{"id":"7dc698d1-b634-4fd4-9b5d-ff4470106ca8","arxiv_id":"2507.11314","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For switched nonlinear systems whose maps are sub-homogeneous and order-preserving on a cone, a nonlinear joint spectral radius is shown to characterize asymptotic stability and bound the worst-case separation growth rate of trajectories.","lead":"This paper defines a nonlinear joint spectral radius for switched systems built from sub-homogeneous, order-preserving maps on cones, and proves it controls asymptotic stability and the growth of trajectory separation. The framework extends classical linear switched-system theory to monotone nonlinear dynamics, with a numerical algorithm and neural-network stability as motivating applications.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The equality ρK(F)=bρK(F) that makes the nonlinear JSR computable rests on an interior-subcone hypothesis (Thm 5.21) that is unverified in general and fails in the paper's own examples; without that hypothesis, bρK can understate the true stability threshold.","rationale":"The reader identified exactly the load-bearing weakness: the interior-subcone condition in Theorem 5.21/Corollary 5.22 is necessary for the advertised equality between the nonlinear JSR and the generalized JSR, and hence for the computable rate bound and the polytopal algorithm's certificate. I reviewed the core stability arguments in Theorems 3.4 and 3.6 and found no internal flaw in the threshold characterization itself; the nonlinear JSR does characterize asymptotic stability and trajectory divergence rates as stated. The genuine soft spot is the passage from ρK to bρK: without the subcone condition, Example 5.24 shows bρK can be strictly smaller than ρK, and Example 5.27 shows the condition can fail for simple linear families even when individual generators have interior eigenvectors. This does not invalidate the central stability theorem, but it means the computational and exactness claims are more limited than the abstract suggests. Since the reader's CONDITIONAL verdict already accounts for this, my stress-test does not move the verdict; it confirms that the condition should be stated as a genuine hypothesis of the computable-equality results and that applications relying on bρK must verify it or impose extra regularity such as uniform Hilbert contractivity.","tokens_in":46887,"tokens_out":9593,"duration_ms":126519,"concrete_test":"Recompute the two assertions of Example 5.24: first verify that every composition f∈Σ_k(F) has ρK(f)=2^{-k}, so bρK(F)=1/2; then verify that for interior points with ∥x∥1=1 one has sup_{f∈Σ_k}∥f(x)∥1=1, so ρK(F)=1. If both hold, the paper's own counterexample confirms that without the Theorem 5.21 subcone condition the generalized JSR is not a stability threshold. As a second check, take the two linear maps of Example 5.27 and test the subcone condition on K′=the positive cone; the product diag(4,1/4) has boundary-only eigenvectors, so if the hypothesis fails but ρK=bρK still holds numerically, Cor. 5.22 is only sufficient, and the algorithm's certificate must explicitly verify K′ before being trusted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central stability theorem (Thm 3.4) is internally coherent, but the advertised claim that the nonlinear JSR is a computable threshold object is carried by Cor. 5.22, which asserts ρK(F)=bρK(F) under the assumption that some closed subcone K′⊂Int(K) contains a dominant eigenvector of every f∈Σ(F). This assumption is load-bearing: Theorem 5.21 uses K′ to construct the bounded set V that converts bρK<1 into asymptotic stability, and the algorithm in Section 6 exploits the equality to certify ρK by finding a dominant spectrum-maximizing product. The hypothesis is not checkable from the generator family alone. Boundedness does not imply it: Example 5.24 gives a bounded family with bρK(F)=1/2 while ρK(F)=1, so using bρK as a stability certificate without extra conditions can be dangerously optimistic. Moreover, the condition can fail even when every generator has an interior dominant eigenvector: in Example 5.27, f1 and f2 have interior eigenvectors, but f1∘f2=diag(4,1/4) has only boundary eigenvectors, so no such K′ exists. Theorem 5.26 provides a sufficient condition via uniform Hilbert contractivity, but the paper does not establish a verifiable characterization of when K′ exists, and Theorem 5.26's proof does not cover the non-contractive case. Thus the exactness statements Cor. 5.22 and the finite-time convergence claims of Theorem 6.4 are conditional on a hypothesis that is neither implied by the main assumptions nor easily certified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a nonlinear joint spectral radius ρ_K(F) for switched systems x_{k+1}=f_{σ(k)}(x_k) whose maps f_i are subhomogeneous and order-preserving on a cone K. The main mathematical results are: Theorem 3.4, which characterizes asymptotic stability of the switched semigroup by ρ_K(F)<1; Theorem 3.6, which bounds trajectory separation under Thompson nonexpansiveness in terms of ρ_K(F); Theorem 