{"id":"d0714daa-2179-4e6a-8428-17ba94d915c0","arxiv_id":"2503.05875","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A rank-one dual-LMI solution yields a slope-restricted nonlinearity and a nonzero equilibrium that prove a discrete-time feedback system is not absolutely stable.","lead":"This paper shows that when a standard matrix-based stability test fails, a special solution of a companion test can be used to construct a nonlinearity that actually destabilizes the system. The result turns an inconclusive 'test failed' answer into a concrete counterexample with an explicit equilibrium point.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Rank-one plus sign-condition hypothesis is not guaranteed or algorithmically enforced; Remark 9 concedes this, and the practical detection claim rests on that unquantified reach.","rationale":"The proof of Theorem 6 is sound: the dual equality plus H=hh^T gives uu^T=vv^T with u=h1, v=Ah1+Bh2; the sign condition is exactly what forces u=v rather than u=-v, making h1 an equilibrium. The pairwise sector inequalities follow from the dual equality and the cone memberships of f,g,X. The odd case is analogous. The only genuinely load-bearing unproven element is whether such rank-one sign-conditioned solutions can be found or are generic. This was the reader's weakest_assumption as well, and the authors' Remark 9 is an explicit admission. Since the paper states a conditional theorem and does not overclaim an algorithm, the verdict ACCEPT remains appropriate; the concern affects practical reach, not mathematical correctness.","tokens_in":14385,"tokens_out":22841,"duration_ms":216781,"concrete_test":"On a random ensemble of systems (A Schur, ‖D‖<1, m≥2) whose primal LMI (4) is infeasible, solve the dual SDP (6) and try to extract a rank-one solution (e.g., by a rank-minimization heuristic or by checking the rank of returned extreme points). Count how often a rank-one H satisfying Pd((Ah1+Bh2)h1^T)≥0 exists. If this fraction is negligible, the paper's 'detect' claim is much weaker than the conditional theorem suggests; if high, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 6 (and Theorem 8) is internally consistent: the proof that a rank-one H forces Ah1+Bh2 = ±h1, and that the sign condition selects +h1, is correct, as is the interpolation argument using (7). The load-bearing assumption is that the dual LMI (6)/(13) admits a rank-one H with Pd((Ah1+Bh2)h1^T) ≥ 0 whenever the primal LMI is infeasible. The paper provides no algorithm to find such H and no guarantee of existence; Remark 9 explicitly says the interpretation and active use of the sign condition are under investigation. The numerical examples only report that the SDP solver returned a solution numerically verified to be rank one; this does not establish a reproducible procedure. Without evidence on how often such solutions exist, the central claim's practical scope is unquantified. This is a limitation of the contribution, not a flaw in the conditional theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies absolute stability of discrete-time LTI feedback systems with repeated slope-restricted nonlinearities, using static O'Shea-Zames-Falb multipliers in an IQC/LMI framework. It derives primal LMIs for absolute stability together with their duals, and the main results (Theorems 6 and 8) state that if the dual LMI admits a rank-one solution H = [h1; h2][h1; h2]^T satisfying the sign condition Pd((Ah1+Bh2)h1^T) >= 0, then there exists a scalar nonlinearity phi_wc in slope[0,1] (odd in Theorem 8) interpolating the data phi_wc(z*_i) = w*_i, and the feedback system with Phi_wc = diag_m(phi_wc) has the nonzero equilibrium h1. The proofs use algebraic identities from the dual equality constraints and an interpolation argument. Numerical examples for both the non-odd and odd cases illustrate the construction.","tokens_in":14500,"tokens_out":8040,"duration_ms":84732,"significance":"If the main result holds, it offers a constructive way to convert infeasibility of a sufficient LMI certificate into a falsifying nonlinearity and an explicit nonzero equilibrium, which is valuable for absolute stability analysis of neural-network and other nonlinear feedback loops. The proof of the conditional theorem is sound, self-contained, and the interpolation argument is elegant. The main caveat is that the result requires a rank-one dual solution with an additional sign condition; the paper gives no guarantee or algorithm for finding such a solution, and Remark 9 explicitly states that the sign condition is under investigation. This limits the practical reach of the detection method but does not invalidate the conditional theorem.","major_comments":[{"comment":"The dual LMI is feasible whenever the primal LMI is infeasible, but the theorems require a rank-one feasible matrix H satisfying the additional sign condition Pd((Ah1+Bh2)h1^T) >= 0. The rank-one constraint is nonconvex and cannot be enforced by a standard SDP solver, and the paper provides no theoretical condition or algorithm guaranteeing that such a solution exists whenever the primal