{"id":"1d5f5f01-c0b7-4622-9d71-d0adb255e775","arxiv_id":"1908.02905","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For polynomial (and immersible analytic) control systems, the set of accessibility singular points is algebraic, and the paper provides algorithms to compute it and the exact accessibility index.","lead":"This paper gives constructive algebraic-geometry algorithms that compute, for polynomial control systems, the exact depth of Lie brackets needed to decide accessibility, along with the full set of singular points. A companion result extends the method to analytic systems that can be immersed into polynomial form.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1's proof rests on a false finite-dimensionality assertion; the required generic constant-rank lemma is unstated, and the algorithms' correctness depends on it.","rationale":"The reader and I converge on the same weakest point. I read the central claim charitably: the algebraic-geometric construction (finite stabilization of the ideal chain, invariance under Lie derivatives, maximal zero-measure invariant set) is internally coherent, and the examples are consistent with it. However, Theorem 3.1's proof contains an explicitly false sentence about V! being finite-dimensional, and the missing plateau lemma is exactly what connects 'generic rank stops growing' to 'pointwise accessibility is uniformly decided by depth n-1'. Because Algorithm 1's initialization and Theorem 4.6's use of a proper IM_q for q<n depend on that connection, this is a real correctness risk. The lemma is standard and very likely provable, so CONDITIONAL rather than REJECT is the right posture. I would not change the reader's verdict. Secondary issues (Theorem 4.10 being stated without proof, and the loose assertion in Theorem 4.6's proof that a proper ideal is zero-measure when the ideal could be the zero ideal) should be cleaned up but do not alter the main verdict.","tokens_in":16433,"tokens_out":27188,"duration_ms":321329,"concrete_test":"Run a computer-algebra search (e.g., Singular/Sage) over all polynomial control-affine systems on R^n, n≤4, with f and g_i of total degree ≤3: compute generic ranks of C_k for k=0,...,n-1 and the generic rank of the full Lie algebra generated by all brackets up to depth 2n, detected by stabilization of the ascending module chain. If any system satisfies generic rank C_{n-1}<n while the full generic rank is n, then Theorem 3.1 and the initialization of Algorithms 1-2 are false. If no counterexample appears, independently prove the omitted plateau lemma: equality of generic ranks of C_k and C_{k+1} implies C_{k+1} is generically contained in the C∞-span of C_k, using constant-rank splitting and the Jacobi identity.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing gap is in the proof of Theorem 3.1 (Section 3). It asserts \"Since V! is an n-dimensional vector space, therefore for some k* ≤ n-1...\", but V!, the Lie algebra of analytic vector fields on R^n, is infinite-dimensional. The argument that actually needs to be supplied is a constant-rank/plateau lemma: if the generic ranks of C_k and C_{k+1} coincide, then on a nonempty open set C_k is involutive and every ad_X h_i is a C∞-linear combination of the chosen generators h_1,...,h_q, so the entire filtration is generically constant. This lemma is true for analytic distributions but is neither stated nor proved; the written justification is factually incorrect. The conclusion \"if dim C_{n-1} ≠ n then the system is non-accessible from every x\" is load-bearing: it justifies Theorem 4.6's invocation of a proper IM_q for q<n and the initialization of Algorithms 1-2 at a low-depth matrix. If a plateau could later be followed by a rank jump, then a polynomial system could have generic rank C_{n-1}<n while the full distribution has generic rank n; Theorem 4.6 with q=n-1 would then yield V((0))=R^n rather than the true S∞, destroying the central construction. The algebraic parts (Lemma 4.1, Theorems 4.2, 4.5, 4.6) are coherent conditional on this lemma. Secondary gap: Theorem 4.10 is stated without proof, but it is less central.