4.7, which sandwiches ρ_K(F) between the JSRs of two homogeneous limit families; Theorem 5.10, a dual representation of the homogeneous JSR via monotone prenorms; and Section 5.2, which introduces a generalized JSR bρ_K(F) and proves equality with ρ_K(F) under a strong interior-subcone hypothesis. A polytopal-type algorithm is proposed in Section 6 with finite-time convergence conditions. The paper is notably transparent about limitations, including counterexamples where bρ_K≠ρ_K (Examples 5.24 and 5.27) and where continuous extremal prenorms fail (Examples 5.16 and 5.19).","tokens_in":47064,"tokens_out":23802,"duration_ms":294586,"significance":"If the main theorems are correct, the paper gives a coherent and natural threshold object for stability of switched monotone cone systems, extending the linear joint spectral radius to nonlinear Perron-Frobenius settings. The core stability theorem and the dual prenorm construction are nontrivial and appear internally consistent. A particular strength is the collection of counterexamples that delineate where the linear Berger-Wang equality fails in the nonlinear world; these examples are an honest and useful contribution. The practical significance is tempered, however, by the fact that the equality ρ_K(F)=bρ_K(F) and the finite-time convergence of the algorithm rest on hypotheses that are not checkable from the generator family alone and can fail in simple cases; the paper would benefit from stating this limitation at the level of the abstract and introduction.","major_comments":[{"comment":"The equality ρ_K(F)=bρ_K(F) and, through it, the exactness and finite-time convergence claims of the polytopal algorithm are proved only under the assumption that there is a closed subcone K'⊂Int(K) containing a dominant eigenvector of every f∈Σ(F). This hypothesis is not checkable from the generator family alone, and the manuscript's own examples show that it cannot be replaced by natural weaker assumptions: Example 5.24 gives a bounded homogeneous family with bρ_K(F)=1/2 and ρ_K(F)=1, and Example 5.27 gives two generators each with an interior dominant eigenvector whose product has only boundary eigenvectors. Since Section 6's algorithm is advertised as computing or certifying ρ_K(F), the paper should state prominently that this computability is conditional on K', explain how one could certify such a subcone in concrete applications, and describe what can be guaranteed when K' is not known to exist. Without such discussion, the algorithmic section overstates the scope of the results.","section":"§5.2, Cor. 5.22; §6"},{"comment":"The claim that Theorem 3.4 yields 'a necessary condition for global convergence to a fixed point' is not correct as stated. Theorem 3.4 concerns asymptotic stability in the sense of convergence to zero, whereas the fixed points of deep equilibrium models and of affine maps with bias are generally nonzero. A concrete counterexample within the paper's own framework is f(x)=x/2+1 on K=R_+. This map is continuous, subhomogeneous, and order-preserving, has the unique globally attracting fixed point x*=2, but ρ_K({f})=1 because ∥f^k∥→2 on the unit ball. Thus ρ_K(F)<1 is not necessary for global convergence to a nonzero fixed point. The application paragraph should be revised to refer instead to the error dynamics around a fixed point or to a separate stability notion for nonzero equilibria.","section":"§3, 'Fixed points and deep equilibrium models'"}],"minor_comments":[{"comment":"In the converse direction, the normality constant δ of the cone is omitted when passing from ∥f(r_1x)∥≤α^k r_2 to a bound on ∥f(x)∥; the argument still works if one writes Cα^k and then notes that (Cα^k)^{1/k}→α, but the displayed chain of inequalities is not correct as written.","section":"§3, proof of Theorem 3.4"},{"comment":"The statement that the lim sup in the definition of bρ_K(F) can be replaced by a supremum is only plausible because any word can be repeated; the text says this follows from Proposition A.7, but Proposition A.7 alone does not give this replacement. A short explicit argument using ρ_K(f^m)=ρ_K(f)^m would improve the presentation.","section":"§5.2, Eq. (16)"},{"comment":"The proof applies Theorem 3.4 to the scaled family F_ε, but Theorem 3.4 assumes boundedness; in the present setting boundedness follows from Remark 5.23 under the same K' hypothesis. This chain of reasoning should be stated explicitly so the corollary is self-contained.","section":"§5.2, proof of Cor. 5.22"},{"comment":"In Example 5.19 the computation of the supremum over Σ_k(F) is slightly confusing because it is written at the point x_0=(1,1) with ∥x_0∥_1=2, whereas the operator norm is an extremum over the unit ball. The final value ρ_K(F)=1/2 is correct, but the intermediate display would be clearer if the unit-normalized point were used.","section":"§6, Example 5.19"},{"comment":"Several typographical errors should be corrected: 'controlloed' in Section 4, 'ans assume' in Proposition 4.2, 'Schouder' in Theorem 5.26, 'Algortihm' in Section 6, and 'sepctrum' in the proof of Theorem 6.4. These do not affect the mathematics.