LMI is infeasible. Since the stated purpose is to draw a definite conclusion from infeasibility, the practical scope of the detection method is unquantified. The numerical examples only report that the solver returned a solution 'numerically verified' to be rank one, without solver details or tolerances. Please either provide a guarantee or a systematic search procedure for rank-one solutions, or substantially soften the claims about detecting destabilizing nonlinearities from infeasibility and clearly state the conditional nature in the abstract and introduction.","section":"§3.2, Theorem 6 and §4.2, Theorem 8"},{"comment":"The numerical demonstrations are load-bearing for the claim that the technical conditions are usable, but they omit the verification data needed to reproduce them. For each example, please report the SDP solver and settings, the numerical values of the second singular value of H (or an equivalent rank-one verification), and the computed margin of the sign condition Pd((Ah1+Bh2)h1^T) >= 0. Without these, the reader cannot distinguish a genuine rank-one solution from a numerically near-rank-one one, nor check that the sign condition actually holds.","section":"§3.4 and §4.4"}],"minor_comments":[{"comment":"In the final sentence of the proof, the condition on phi_wc is written as phi_wc(-z*_i) = w*_i, but it should be phi_wc(-z*_i) = -w*_i to match the statement of the theorem and the preceding argument.","section":"§4.2, proof of Theorem 8(ii)"},{"comment":"The phrase 'To prove that rank(H) = 1 can happen' is misleading: the paragraph actually proves H is nonzero and h1 is nonzero under the rank-one hypothesis. Please rephrase to describe what is being shown.","section":"§3.2, proof of Theorem 6(i)"},{"comment":"The proof of assertion (iii) is omitted with 'Omitted since the proof is exactly the same as the proof of (iii) in Theorem 6.' Since the paper aims to be self-contained, it would be better to include the short argument or at least spell out the identical steps.","section":"§4.2, Theorem 8(iii)"},{"comment":"The figure captions are minimal. Please state explicitly which curve is phi_wc and which is the identity line, and distinguish the trajectories from x(0) = h1 and from the other initial condition.","section":"Figures 2-5"},{"comment":"The reference to Gyotoku et al. (2025) is an arXiv preprint; if a published version exists, please update the citation, and if not, consider noting that the continuous-time counterpart is a preprint.","section":"§1 and §4.3"}],"recommendation":"major_revision","confidential_remarks":"The conditional theorems appear correct, and the authors are honest about the sign condition being under investigation. The main editorial question is whether the paper's contribution is sufficiently broad: the practical detection claim is currently supported only by two numerical instances of a nonconvex rank-one condition, with no evidence about how often such solutions exist. In my view this warrants a major revision so that the claims are aligned with the actual scope, rather than rejection, because the conditional result is a valid contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper does something genuinely useful: it turns an infeasible IQC-based stability LMI into a constructive instability certificate for discrete-time feedback systems. Theorems 6 and 8 prove that if the dual LMI has a rank-one solution H = [h1;h2][h1;h2]^T and the sign condition Pd((Ah1+Bh2)h1^T) >= 0 holds, then there is a slope-[0,1] repeated nonlinearity (odd, in Theorem 8) that makes the system have a nonzero equilibrium h1, so absolute stability fails. I checked the proofs; they are sound. The interpolation step that converts the dual equalities into a well-defined phi_wc is careful, and the construction of the piecewise-linear destabilizing nonlinearity is explicit.\n\nThe genuinely new part relative to the authors' continuous-time paper is the discrete-time formulation and the extra sign condition, which they flag in Remark 9. That is a legitimate extension, not a restatement. The numerical examples show the construction working, and the text is honest that H is only numerically verified to be rank one.\n\nThe soft spots are real but proportionate. The theorem is conditional on the dual LMI having a rank-one solution that also satisfies the sign condition. The paper gives no algorithm to find such a solution and no guarantee that one exists whenever the primal LMI is infeasible; Remark 9 says the interpretation and active use of the sign condition are under investigation. So the practical detection claim is unquantified. That is a limitation of the contribution, not a flaw in the conditional theorem. Also, the results are restricted to slope[0,1] and ||D||<1; the abstract's phrase 'rank condition' is looser than the theorem statements, but the body is explicit.\n\nThis paper is for people working on absolute stability via IQCs, especially those analyzing neural-network or optimization-algorithm feedback loops. It deserves a serious referee. The math holds up, the authors are straightforward about the open points, and the result gives a constructive way to turn a dead-end LMI into a counterexample. Send it to review.