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies finite determination of accessibility and strong accessibility for polynomial control-affine systems and for analytic systems that are immersible into polynomial systems. It introduces the accessibility index r* as the maximal Lie-bracket depth needed to decide accessibility at every point, and the set S∞ of accessibility singular points. The central results are: S∞ is an algebraic set and equals the zero set of the smallest real radical ideal, containing the ideal of maximal minors of a bracket matrix, that is invariant under L_f, L_g1, ..., L_gm (Theorems 4.5 and 4.6); S∞ is the maximal zero-measure forward-invariant set of the system (Theorem 4.2); the accessibility index is finite and can be found by the invariant-ideal stabilization (Theorem 4.5(b)); and module-theoretic stabilization gives an upper bound (Theorems 4.9, 4.10). Section 5 extends the results to analytic systems via immersion. The paper also gives constructive algorithms and several worked examples, including a unicycle and a driven pendulum.","tokens_in":16690,"tokens_out":12652,"duration_ms":140731,"significance":"If the results are established, the paper gives a constructive and, in principle, implementable finite test for accessibility of polynomial control systems, along with an exact description of the singular set, going beyond the very large degree bounds in the earlier literature. The identification of S∞ as a maximal zero-measure invariant set and the invariant-ideal characterization are appealing and the worked examples demonstrate the computational idea. The paper does not rely on fitted parameters or circular benchmarks; the Sussmann-Jurdjevic rank condition is used externally. The main reservation is that the proof of the foundational generic-to-pointwise reduction, Theorem 3.1, contains a false finite-dimensionality assertion and omits the needed constant-rank lemma, and a secondary gap appears in the proof of Theorem 4.6. These issues appear repairable, but they are load-bearing for the main construction.","major_comments":[{"comment":"The proof asserts 'Since V! is an n-dimensional vector space', but V!, the Lie algebra of analytic vector fields on R^n, is infinite-dimensional, so the existence of a plateau k* ≤ n−1 cannot be concluded by finite-dimensionality of V!. More importantly, the step from equal generic ranks of C_k and C_{k+1} to membership ad_X h_i ∈ span_{C∞}{h_1,...,h_q} requires a constant-rank/plateau lemma: on a nonempty open set where C_k has constant rank equal to its generic rank, the new brackets lie in the C∞-span of the chosen generators, so the entire filtration is generically constant. This lemma is not stated or proved. The subsequent conclusion that dim C_{n−1} ≠ n implies non-accessibility from every point is load-bearing, since Theorem 4.6 and the initialization of Algorithms 1 and 2 rely on the existence of q < n with IM_q proper. The claim is plausible for analytic distributions, but the written proof is invalid as it stands and the needed lemma must be supplied.","section":"Section 3, proof of Theorem 3.1"},{"comment":"The proof states that because the ideal \\bar I_{M_q} is proper, its zero set V(\\bar I_{M_q}) is a zero-measure set. This implication is false in real algebraic geometry: the zero ideal is proper but V((0)) = R^n, which is not zero-measure. The intended argument works if \\bar I_{M_q} contains a nonzero polynomial, which is guaranteed by the standing assumption that the generic rank of C_q is full, since then at least one n×n minor is a nonzero polynomial. This additional justification should be supplied; without it, the proof of the inclusion V(\\bar I_{M_q}) ⊂ S∞ via Theorem 4.2 is incomplete.","section":"Section 4.1, proof of Theorem 4.6"}],"minor_comments":[{"comment":"Theorem 4.10 is stated without proof. Since it is the strong-accessibility analogue of Theorem 4.9, the omission is likely routine, but the paper should either provide the proof or explicitly state that it follows by replacing C#_k with C#_k^0 and the operators ad_f, ad_{g_i} with the appropriate strong-accessibility operators.","section":"Section 4.2, Theorem 4.10"},{"comment":"The proof says that without loss of generality T_i(x) = x_i for i ≤ n and that an injective immersion is an embedding. An injective immersion need not be an embedding in general, and the 'without loss' reduction needs justification. The argument can be reformulated using local coordinates and the fact that T-relatedness commutes with Lie brackets, so the claimed rank preservation is valid, but the current wording is imprecise.","section":"Section 5, proof of Theorem 5.1"},{"comment":"The constants a_n in the definition of β(x1) are not specified beyond convergence of the product. The example would be clearer if a concrete choice such as a_n = 2^n is given and the claim about the sets S_k is verified in a few lines.","section":"Example 4, Section 5"},{"comment":"There is a typo: 'We chack the invariance' should be 'We check the invariance'. Similar minor language issues occur elsewhere, e.g., 'some care should be payed' in Remark 1.","section":"Example 2, Section 4.1"},{"comment":"In line 7, the notation J ← J ∪ ⟨L_{X_j}(z_i)⟩ mixes generators and ideals; it would be clearer to write that the new generator L_{X_j}(z_i) is added to the ideal, or to define J as the ideal generated by all such derivatives.","section":"Algorithm 2, Section 4.