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is mathematically substantial and the core stability and duality results appear sound. My main reservation is that the advertised computational and application-level claims outrun the proved statements: the equality ρ_K=bρ_K is conditional on an uncheckable interior-subcone hypothesis, and the deep-equilibrium application is contradicted by simple affine maps with nonzero fixed points. Both issues are fixable by rewriting the relevant passages, so I am not recommending rejection, but the manuscript should not be accepted in its current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core stability theorem (Thm 3.4) is sound and the paper is honest about its debt to prior work: Remark 5.5 discloses that the nonlinear JSR is a particular case of the competitive spectral radius of Akian, Gaubert and Marchesini. That is handled properly, not buried. The genuinely new material is Theorem 4.7 (the two-sided bound via the homogeneous limits f0 and f∞), the monotone-prenorm duality in Theorem 5.10, the generalized-JSR equality conditions around Corollary 5.22, the continuity results in 5.31, and the polytopal algorithm. I read the main proofs with the nonlinear Perron-Frobenius toolkit and they hold together; in particular there is no circularity in Theorem 5.10, which constructs the extremal prenorm from the semigroup rather than assuming it.\n\nThe soft spot is the computability claim. The advertised \"computable threshold\" is carried by Cor 5.22, asserting ρK(F) = bρK(F) under the existence of a subcone K′ contained in Int(K) containing a dominant eigenvector of every map in Σ(F). That hypothesis is load-bearing and is not checkable from the generator family alone. Example 5.24 is the smoking gun: a bounded family with bρK = 1/2 and ρK = 1. Use bρK as a stability certificate without extra conditions and you can be off by a factor of two in the wrong direction. Example 5.27 shows the hypothesis can fail even when every generator has an interior eigenvector, because a product only has boundary eigenvectors. The authors are candid about both examples, but they never give a verifiable characterization of when K′ exists; Theorem 5.26 offers a Hilbert-contractive sufficient condition, but its K′′ is asserted to be a subcone without proof, and the non-contractive case is left open. So the exactness statements in Cor 5.22 and the finite-time claims in Theorem 6.4 are conditional in a way the abstract does not convey.\n\nThe neural-network section overreaches: \"tight,\" \"sharper than L2 bounds,\" and uniform Jacobian bounds are asserted, not derived or tested, and the algorithm ships no code. Theorem 6.4 is proved via three sketched claims, which I could not fully verify line by line.\n\nWho this is for: people working on switched monotone systems and nonlinear Perron-Frobenius theory. The central stability result and the negative examples are valuable; the exactness framing needs revision. This deserves a serious referee: accept the theory, require the authors to flag the subcone hypothesis as an open condition or prove it from the data, and strip the unverified engineering claims.","headline":"Solid core stability theory with honest disclosure of prior work; the computability claim rests on a subcone hypothesis that fails in the paper's own examples, so the exactness framing needs tightening.","tokens_in":47876,"tokens_out":4573,"would_cite":true,"duration_ms":47555,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37B25","47H07","15A18"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper defines a nonlinear joint spectral radius on cones and proves it is the exact threshold for worst-case stability of switched subhomogeneous monotone systems: $\\rho_K(F)<1$ if and only if the system is asymptotically stable.","keywords":["joint spectral radius","nonlinear Perron-Frobenius theory","switched dynamical systems","order-preserving maps","subhomogeneous maps","Thompson metric","asymptotic stability","deep neural networks"],"falsifier":"Run the paper's Example 5.24: a bounded subhomogeneous family on $\\mathbb{R}_+^2$ with $\\rho_K(F)=1$ and $\\hat\\rho_K(F)=1/2$. The generalized-radius test would incorrectly certify asymptotic stability, while Theorem 3.4 says $\\rho_K=1$ means the family is not asymptotically stable; any proposed relaxation of the interior-subcone condition must explain this example. A more decisive test would be to check whether equality can fail for a bounded family that is additionally equicontinuous — the open question the authors flag.","tokens_in":46470,"feed_emoji":"📐","tokens_out":9333,"duration_ms":101433,"temperature":0.7,"pith_summary":"This paper introduces a nonlinear joint spectral radius for families of subhomogeneous, order-preserving maps acting on a cone, and argues that it is the correct worst-case growth indicator for switched discrete-time systems. The central result is a threshold theorem: the family is asymptotically stable under every switching sequence if and only if this nonlinear joint spectral radius