{"id":"065718b8-c263-4db5-822f-0558c5886d2b","arxiv_id":"2509.02787","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The generalized and joint spectral radii coincide for bounded, equicontinuous families of order-preserving homogeneous maps on polyhedral cones.","lead":"This paper proves that two standard ways of measuring the growth rate of repeated nonlinear maps on a cone always agree, for bounded, equicontinuous families on polyhedral cones. It is the natural nonlinear analogue of the Berger-Wang theorem and leaves the non-polyhedral case as an open question.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; Theorem 2.1 proof is sound, with only a repairable terseness in Lemma 2.5.","rationale":"The reader's verdict (ACCEPT, moderate confidence) matches my reading of the paper. The central theorem is proved by an induction over the finitely many parts of a polyhedral cone. Lemma 2.4 correctly shows that a normalized limit of large orbit points lies in a strictly lower part, and Lemma 2.5 uses equicontinuity of the composed family A^m together with the lower-part spectral-radius bound to rule out supercritical growth in P. I checked the scaling step, the use of condition G, the compactness arguments, and the boundedness of O_m. The only place where the written proof is more terse than ideal is the passage in Lemma 2.5 that selects y_k with an expanding m-block; but this is repairable by a compactness argument and does not undermine the theorem. Section 6's proof of the subradii identities is also sound, though it invokes [5, Lemma 1] in an unnecessarily strong form; the needed inequality follows directly from the definition of the infimum and r(g^k) = r(g)^k. No counterexample or hidden assumption was found, so the verdict should remain unchanged.","tokens_in":15338,"tokens_out":27557,"duration_ms":250435,"concrete_test":"Independently formalize the selection step in Lemma 2.5: prove that if O_m is unbounded then there exist y_k in O_m with ||y_k|| >= k and sup_{f in A^m} ||f(y_k)|| / ||y_k|| >= 1, using only boundedness of A and pointwise equicontinuity of A^m. If this step fails, the induction in Theorem 2.1 needs repair; if it succeeds, the central proof is complete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing defect in the central claim. The proof of Lemma 2.5 has a terse step: after assuming O_m is unbounded, it asserts the existence of y_k in O_m with ||y_k|| >= k and some f in A^m satisfying ||f(y_k)|| >= ||y_k||. This is not immediate from unboundedness alone, but it is justifiable: if every f in A^m shrank every sufficiently large y in O_m, then along any trajectory from x the norms would eventually decrease once above a threshold and could never re-exceed a bounded level, so O_m would be bounded. The subsequent contradiction using equicontinuity of A^m and the lower-part bound ||f||_Q <= beta is then valid. The induction over parts and the use of polyhedrality/condition G are sound. The invocation of [5, Lemma 1] in Theorem 6.3 is unnecessary (Fekete already gives rhat_* = lim_m beta_m^{1/m}) and is more a presentation shortcut than an error.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a nonlinear Berger-Wang theorem: for a bounded, equicontinuous family of order-preserving, positively homogeneous maps on a closed polyhedral cone in a finite-dimensional normed space, the generalized spectral radius equals the joint spectral radius (Theorem 2.1). The proof proceeds by introducing a partial joint spectral radius on the parts of the cone, defining a preorder on parts determined by the dynamics of the family, and proving the key estimate Lemma 2.5 by contradiction. The paper then gives a continuity result for the joint spectral radius with respect to the Hausdorff metric (Theorem 3.3), studies conditions under which the generated semigroup is bounded when the generalized spectral radius is one (irreducible subadditive families and primitive families, Theorems 4.3 and 4.5), constructs extremal and Barabanov norms for irreducible subadditive families on the standard cone (Theorems 5.1 and 5.2), and extends the notions of joint and generalized spectral subradii to homogeneous maps on