\n\nBest,","headline":"Solid conditional result: dual-LMI rank-one plus a sign condition constructively certifies instability in discrete-time, but the unquantified reach of that condition limits the practical claim.","tokens_in":15088,"tokens_out":2508,"would_cite":true,"duration_ms":23397,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93C55","93D05","93D09","90C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"A rank-one solution of the dual LMI furnishes a constructive proof that the system is not absolutely stable.","keywords":["absolute stability","discrete-time feedback systems","slope-restricted nonlinearities","integral quadratic constraints","O'Shea-Zames-Falb multipliers","dual linear matrix inequalities","destabilizing nonlinearity detection","rank-one dual solution"],"falsifier":"Take any system $(A,B,C,D)$ with $A$ Schur stable and $\\|D\\|<1$ where the dual LMI (6) is feasible with a rank-one $H$ satisfying the sign condition, and verify by independent means (e.g., a full-block IQC test or exhaustive simulation) whether the closed-loop system with every slope[0,1] repeated nonlinearity is globally asymptotically stable; if such a system is found to be absolutely stable, Theorem 6's construction of a destabilizing nonlinearity would be contradicted.","tokens_in":14172,"feed_emoji":"🎯","tokens_out":9896,"duration_ms":82100,"temperature":0.7,"pith_summary":"This paper turns a previously inconclusive outcome in absolute stability analysis into a constructive proof of instability. The standard IQC-based test for a discrete-time feedback system with slope-restricted nonlinearities is a linear matrix inequality (LMI); if the LMI is feasible, stability follows, but if it is infeasible, nothing could be concluded. The authors analyze the dual LMI, which is feasible exactly when the primal LMI is infeasible, and show that a rank-one dual solution, together with a sign condition, lets them build a specific slope-restricted nonlinearity that makes the closed-loop system have a nonzero equilibrium. That equilibrium proves the system is not absolutely stable over the assumed nonlinearity class.","feed_headline":"Rank-one dual LMI builds a destabilizing nonlinearity","feed_subtitle":"A feasible rank-one dual solution plus one sign condition constructs a slope-restricted destabilizer and a nonzero equilibrium.","key_machinery":"The engine is the dual of the IQC-stability LMI, obtained via Lagrange duality. On that dual feasible set, a rank-one factor $H = [h_1;\\,h_2][h_1;\\,h_2]^T$ supplies a candidate equilibrium $h_1$ and a candidate input-output pair $(z^*, w^*) = (Ch_1+Dh_2,\\, h_2)$. The constraint $(Ah_1+Bh_2)(Ah_1+Bh_2)^T = h_1 h_1^T$ forces $Ah_1+Bh_2 = \\pm h_1$, and the sign condition $P_d((Ah_1+Bh_2)h_1^T) \\ge 0$ selects the plus branch so that $h_1$ is a fixed point. The second dual equality collapses to $w^*(z^*-w^*)^T = 1_m f^T + g 1_m^T + X$ with $f,g \\ge 0$ and $X$ a Z-matrix with zero diagonal; that identity is exactly what guarantees the interpolation points $(z^*_i, w^*_i)$ have slopes between 0 and 1 (and, in the odd case, the symmetry), so an admissible destabilizing nonlinearity can be pieced together.","core_discovery":"The paper's central claim is Theorem 6 (and its odd-symmetry counterpart, Theorem 8). Suppose the dual LMI (6) (or (13)) has a feasible solution H of rank one, written as $H = [h_1;\\,h_2][h_1;\\,h_2]^T$, and suppose the entrywise condition $P_d((Ah_1+Bh_2)h_1^T) \\ge 0$ holds. Then $h_1$ is nonzero, and there is a nonlinearity $\\varphi_{wc} \\in \\mathrm{slope}[0,1]$ (odd in the second case) that interpolates the data $\\varphi_{wc}(Ch_1+Dh_2) = h_2$. The feedback system with this nonlinearity has $h_1$ as a nonzero equilibrium point, so it is not globally asymptotically stable; because $\\varphi_{wc}$ lies in the assumed class of slope-restricted repeated nonlinearities, the original system is never absolutely stable over that class. The dual LMI's feasibility therefore upgrades LMI infeasibility from 'inconclusive' to 'definitively not absolutely stable' whenever a rank-one solution with the sign condition exists.","pith_inferences":["If a rank-one dual solution can be found systematically by the solver (for instance by adding a rank penalty), the result becomes an automated instability test that runs whenever the primal LMI fails, turning every LMI infeasibility into either a constructive instability certificate or a reason to refine the multiplier class.","The same Lagrange-duality technique could be applied to other IQC-based sufficient conditions (e.g., with dynamic multipliers or sector-bounded nonlinearities), where a rank-one dual solution might similarly expose a destabilizing element in the assumed class.","The sign condition $P_d((Ah_1+Bh_2)h_1^T) \\ge 0$ is the discrete-time analogue of a condition that was automatically satisfied in continuous time; understanding when it holds may reveal which systems are prone to having hidden equilibria at