phi(x)), using compactness of Phi^{-1}(gamma([0,T])) for every compact base trajectory segment. Then re-derive the same statement for C^nu etale trajectories by checking that along each constant piece of a pwc etale open-loop, the trajectory satisfies the same ODE and the compactness argument applies. If the equality can be shown, the transfer is valid; if a counterexample with a proper submersion and a Phi-related C^nu vector field with unequal maximal intervals exists, Theorem 3's proof collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 3) rests on the assertion in its proof that properness of Phi makes the trajectory-preserving morphism from Theorem 1 global in time, citing Corollary 1 of [7]. That corollary is proved for ordinary C^1 control systems, whereas Definition 5 and Proposition 5 are stated for C^nu tautological systems with etale sheaf trajectories. The paper gives no argument that the maximal-interval equality J_{Phi#Y}(s,x)=J_Y(s,Phi(x)) holds for etale trajectories under proper Phi, nor does it show that Grasse's hypotheses transport through Propositions 3-4 and 7-8. Without global-in-time, the lift of a piecewise constant etale trajectory of H starting from x0 may fail to exist on the full time interval, so the endpoint in the fiber over the target point is not reached and Proposition 5 cannot be applied. Proposition 5's proof is also omitted ('analogous to Theorem 5 in [7]'), so the key reachability implication is not independently established. The gap is not a demonstrated contradiction but a missing proof at the point where the main theorem's conclusion depends on it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies morphisms of tautological control systems in the framework introduced by Lewis. Given a C^ν tautological control system H on a manifold N and a map Φ : M → N, the author asks (i) when there exists a tautological control system G on M with a trajectory-preserving morphism (Φ, Φ#) from G to H, and (ii) when reachability of H implies reachability of G. The main result, Theorem 3, asserts that if H is globally generated by linearly independent vector fields, Φ is a proper submersion, y0 ∈ Φ(M), Φ(M) is reachable from y0 by piecewise constant etale open-loop trajectories, and the fiber Φ^{-1}(y0) is connected, then there exists a tautological control system G on M, with a trajectory-preserving morphism, that is reachable from every point of the fiber. Section 3 also contains a real-analytic existence theorem (Theorem 2), a small-time local controllability result (Proposition 6), and a discussion of the relation between lifting and morphisms (Propositions 7–8). Section 4 applies these ideas to second-order type control systems.","tokens_in":16660,"tokens_out":9322,"duration_ms":86441,"significance":"If the main results are correct, the paper would extend Grasse's lifting theorem for ordinary C^1 control systems to tautological control systems with sheaf-theoretic and etale trajectories, while also addressing global constructions and some real-analytic cases. The framework is potentially valuable because it distinguishes local and global issues and avoids a fixed parametrisation by controls. The paper is clearly organized and honestly states limitations in Remarks 3 and 5. However, the proof of the central reachability theorem depends on several unproved or insufficiently justified assertions, including an omitted proof of Proposition 5, an unjustified transfer of a global-in-time property from ordinary control systems, and a construction of the morphism in Theorem 1 that may not satisfy the linearity requirement in Definition 3. These gaps must be repaired before the main claims can be accepted.","major_comments":[{"comment":"The proof of Proposition 5 is omitted entirely, with the sentence 'The proof is analogous to the proof for Theorem 5 in [7] so we omit it here.' This proposition is the key step that turns a trajectory-preserving morphism into a reachability implication for the lifted system, and Theorem 3 invokes it directly. The tautological/etale setting differs from the ordinary C^1 setting of Grasse's Theorem 5, so an analogy is not a proof. A complete proof must be supplied, showing in particular how the reachability of H by the piecewise constant etale open-loop subsystem transfers to reachability of G under the stated assumptions.","section":"Section 3.2, Proposition 5"},{"comment":"The proof states: 'For Φ is a proper mapping, by Corollary 1 in [7], we know the trajectory-preserving morphism from G to H is global in time.' Corollary 1 of Grasse is proved for ordinary C^1 control systems, whereas Definition 5 and Proposition 5 are formulated for C^ν tautological systems with etale sheaf trajectories. No argument is given that the maximal-interval equality J_{Φ#(Y)}(s,x)=J_Y(s,Φ(x)) holds for etale trajectories under properness of Φ, and no bridge is provided through Propositions 3–4 and 7–8. Without global-in-time, Proposition 5 cannot be applied, so the reachability conclusion of Theorem 3 is not established.","section":"Section 3.2, proof of Theorem 3"},{"comment":"The construction of Φ# defines, for each Y ∈ G(N), a vector field X by patching local lifts with a partition of unity. This does not demonstrate the linearity of the resulting map Y ↦ Φ#(Y) required in