wedges (Section 6), with applications to selectable stability. A counterexample (Example 2.7) shows that the equicontinuity hypothesis in Theorem 2.1 cannot be dropped, and the paper leaves open the non-polyhedral equicontinuous case.","tokens_in":15493,"tokens_out":37755,"duration_ms":332744,"significance":"If the results stand, the paper gives a substantial nonlinear generalization of the classical Berger-Wang theorem, replacing linear maps on a vector space by order-preserving homogeneous maps on a cone. The proof strategy via partial spectral radii and induction on the preorder of parts of a polyhedral cone is natural and appears sound. The paper is careful to identify the role of each hypothesis: Example 2.6 distinguishes boundedness from equicontinuity, and Example 2.7 shows that equicontinuity is genuinely needed for the Berger-Wang formula. The boundedness and Barabanov norm results for subadditive and primitive families are useful extensions of the linear theory, and the subradius results answer a natural question in the wedge setting. The paper is self-contained, gives complete proofs of the main claims, and clearly states the open non-polyhedral case. These strengths make the paper a valuable contribution to nonlinear Perron-Frobenius theory and spectral radius theory on cones.","major_comments":[],"minor_comments":[{"comment":"The step after the definition of O_m should be expanded: unboundedness alone does not immediately produce a sequence y_k∈O_m with ∥f(y_k)∥≥∥y_k∥≥k for some f∈A^m. The authors should add the short argument that if every sufficiently large y∈O_m satisfied ∥f(y)∥<∥y∥ for all f∈A^m, then boundedness of A^m would force O_m to be bounded, contradicting the assumption.","section":"Lemma 2.5"},{"comment":"The use of [5, Lemma 1] to 'assume' that r̂_*(A)=β_N^{1/N} and r_*(A)=γ_N^{1/N} is not justified as stated, since such infima need not be attained; the proof should be rewritten with standard ε-approximations. In addition, the equality r_*(A)=liminf_m γ_m^{1/m} should be justified from the property γ_{mn}≤γ_m^n, which is not stated explicitly.","section":"Theorem 6.3"},{"comment":"The reduction to r(A)=1 by scaling should note that primitivity rules out r(A)=0: for any nonzero x there is f∈A^+ with f(x)∈intK, and then f(x)≥cx for some c>0, which forces r(f)≥c and hence r(A)>0.","section":"Corollary 4.6"},{"comment":"For clarity, the statement about eigenvalues should say 'the only possible cone eigenvalues' and note that the eigenvalue 2^{-m} may have interior eigenvectors as well as the boundary eigenvector e2.","section":"Example 2.7"},{"comment":"The application of Theorem 2.1 to the limiting family A should explicitly note that A is bounded and equicontinuous by compactness, so the Berger-Wang equality is available for the limiting family.","section":"Theorem 3.3"},{"comment":"There are minor typographical issues, including 'Berger-W ang' in the running header, the spacing in 'Arzel` a-Ascoli', and 'competive' in reference [1].","section":"General"}],"recommendation":"minor_revision","confidential_remarks":"The paper is within the journal's scope and the references to prior work, especially [7], are appropriate. The central theorem is sound; the requested changes are local clarifications rather than substantive revisions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main theorem is real. Lins and Peperko extend the Berger-Wang formula to bounded equicontinuous families of order-preserving homogeneous maps on polyhedral cones, and they do it without the eigenvector-in-interior condition that the recent Deidda–Guglielmi–Tudisco preprint needs. The proof is a clean induction over the parts of the cone using partial spectral radii. I checked the key step in Lemma 2.5: the reader is right that it is terse, but it is also repairable. If every map in A^m shrank every sufficiently large point of O_m, the orbit would be bounded, so one can pick y_k with a non-contracting A^m step. The rest of the proof goes through.