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a plausible extension of the authors' earlier work, and the relationship to their discrete-time paper [11] is properly acknowledged. The main concern is the proof gap in Theorem 3.1, which is foundational for the algorithms; this is fixable with a standard constant-rank lemma, so I recommend major revision rather than rejection. I see no issue with the fit between the paper and the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a real contribution with a fixable proof bug. The paper gives constructive finite tests for exact accessibility index and characterizes S∞ as an algebraic set, which is genuinely new relative to the Gabrielov-Risler upper bounds. The algebraic machinery—Noetherian stabilization, invariant ideals, descending chains—is coherent, and the examples are worked in enough detail to be credible. The central characterizations (Theorems 4.2, 4.5, 4.6 and Algorithm 2) are plausible and mutually consistent.\n\nThe soft spot is Theorem 3.1. The proof literally says V! is an n-dimensional vector space; that is false, and the gap matters because the theorem is used to justify starting at q<n and to argue that generic accessibility implies accessibility outside a zero-measure set. The missing step is a constant-rank lemma: for analytic vector fields, if generic ranks of C_k and C_{k+1} coincide, then on a nonempty open set the extra brackets lie in the smooth span of the generators, so all higher levels have the same generic rank. That lemma is standard and true in this analytic setting, so I do not think the theorem is wrong; but as written the proof does not establish it. This is a repairable flaw, not a fatal one.\n\nOther issues are minor. Theorem 4.10 is asserted without proof; it mirrors Theorem 4.9, so it is acceptable as a stated claim, but should either be proved or explicitly labelled as a conjecture. The line that the strong accessibility index equals the accessibility index or one greater is stated without derivation; likely true but needs a sentence. The immersion extension in Theorem 5.1 assumes an injective immersion gives an embedding onto its image; that is not generally true, though the local statement is fine for the examples. The citation pattern looks fair: prior work on bounds is cited, and the self-citation [11] is relevant because it is the discrete-time analogue.\n\nWho is this for? People working on nonlinear controllability, singular distributions, or algebraic methods in control. The algorithms are useful for small polynomial systems; the real-radical step in Algorithm 1 is heavy, but Algorithm 2 avoids it.\n\nRecommendation: send to serious peer review. The result deserves referee time. Ask the referee to require a repaired proof of Theorem 3.1 using the constant-rank lemma, a proof or clear labelling for Theorem 4.10, and a short justification of the strong-accessibility index statement. After those revisions, I would accept.","headline":"A genuinely new finite test for exact accessibility index with a flawed but repairable proof of the key plateau theorem.","tokens_in":17278,"tokens_out":2651,"would_cite":true,"duration_ms":30772,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B05","93C10","14P10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for polynomial systems, the accessibility singular set $S_\\infty$ is an algebraic set — the maximal zero-measure forward-invariant set — and equals the zero set of the smallest real radical ideal, invariant under…","keywords":["accessibility","strong accessibility","singular points","polynomial systems","Lie brackets","accessibility index","real algebraic geometry","system immersion"],"falsifier":"For Theorem 3.1, search for an analytic distribution generated by $h_1,\\ldots,h_q$ whose pointwise span has constant rank $q$ but where $\\mathrm{ad}_X h_i$ is not in the smooth span of $h_1,\\ldots,h_q$ at some point; such an example would break the generic-to-pointwise reduction. For the polynomial main theorem, run Algorithm 1 on a polynomial system and compare its output $V(I)$ with a direct numerical evaluation of $\\dim C(x)$ on a fine grid: any mismatch between the set where $\\dim C(x)<n$ and the computed algebraic set would falsify Theorem 4.5.","tokens_in":16178,"feed_emoji":"⚙️","tokens_out":11752,"duration_ms":119369,"temperature":0.7,"pith_summary":"For a polynomial control system, this paper proves a finite test for accessibility: you never need to compute infinitely many Lie brackets. The set of states from which the system is not accessible is an algebraic set, and it is exactly the largest zero-measure set that the system's trajectories cannot escape. The paper identifies this set as the zero set of the smallest ideal, invariant under the system's vector fields, that contains the maximal minors of the matrices built from Lie brackets up