Definition 3(ii)(a), which calls for a family L_V of continuous linear mappings. Moreover, the proof only constructs Φ# for global sections in G(N), not for sections over arbitrary open V ⊆ N, and no extension to a compatible family L_V is provided. Since Theorem 3 depends on Theorem 1 for the existence of a trajectory-preserving morphism, this gap is load-bearing.","section":"Section 3.1, Theorem 1"},{"comment":"In the proof of Proposition 6, the sentence 'By the mapping Φ#, we know that x is reachable from some point \\bar{x} ∈ Φ^{-1}(y0) in time at most t' lifts a trajectory of H to a trajectory of G without invoking global-in-time or Proposition 5. This step is not justified; reachability of H does not automatically imply reachability of G unless the trajectory-preserving morphism is global in time and the fiber reachability assumption holds. Proposition 6 is a stated result, so its proof must be completed or its hypotheses strengthened.","section":"Section 3.2, Proposition 6"}],"minor_comments":[{"comment":"The manuscript contains frequent typographical errors and broken hyphenation (e.g., 'sy s-tems' in the abstract, 'lea ds' in the abstract). A careful proofreading pass is needed.","section":"Abstract and general"},{"comment":"The statement 'Let Φ : M → N be a real analytic vector bundle' is unclear: a vector bundle is typically a triple (E, π, M), not a mapping between two manifolds. Please clarify whether Φ is a vector bundle projection or a submersion with additional vector-bundle structure.","section":"Theorem 2"},{"comment":"In the proof of Proposition 7, the well-definedness of Φ#_V on the subset {F^u_2|V : u ∈ C_2} does not by itself give a continuous linear map on Γ^ν(TV). The argument needs to construct an extension to the whole space and verify the linearity and compatibility conditions of Definition 3(ii).","section":"Section 3.3, Proposition 7"},{"comment":"The hypothesis 'T(φ(P)) ⊆ R_{HΣ2}((z0,0), G_{HΣ2}, pwc)' should be stated more precisely; as written it is ambiguous whether the reachable set is taken in the tangent bundle TQ or in the image φ(P).","section":"Section 4, Theorem 5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript appears to be an early preprint with several incomplete proofs. The main theorem's reachability claim rests on Proposition 5 and on a global-in-time assertion imported from Grasse's ordinary control system setting without proof; these are not routine gaps and require substantial additional work. The editors may also wish to verify that the literature on tautological control systems and lifting is represented accurately, particularly the claims about the transpose mapping (dΦ)^T in Remark 1 and the use of Corollary 1 in [7]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper extends Grasse's lifting results to Lewis's tautological control systems, and that is a real contribution. The sheaf framework is a natural fit for the local-global distinction that Grasse's local construction leaves unresolved, and the real-analytic version via connections (Theorem 2) is genuinely new. The second-order type application is also a real plus: the control count drops from 2n−2m+k to n−m+k, and the lifted system is global rather than local. If the main theorem holds, it is a useful addition to the tautological control systems literature, though not a field-shifting one.\n\nThe main theorem is plausible, but the proof as written has three soft spots. First, Proposition 5 is the load-bearing reachability transfer, and its proof is omitted with a reference to an analogous result in [7]. In a paper whose central theorem depends on that proposition, an omitted proof is a real gap. Second, Theorem 3 imports 'global in time' from Corollary 1 of [7], which is proved for ordinary C^1 control systems. No argument shows that the maximal-interval equality J_{Φ#(Y)}(s,x)=J_Y(s,Φ(x)) carries over to tautological systems with etale sheaf trajectories under properness. This is exactly the step where the reachability transfer could fail, so it cannot be taken on faith. Third, the construction of Φ# in Theorem 1 uses a partition of unity chosen anew for each Y. As written, that does not obviously give the linear map required by Definition 3, nor even a well-defined map on restricted sections. This particular issue is fixable by fixing a splitting of TΦ and a fixed partition of unity, but the paper does not do that.\n\nNone of these are demonstrated contradictions; they are missing arguments at critical points. The framework and the statements of the theorems are coherent, and the author clearly knows the literature. Proposition 7 is essentially a definitional equivalence, but it is harmless. The examples, especially the Möbius band, are illustrative.