is strictly less than one. A companion estimate shows that the same number controls how fast two nearby trajectories separate, giving a computable Lipschitz-type bound along every composition. The authors also develop dual characterizations in terms of monotone prenorms, compare the joint spectral radius with a generalized version built from cone spectral radii of compositions, and give a polytopal-type algorithm that can compute the value exactly under spectral-maximizing conditions. A reader should care because this extends a classical linear tool to nonlinear Perron-Frobenius type systems and gives concrete stability and robustness certificates for switched networks, including nonnegative deep neural networks.","feed_headline":"One cone-based number decides switched nonlinear stability","feed_subtitle":"The nonlinear joint spectral radius below 1 means every switching sequence converges, and bounds trajectory separation.","key_machinery":"The load-bearing object is the cone joint spectral radius $\\rho_K(F)$, defined through the induced norm of subhomogeneous maps on a cone; its threshold behavior is carried by the order-preserving property, which makes each map non-expansive in Thompson's metric and connects Euclidean trajectory growth to the metric via standard cone-norm bounds. Two associated objects do most of the work: the asymptotic homogeneous maps $f_0(x)=\\lim_{c\\to 0} f(cx)/c$ and $f_\\infty(x)=\\lim_{c\\to\\infty} f(cx)/c$, whose joint spectral radii bracket $\\rho_K(F)$ for a subhomogeneous family (Theorem 4.7); and the generalized cone joint spectral radius $\\hat\\rho_K(F)$ based on cone spectral radii of compositions, whose equality with $\\rho_K(F)$ is established under the interior-subcone eigenvector condition and used in the polytopal-type algorithm. The algorithm itself builds an extremal finitely generated monotone prenorm whose vertices are images of a conjectured spectrum-maximizing product, mirroring the linear polytope method (Theorems 5.10, 6.4, Lemma 6.2).","core_discovery":"On the paper's own terms, the discovery is that the classical joint spectral radius threshold survives nonlinearity: for a bounded family $F$ of continuous subhomogeneous maps on a solid closed cone $K$, $\\rho_K(F)<1$ if and only if the switched system $x_{k+1}=f_{\\sigma(k)}(x_k)$ is asymptotically stable for every switching sequence (Theorem 3.4). When the maps are also order-preserving and thus non-expansive in Thompson's metric, the same radius bounds trajectory separation: $\\|f(x)-f(y)\\| \\leq C(x,y,\\epsilon)(\\rho_K(F)+\\epsilon)^k\\|x-y\\|$ for every composition of length $k$ (Theorem 3.6). The paper further proves that in the homogeneous case the nonlinear JSR has a dual formula as the infimum over monotone prenorms of their induced suprema, and that it coincides with the generalized cone JSR — the limsup of $k$-th roots of cone spectral radii of length-$k$ compositions — whenever a fixed subcone inside the interior contains a dominant eigenvector of every composition (Corollary 5.22). Two explicit examples show that boundedness of the family alone is not enough for that equality, in contrast to the linear Berger-Wang theorem.","pith_inferences":["A practical consequence not stated in the paper is that the interior-subcone eigenvector condition is the real gatekeeper for spectral-radius certificates: without it, the easy test can be off by an arbitrary factor, as in the paper's Example 5.24 where $\\rho_K(F)=1$ while $\\hat\\rho_K(F)=1/2$.","The framework suggests a testable design rule for neural network architectures: keep activations and maps such that all compositions share an invariant interior subcone, which would make the generalized JSR computable and the Lipschitz bound tight, extending what the paper proves for perturbed families in Example 5.25.","The open equicontinuity question could be probed numerically: if a bounded, equicontinuous, homogeneous order-preserving family with $\\hat\\rho_K<\\rho_K$ exists, the analogy with the linear Berger-Wang theorem fails more strongly than the examples shown; the paper leaves this unresolved.","Because $f_0$ can degenerate to infinity when constant terms dominate, applying the framework to networks with biases requires shifting the cone or rescaling around a working point; the paper does not develop that variant."],"forward_implications":["If $\\rho_K(F)<1$, every trajectory of the switched system converges to zero under any admissible switching rule; if $\\rho_K(F)\\geq 1$, the family is not asymptotically stable at the uniform rate.","The trajectory-separation bound (Theorem 3.6) turns the nonlinear JSR into a worst-case Lipschitz certificate: a small change in an interior input cannot be amplified by more than $(\\rho_K(F)+\\epsilon)^k$ up to a constant, across all layer choices.","For subhomogeneous families, the JSR is sandwiched between the JSRs of the two homogeneous asymptotic families, so stability bounds can be computed from homogeneous maps without simulating all switching sequences.","When the interior-subcone condition holds, $\\rho_K(F)=\\hat\\rho_K(F)$, so