exactly the boundary of the multiplier class.","A testable extension is to check numerically whether, for random stable (A,B,C,D), the set of rank-one dual solutions with the sign condition is nonempty whenever the primal LMI is infeasible; if gaps exist, the detection test is conservative."],"forward_implications":["Whenever the dual LMI (6) is feasible with a rank-one $H$ satisfying the sign condition, the system is provably not absolutely stable over slope[0,1] repeated nonlinearities, not merely possibly unstable.","The same rank-one certificate for the odd-symmetry dual LMI (13) proves the system is not absolutely stable even over the smaller class of odd slope-restricted nonlinearities.","The constructed $\\varphi_{wc}$ is explicit piecewise-linear and can be extracted from the dual solution, so the certificate is constructive: it yields both a witness nonlinearity and a witness nonzero equilibrium.","Because the dual LMI is feasible if and only if the primal LMI is infeasible, the theorem provides a rigorous way to interpret primal infeasibility in terms of actual instability, closing the gap left by the sufficiency-only primal condition.","The restriction to slope[0,1] and $\\|D\\|<1$ is used to keep the nonlinearities nonexpansive and the feedback well-posed; the authors state that extending to general slope[$\\mu$,$\\nu$] remains open."],"supporting_citations":[{"why":"Supplies the IQC framework and the basic stability condition (Proposition 1) that the primal LMI is built on.","marker":"Megretski and Rantzer (1997)"},{"why":"Provides the static O'Shea-Zames-Falb multiplier constraints for discrete-time slope-restricted and repeated nonlinearities used in Lemma 4.","marker":"Fetzer and Scherer (2017)"},{"why":"Gives the duality relation that the dual LMI is feasible if and only if the primal LMI is infeasible, the foundation for interpreting dual feasibility.","marker":"Scherer (2006)"},{"why":"The continuous-time counterpart whose rank-one detection idea this paper extends to discrete-time systems.","marker":"Gyotoku et al. (2025)"},{"why":"The earlier continuous-time ReLU-nonlinearity study that introduced the rank-condition approach to concluding non-global asymptotic stability.","marker":"Yuno et al. (2024b)"},{"why":"The linear-algebra fact used in the proof that $(Ah_1+Bh_2)(Ah_1+Bh_2)^T = h_1 h_1^T$ with $h_1\\neq 0$ implies $Ah_1+Bh_2 = \\pm h_1$.","marker":"Ebihara (2012)"}],"fun_headline_variants":["Rank-one dual LMI certifies instability","One dual LMI rank-one detects destabilizer","Dual LMI rank-one reveals no absolute stability","LMI infeasibility becomes instability proof","Rank-one dual solution constructs destabilizing nonlinearity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the dual LMI's feasible set actually contains a rank-one solution $H = [h_1;\\,h_2][h_1;\\,h_2]^T$ that also satisfies the sign condition $P_d((Ah_1+Bh_2)h_1^T) \\ge 0$; the theorems say nothing when no such solution exists, and the paper offers no algorithm or guarantee that one will be present.","fun_headline_variants_meta":{"raw":{"variants":["Rank-one dual LMI certifies instability","One dual LMI rank-one detects destabilizer","Dual LMI rank-one reveals no absolute stability","LMI infeasibility becomes instability proof","Rank-one dual solution constructs destabilizing nonlinearity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000204,"raw_usage":{"total_tokens":1403,"prompt_tokens":975,"completion_tokens":428,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":591,"completion_tokens_details":{"reasoning_tokens":358}},"tokens_in":591,"tokens_out":428,"duration_ms":4855,"temperature":1.0,"reasoning_tokens":358,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T17:41:36.602571+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any system $(A,B,C,D)$ with $A$ Schur stable and $\\|D\\|<1$ where the dual LMI (6) is feasible with a rank-one $H$ satisfying the sign condition, and verify by independent means (e.g., a full-block IQC test or exhaustive simulation) whether the closed-loop system with every slope[0,1] repeated nonlinearity is globally asymptotically stable; if such a system is found to be absolutely stable, Theorem 6's construction of a destabilizing nonlinearity would be contradicted.","supporting_citations":[{"cited_title":"and Rantzer, A","cited_arxiv_id":null,"evidence_quote":"Supplies the IQC framework and the basic stability condition (Proposition 1) that the primal LMI is built on."},{"cited_title":"and Scherer, C.W","cited_arxiv_id":null,"evidence_quote":"Provides the static O'Shea-Zames-Falb multiplier constraints for discrete-time slope-restricted and repeated nonlinearities used in Lemma 4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the duality relation that the dual LMI is feasible if and only if the primal LMI is infeasible, the foundation for interpreting dual feasibility."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The linear-algebra fact used in the proof that $(Ah_1+Bh_2)(Ah_1+Bh_2)^T = h_1 h_1^T$ with $h_1\\neq 0$ implies $Ah_1+Bh_2 = \\pm h_1$."}],"review_version":1}