\n\nWho gets value from this? Researchers working on tautological control systems, lifting theory, or real-analytic geometric control. A serious referee should engage with it, but the verdict should be 'revise' with a demand for a full proof of Proposition 5, a proof of the global-in-time transfer, and a repaired construction of Φ#. If those are supplied, the paper would be a solid, if niche, addition. As it stands, the main theorem is not fully proved.","headline":"A plausible extension of Grasse-style lifting to tautological control systems, with genuinely new real-analytic and second-order results, but the main reachability theorem depends on an omitted proof and an unproven transfer of global-in-time from ordinary systems.","tokens_in":17142,"tokens_out":3153,"would_cite":false,"duration_ms":36708,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B03","93B17","93C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes sufficient conditions under which the reachability of a tautological control system on a base manifold is inherited by a lifted system on another manifold, via trajectory-preserving morphisms along proper…","keywords":["Tautological control system","Morphism","Reachability","Trajectory lifting","Etale open-loop system","Second-order type","Sheaf theory","Proper submersion"],"falsifier":"Build an explicit example of a proper submersion $\\Phi: M \\to N$ and a tautological control system $H$ on $N$, globally generated by independent vector fields, for which the trajectory-preserving lift $\\Phi^\\#(Y)$ of some $Y$ has an integral curve that dies before its projection through $\\Phi(x)$. If such an example exists, Theorem 3 fails, because its proof needs the morphism to be global in time; if none exists, proving the equal-maximal-interval property would complete the argument.","tokens_in":16250,"feed_emoji":"⚙️","tokens_out":13884,"duration_ms":129022,"temperature":0.7,"pith_summary":"Tautological control systems describe dynamics as sheaves of vector fields rather than as equations with a chosen control parameterization, so local and global questions can be separated cleanly. This paper asks when such a system on a base manifold can be pulled back to another manifold through a smooth map $\\Phi: M \\to N$, preserving trajectories, and when reachability of the base system forces reachability of the lifted one. The main theorem says that if the base system is globally generated by pointwise linearly independent vector fields, $\\Phi$ is a proper submersion (smooth map with surjective derivative and compact fibers), the base is reachable from a point $y_0$ by piecewise constant etale open-loop trajectories, and the fiber $\\Phi^{-1}(y_0)$ is connected, then a trajectory-preserving lifted system exists and is reachable from every point of that fiber. A sympathetic reader should care because this gives a global, control-parameter-free analogue of classical trajectory lifting, with smooth and real-analytic versions, and it improves the input count for second-order mechanical systems.","feed_headline":"Reachability lifts through proper submersions with connected fibers","feed_subtitle":"Reachability of the base passes to every point of a connected fiber when the morphism is global in time.","key_machinery":"The central object is the trajectory-preserving morphism $(\\Phi, \\Phi^\\#)$: a map $\\Phi \\in C^r(M,N)$ together with continuous linear maps $\\Phi^\\#_V$ that send every local section $Y$ of the base presheaf over an open set $V \\subseteq N$ to a vector field on $\\Phi^{-1}(V)$ satisfying $T_x\\Phi(\\Phi^\\#(Y)(x)) = Y(\\Phi(x))$ at each $x$. This tangent identity is exactly what makes integral curves of the lifted field project to integral curves of $Y$. The proof's second ingredient is the presheaf of vertical fields $\\mathcal{H}(U)=\\{X \\in \\Gamma^\\nu(TU) \\mid X(x) \\in \\ker T_x\\Phi\\}$; adding $\\mathcal{H}$ to the lifted presheaf lets trajectories run freely inside the fibers, so a connected fiber becomes a reachability set. The etale open-loop subsystem, built from locally integrable sections of the sheaf of time-varying vector fields, is the trajectory class in which reachability is measured.","core_discovery":"The core discovery is that reachability passes upward along a trajectory-preserving morphism when three conditions hold: the base system's reachable set covers the image of the map, the morphism is global in time, and each fiber is a reachability set. Theorem 3 realizes these conditions for proper submersions by adjoining vertical vector fields tangent to the fibers: connectedness of $\\Phi^{-1}(y_0)$ makes that fiber path connected, hence a reachability set for the vertical subsystem, while properness of $\\Phi$ is used to ensure the lifted morphism is global in time via a corollary imported from the classical lifting theory. With the fiber a reachability set and the base reachable from $y_0$, Proposition 5 yields reachability of the lifted system from every $x_0 \\in \\Phi^{-1}(y_0)$ by its piecewise constant etale open-loop subsystem. The same construction also yields small-time local controllability, and for second-order type systems the lifted system can be chosen within the same class with fewer inputs than the local classical construction.","pith_inferences":["One could replace the imported properness condition by an explicit 'equal maximal interval' hypothesis on the lifted morphism; the reachability argument would then go through unchanged and would no longer depend on the unproved transfer from $C^1$ systems.","The same construction suggests a recipe for observer or synchronization designs: whenever a projection has connected fibers and the base system is reachable, augmenting the lifted dynamics with arbitrary vertical fields should make the whole fiber reachable regardless of the control parametrization.","For