the radius can be certified by checking spectral radii of length-$k$ compositions, and the proposed algorithm terminates finitely for finite families with a dominant spectrum-maximizing product.","Applied to nonnegative neural networks, the framework yields conditions for fixed-point attraction in deep equilibrium models and bounds on gradient growth, stated as a necessary condition $\\rho_K<1$ for global convergence."],"supporting_citations":[{"why":"Supplies the classical definition of the joint spectral radius and the linear theory of stability thresholds and generalized JSR that this paper extends to subhomogeneous cone maps.","marker":"[40]"},{"why":"Provides the nonlinear Perron-Frobenius toolbox: cones, Thompson and Hilbert metrics, cone spectral radius, and eigenvector results used throughout the main theorems.","marker":"[46]"},{"why":"Gives the Berger-Wang theorem that bounded linear semigroups have JSR equal to the generalized JSR, the result whose nonlinear analogue is established under an additional subcone condition.","marker":"[10]"},{"why":"Gives the Rota-Strang dual characterization of the linear JSR as an infimum over norms, which Theorem 5.10 generalizes to monotone prenorms.","marker":"[58]"},{"why":"Supplies the analytic-geometric proof of the generalized spectral radius theorem and the reducibility discussion underlying Proposition 5.14 on extremal prenorms.","marker":"[21]"},{"why":"Provides the linear polytopal algorithm whose framework is generalized in Section 6 to homogeneous cone maps.","marker":"[30]"},{"why":"Contains complex polytope extremality results used as the model for the finite-convergence conditions in Theorem 6.4.","marker":"[32]"}],"fun_headline_variants":["Nonlinear JSR: one number says all switching sequences converge","Cone JSR: stability and trajectory separation in one number","Nonlinear JSR <1: every switching sequence converges","From linear to nonlinear: JSR threshold holds for cone maps"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The main extension beyond the linear theory assumes that all long compositions of the maps share a dominant eigenvector lying in a fixed region strictly inside the cone; without this assumption, boundedness alone does not make the easy spectral-radius-based test correct.","fun_headline_variants_meta":{"raw":{"variants":["Nonlinear JSR: one number says all switching sequences converge","Cone JSR: stability and trajectory separation in one number","Nonlinear JSR <1: every switching sequence converges","From linear to nonlinear: JSR threshold holds for cone maps"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000848,"raw_usage":{"total_tokens":3706,"prompt_tokens":982,"completion_tokens":2724,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":598,"completion_tokens_details":{"reasoning_tokens":2654}},"tokens_in":598,"tokens_out":2724,"duration_ms":23080,"temperature":1.0,"reasoning_tokens":2654,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:19:37.574946+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the paper's Example 5.24: a bounded subhomogeneous family on $\\mathbb{R}_+^2$ with $\\rho_K(F)=1$ and $\\hat\\rho_K(F)=1/2$. The generalized-radius test would incorrectly certify asymptotic stability, while Theorem 3.4 says $\\rho_K=1$ means the family is not asymptotically stable; any proposed relaxation of the interior-subcone condition must explain this example. A more decisive test would be to check whether equality can fail for a bounded family that is additionally equicontinuous — the open question the authors flag.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the classical definition of the joint spectral radius and the linear theory of stability thresholds and generalized JSR that this paper extends to subhomogeneous cone maps."},{"cited_title":"Lemmens and R","cited_arxiv_id":null,"evidence_quote":"Provides the nonlinear Perron-Frobenius toolbox: cones, Thompson and Hilbert metrics, cone spectral radius, and eigenvector results used throughout the main theorems."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Berger-Wang theorem that bounded linear semigroups have JSR equal to the generalized JSR, the result whose nonlinear analogue is established under an additional subcone condition."},{"cited_title":"Rota and G","cited_arxiv_id":null,"evidence_quote":"Gives the Rota-Strang dual characterization of the linear JSR as an infimum over norms, which Theorem 5.10 generalizes to monotone prenorms."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the analytic-geometric proof of the generalized spectral radius theorem and the reducibility discussion underlying Proposition 5.14 on extremal prenorms."},{"cited_title":"Guglielmi and V","cited_arxiv_id":null,"evidence_quote":"Provides the linear polytopal algorithm whose framework is generalized in Section 6 to homogeneous cone maps."},{"cited_title":"Guglielmi, F","cited_arxiv_id":null,"evidence_quote":"Contains complex polytope extremality results used as the model for the finite-convergence conditions in Theorem 6.4."}],"review_version":1}