to a certain depth. Because the polynomial ring is Noetherian, the chains of ideals and algebraic sets stabilize, so the accessibility index — the maximum bracket depth needed anywhere in the state space — is finite and can be computed by the proposed algorithms. The same results hold for strong accessibility, and for analytic systems that immerse into polynomial systems.","feed_headline":"A finite Lie-bracket depth settles accessibility everywhere","feed_subtitle":"For polynomial and immersible analytic systems, singular points form an algebraic set computed by invariant ideals.","key_machinery":"The central objects are the bracket matrices $M_k$ and the ideals $I_{M_k}$ generated by all $n\\times n$ minors. Each variety $V(I_{M_k})$ is exactly the order-$k$ singular set $S_k$, and because each $M_k$ is a submatrix of $M_{k+1}$, the ideals ascend while the algebraic sets descend. Noetherianity of the polynomial ring forces the chain to stabilize, and invariance of a real radical ideal under $L_f,L_{g_1},\\ldots,L_{g_m}$ characterizes forward-invariance of the corresponding algebraic set; combining these two facts yields $S_\\infty$ and $r^*$. The real radical operation reconnects an ideal to its real zero set, which is what guarantees that the computed algebraic set is exactly the non-accessible states rather than some complex variety.","core_discovery":"For a polynomial system of the form $\\dot x=f(x)+\\sum_{i=1}^m u_i g_i(x)$, the paper proves that the set $S_\\infty$ of accessibility singular points is the maximal zero-measure forward-invariant set of the system, and that it is the algebraic set $V(I)$ where $I$ is the smallest real radical ideal containing the ideal $I_{M_q}$ of all $n\\times n$ minors of the bracket matrix $M_q$ ($q<n$) and invariant under $L_f,L_{g_1},\\ldots,L_{g_m}$. The accessibility index $r^*$ is therefore the smallest $k$ for which the real radical of $I_{M_k}$ is invariant under those Lie derivatives; Algorithm 1 computes both $S_\\infty$ and $r^*$ once the generic rank is known. A cheaper variant, Theorem 4.6, shows that the zero set of the smallest invariant ideal containing $I_{M_q}$ already equals $S_\\infty$. For generically strongly accessible polynomial systems, $S^*_\\infty=S_\\infty$, so the same computation covers strong accessibility; for analytic systems immersible into polynomial systems, the singular set is pulled back through the immersion, making the accessibility index finite there as well.","pith_inferences":["The invariance-of-ideals criterion indicates a path toward certifying accessibility symbolically for larger systems: once ideal-membership computations are feasible, the same construction can output a polynomial certificate that every non-accessible state lies in a given algebraic set.","Since $S_\\infty$ is the maximal zero-measure forward-invariant set, motion-planning and global stabilization algorithms could treat $S_\\infty$ as an obstruction set and plan around it, analogous to singular configurations in non-holonomic robotics.","The equality $S^*_\\infty=S_\\infty$ for generically strongly accessible systems suggests that in polynomial systems, strong-accessibility failures do not have a separate geometric mechanism; controller synthesis needing strong accessibility can reuse the accessibility computation and only adjust the bracket depth by at most one.","If the constant-rank gap in Theorem 3.1 is closed, the same immersion-based pipeline would apply to a broad class of analytic systems with trigonometric and rational terms, making the algebraic methods a general substitute for pointwise rank computations."],"forward_implications":["Accessibility from every point of a polynomial system can be decided after computing Lie brackets only up to the finite depth $r^*$, and Algorithm 1 returns both $r^*$ and the whole singular set $S_\\infty$.","Because $S_\\infty$ is the maximal zero-measure forward-invariant set, any trajectory starting in $S_\\infty$ stays in $S_\\infty$; avoiding initialization on $S_\\infty$ becomes an explicit computational step rather than a pointwise search.","For generically strongly accessible polynomial systems, the singular sets for accessibility and strong accessibility coincide, so a single ideal computation serves both notions.","For analytic systems immersible into polynomial systems, the accessibility index is finite and $S_\\infty$ is obtained by intersecting the polynomial singular set with the image of the immersion; the unicycle and pendulum examples show this covers systems with trigonometric dynamics.","The algorithms give exact values or upper bounds that can be much smaller than earlier degree-of-nonholonomy estimates; in the worked example, depth 2 replaces a prior bound of 22."],"supporting_citations":[{"why":"Supplies the pointwise accessibility