real-analytic systems, the failure of global finite generation for the vertical distribution suggests that global reachability lifting may be controlled by a sheaf-cohomology obstruction; checking whether vanishing of the relevant cohomology restores the global result would be a natural next step."],"forward_implications":["Under the hypotheses of Theorem 3, from any single lift $x_0$ of a reachable base point $y_0$, the lifted system reaches every point of $M$; reachability is inherited fiber-by-fiber, not just pointwise.","When the vertical distribution is globally finitely generated (the smooth case), the lifted system can be chosen globally generated, so the result applies to ordinary control systems through the correspondence established in Propositions 7 and 8.","Small-time local controllability of the base system from $y_0$ lifts to small-time local controllability of the lifted system from every point of the connected fiber (Proposition 6).","For second-order type systems, the lifted system can be chosen second-order type with $n-m+k$ inputs (fiber dimension plus base controls) rather than $2n-2m+k$, and the construction gives a globally defined system rather than only a local one (Theorem 5 and Remark 7)."],"supporting_citations":[{"why":"Supplies the ordinary-control-system lifting theorem whose global-in-time corollary is used to apply Proposition 5 in Theorem 3.","marker":"[7]"},{"why":"Defines tautological control systems, presheaves of vector fields, trajectories, and trajectory-preserving morphisms, so all statements in the paper rest on it.","marker":"[8]"},{"why":"Supplies the sheaf of time-varying vector fields and the etale open-loop trajectory framework used for reachability by piecewise constant etale open-loop subsystems.","marker":"[11]"},{"why":"Establishes that smooth distributions are globally finitely generated, used to construct a globally generated presheaf spanning the kernel of the derivative of the submersion.","marker":"[12]"},{"why":"Provides vertical lifts and affine connection control systems used in the second-order type application and examples.","marker":"[14]"}],"fun_headline_variants":["Reachability lifts via proper submersions","Connected fibers make reachability lift","Lifting reachability through tautological morphisms","Morphisms transfer reachability in control systems","Reachability lifts along proper submersions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire reachability-lifting argument assumes without proof that a trajectory lifted through a proper submersion $\\Phi: M \\to N$ exists exactly as long as its projection, a 'global in time' fact imported from the classical setting of $C^1$ control systems and never proved for the sheaf-theoretic etale trajectories used here.","fun_headline_variants_meta":{"raw":{"variants":["Reachability lifts via proper submersions","Connected fibers make reachability lift","Lifting reachability through tautological morphisms","Morphisms transfer reachability in control systems","Reachability lifts along proper submersions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000373,"raw_usage":{"total_tokens":1967,"prompt_tokens":890,"completion_tokens":1077,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":506,"completion_tokens_details":{"reasoning_tokens":1009}},"tokens_in":506,"tokens_out":1077,"duration_ms":8406,"temperature":1.0,"reasoning_tokens":1009,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:23:36.297588+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Build an explicit example of a proper submersion $\\Phi: M \\to N$ and a tautological control system $H$ on $N$, globally generated by independent vector fields, for which the trajectory-preserving lift $\\Phi^\\#(Y)$ of some $Y$ has an integral curve that dies before its projection through $\\Phi(x)$. If such an example exists, Theorem 3 fails, because its proof needs the morphism to be global in time; if none exists, proving the equal-maximal-interval property would complete the argument.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the ordinary-control-system lifting theorem whose global-in-time corollary is used to apply Proposition 5 in Theorem 3."},{"cited_title":"Springer B riefs in Electrical and Computer Engineering-Control, Automation and Robotics","cited_arxiv_id":null,"evidence_quote":"Defines tautological control systems, presheaves of vector fields, trajectories, and trajectory-preserving morphisms, so all statements in the paper rest on it."},{"cited_title":"Springer Briefs in Mathematics","cited_arxiv_id":null,"evidence_quote":"Supplies the sheaf of time-varying vector fields and the etale open-loop trajectory framework used for reachability by piecewise constant etale open-loop subsystems."},{"cited_title":"Smooth dist ributions are finitely generated","cited_arxiv_id":null,"evidence_quote":"Establishes that smooth distributions are globally finitely generated, used to construct a globally generated presheaf spanning the kernel of the derivative of the submersion."},{"cited_title":"Modeling, Analysis and Design for Simple Mechanical Control Systems","cited_arxiv_id":null,"evidence_quote":"Provides vertical lifts and affine connection control systems used in the second-order type application and examples."}],"review_version":1}