rank condition $\\dim C(x_0)=n$ that defines accessibility and drives Definitions 1-3.","marker":"[1]"},{"why":"Supplies the Noetherian and algebraic-geometry facts used to show each $S_k$ is algebraic and the chain stabilizes.","marker":"[3]"},{"why":"Provides the density lemma for piecewise constant inputs used in Lemma 4.4, plus earlier degree-of-nonholonomy bounds this paper improves.","marker":"[4]"},{"why":"Earlier planar degree-of-nonholonomy bound used as the comparison point in Example 2.","marker":"[5]"},{"why":"Noetherian module property used to stabilize the ascending chain of modules $C^\\#_k$ in Theorem 4.9.","marker":"[12]"},{"why":"Maximal integral manifold property used to show each singular leaf is forward-invariant in Theorem 4.2.","marker":"[14]"},{"why":"Immersion technique that extends the polynomial results to analytic systems in Section 5.","marker":"[19]"},{"why":"Real radical and ideal-variety correspondence used to pass between ideals and singular sets in Theorem 4.5.","marker":"[21]"}],"fun_headline_variants":["Finite bracket depth decides accessibility for all points","Accessibility singular points form an algebraic set","Invariant ideals compute all accessibility singularities","Algorithms give finite accessibility depth for systems","Algebraic geometry settles accessibility and strong accessibility"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing step is Theorem 3.1's reduction from generic accessibility to accessibility at every point outside a zero-measure set; it assumes that once the rank of the analytic bracket filtration stops growing, the next brackets are smooth linear combinations of the previous ones, but the stated justification ('the Lie algebra of analytic vector fields is $n$-dimensional') is false because that Lie algebra is infinite-dimensional.","fun_headline_variants_meta":{"raw":{"variants":["Finite bracket depth decides accessibility for all points","Accessibility singular points form an algebraic set","Invariant ideals compute all accessibility singularities","Algorithms give finite accessibility depth for systems","Algebraic geometry settles accessibility and strong accessibility"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000815,"raw_usage":{"total_tokens":3550,"prompt_tokens":903,"completion_tokens":2647,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":519,"completion_tokens_details":{"reasoning_tokens":2580}},"tokens_in":519,"tokens_out":2647,"duration_ms":21130,"temperature":1.0,"reasoning_tokens":2580,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:31:32.758128+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For Theorem 3.1, search for an analytic distribution generated by $h_1,\\ldots,h_q$ whose pointwise span has constant rank $q$ but where $\\mathrm{ad}_X h_i$ is not in the smooth span of $h_1,\\ldots,h_q$ at some point; such an example would break the generic-to-pointwise reduction. For the polynomial main theorem, run Algorithm 1 on a polynomial system and compare its output $V(I)$ with a direct numerical evaluation of $\\dim C(x)$ on a fine grid: any mismatch between the set where $\\dim C(x)<n$ and the computed algebraic set would falsify Theorem 4.5.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the pointwise accessibility rank condition $\\dim C(x_0)=n$ that defines accessibility and drives Definitions 1-3."},{"cited_title":"Gabrielov, A","cited_arxiv_id":null,"evidence_quote":"Supplies the Noetherian and algebraic-geometry facts used to show each $S_k$ is algebraic and the chain stabilizes."},{"cited_title":"Gabrielov, Multiplicities of zeroes of polynomials on trajectories of polynomial vector fields and bounds on degree of nonholonomy, volume 2, Amer","cited_arxiv_id":null,"evidence_quote":"Provides the density lemma for piecewise constant inputs used in Lemma 4.4, plus earlier degree-of-nonholonomy bounds this paper improves."},{"cited_title":"Risler, A bound for the degree of nonholonomy in the plane, Theoretical Computer Science 157 (1996) 129–136","cited_arxiv_id":null,"evidence_quote":"Earlier planar degree-of-nonholonomy bound used as the comparison point in Example 2."},{"cited_title":"Zariski, P","cited_arxiv_id":null,"evidence_quote":"Noetherian module property used to stabilize the ascending chain of modules $C^\\#_k$ in Theorem 4.9."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Maximal integral manifold property used to show each singular leaf is forward-invariant in Theorem 4.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Immersion technique that extends the polynomial results to analytic systems in Section 5."},{"cited_title":"Bochnak, M","cited_arxiv_id":null,"evidence_quote":"Real radical and ideal-variety correspondence used to pass between ideals and singular sets in Theorem 4.5."}],"review_version":1}