\n\nWhat is genuinely new: Theorem 2.1 and Example 2.7, which shows equicontinuity is necessary. The boundedness results in Section 4, especially Theorem 4.5 for primitive families on arbitrary solid cones, are useful. Lemma 2.8, showing subadditive families are equicontinuous on polyhedral cones, is a neat observation. Section 5's extremal and Barabanov norms are a solid extension of the matrix theory, and Section 6 cleanly extends spectral subradii to wedges.\n\nSoft spots are minor. Theorem 6.3 invokes [5, Lemma 1] where Fekete's lemma would do, and the wording 'assume' is slightly misleading, but there is no mathematical gap. The non-polyhedral case is left open as Question 2.10; given how heavily the proof uses finitely many parts, that is honest, not a flaw. The bibliographic coverage looks fair: they cite the competing preprint [7] and situate their improvement clearly.\n\nWho this is for: anyone working in nonlinear Perron–Frobenius theory, and people using joint spectral radius bounds for neural networks or switched systems. It deserves a serious referee and, in my view, acceptance. I would ask the authors to expand the proof of Lemma 2.5 a bit and drop the unnecessary citation in Theorem 6.3, but neither is load-bearing.","headline":"A real extension of the Berger-Wang formula to order-preserving homogeneous maps on polyhedral cones, with a mostly solid proof and only minor presentation gaps.","tokens_in":16052,"tokens_out":2297,"would_cite":true,"duration_ms":20041,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47H07","47J10","15A80","15B48"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves the Berger–Wang formula for bounded, equicontinuous families of order-preserving homogeneous maps on polyhedral cones: generalized and joint spectral radii coincide.","keywords":["joint spectral radius","generalized spectral radius","Berger-Wang formula","order-preserving homogeneous maps","polyhedral cones","cone spectral radius","spectral subradius","Barabanov norms"],"falsifier":"To test the theorem's necessity, compute the two radii for Example 2.7: $\\mathcal A=\\{f_\\lambda\\}_{0<\\lambda<1}$ on $\\mathbb R^2_{\\ge0}$ with $f_\\lambda(x_1,x_2)=(x_1^\\lambda x_2^{1-\\lambda},\\tfrac12 x_2)$ gives $r(\\mathcal A)=\\tfrac12$ but $\\hat r(\\mathcal A)=1$. This family is bounded but not equicontinuous — the slope in the $x_2$-direction blows up as $x_1\\to0$ — so it isolates exactly the hypothesis in Theorem 2.1 that must fail for the Berger–Wang equality to break. A genuine counterexample to the theorem would have to be a bounded equicontinuous family on a polyhedral cone with unequal radii, and checking equicontinuity is the gate.","tokens_in":15107,"feed_emoji":"📐","tokens_out":11861,"duration_ms":101222,"temperature":0.7,"pith_summary":"This paper proves a nonlinear version of the classical Berger–Wang formula. For any bounded, equicontinuous family of order-preserving, positively homogeneous maps on a closed polyhedral cone in a finite-dimensional normed space, the generalized spectral radius (built from the cone spectral radii of single maps) equals the joint spectral radius (the exponential growth rate of norms of long products). The equality matters because it turns a difficult worst-case product-growth problem into an eigenvalue problem for individual maps. The proof advances through the finitely many parts of the cone, using a preorder induced by the family and partial spectral radii, with equicontinuity as the key regularity input. The paper then uses the equality to prove continuity of the joint spectral radius, semigroup boundedness criteria, existence of Barabanov norms, and an extension of spectral subradii to wedges.","feed_headline":"Two spectral radii agree for cone maps on polyhedral cones","feed_subtitle":"Worst-case product growth equals single-map eigenvalue growth for bounded equicontinuous cone-map families.","key_machinery":"The argument is carried by the decomposition of the polyhedral cone $K$ into its parts — the equivalence classes of nonzero vectors under mutual comparability, $x\\sim y$ iff $\\alpha x\\le y\\le \\beta x$ for some positive $\\alpha,\\beta$. Because $K$ is polyhedral, there are finitely many parts, and their closures are the faces. On the set of parts the authors define a preorder $P\\ge_{\\mathcal A} Q$ iff some composition $f\\in\\mathcal A^*$ sends a point of $P$ above a point of $Q$. The key estimate is the partial joint spectral radius $\\hat r(\\mathcal A,P)$, the exponential growth rate of $\\|f\\|_P$ over $f\\in\\mathcal A^m$. Lemma 2.4 shows that when $r(\\mathcal A)<1$, the limiting direction of any diverging orbit from $P$ lies in a part strictly below $P$; Lemma 2.5 then uses equicontinuity of $\\mathcal A^m$ (Lemma 1.4) to show that $\\hat r(\\mathcal A,P)>r(\\mathcal A)$ is impossible for a minimal part. Polyhedrality enters through condition G — for every $x\\in K$ and $c<1$, sufficiently close points $y\\in K$ satisfy $y\\ge cx$ — which upgrades convergence of directions to inequalities $f_k(x)\\ge c y$, and through finiteness of the preorder, which makes the minimal-part contradiction valid.","core_discovery":"The paper's central claim is Theorem 2.1: if $K$ is a closed polyhedral cone in a finite-dimensional normed space and $\\mathcal A$ is a bounded, equicontinuous family of order-preserving, homogeneous maps $K\\to K$, then $$r(\\mathcal A)=\\hat r(\\mathcal A).$$ The inequality $r(\\mathcal A)\\le \\hat r(\\mathcal A)$ is immediate from $r(f)\\le\\|f\\|$. The reverse inequality is proved by contradiction on the preorder of parts of $K$: for a minimal part $P$ with $\\hat r(\\mathcal A,P)>r(\\mathcal A)$, Lemma 2.5 shows equicontinuity forces any unbounded sequence of iterates from a point in $P$ to approach a strictly lower part $Q$, whose growth is already bounded by $\\beta<1$ after rescaling to $r(\\mathcal A)<1$. This contradiction eliminates all parts, and with them $\\hat r(\\mathcal A)$. The same circle of ideas yields the paper's corollaries: Hausdorff continuity of the joint spectral radius for compact families of order-preserving homogeneous maps on polyhedral cones; boundedness of the generated semigroup when $r(\\mathcal A)=1$ for irreducible subadditive families on $\\mathbb R_{\\ge0}^n$ or for primitive families on any solid cone; existence of monotone extremal norms and Barabanov norms for irreducible subadditive families; and equality of the generalized and joint spectral subradii for homogeneous maps on wedges.","pith_inferences":["I infer that the proof strategy is likely to extend to non-polyhedral cones whose parts form a well-founded poset under $\\ge_{\\mathcal A}$ and which satisfy condition G; the paper itself leaves the general non-polyhedral case as an open question.","I infer that for applications such as neural-network stability, the practical message is to check equicontinuity — equivalently, an eventual Lipschitz-type bound on the family — before using the spectral-radius equality as a stability certificate, since Example 2.7 shows a bounded family with kinked maps can evade the formula.","I infer that the equality for spectral subradii on wedges opens a nonlinear route to the mortality problem for inclusions: deciding whether there is a composition product with exponentially decaying norm, and the paper's selectable-stability theorem gives a verifiable criterion.","I infer that Lemma 2.8 suggests subadditivity is a sufficient structural condition to replace equicontinuity; extending that lemma beyond polyhedral cones would yield a broad Berger–Wang formula for subadditive order-preserving homogeneous maps, a testable conjecture."],"forward_implications":["When $r(\\mathcal A)<1$, all long products $f\\in\\mathcal A^m$ have norms decaying exponentially, so the family is uniformly asymptotically stable; this follows from the equality $r(\\mathcal A)=\\hat r(\\mathcal A)$ together with Lemma 1.1.","The joint spectral radius is continuous with respect to Hausdorff convergence of compact families of order-preserving homogeneous maps on a polyhedral cone, so close approximations of a family have close growth rates.","For irreducible subadditive families on $\\mathbb R_{\\ge0}^n$, $r(\\mathcal A)=1$ implies the entire semigroup $\\mathcal A^+$ is bounded; for primitive families on any solid cone the same conclusion holds without subadditivity.","Irreducible subadditive families admit monotone extremal norms, and compact ones admit Barabanov norms, yielding Lyapunov functions that characterize stability of the nonlinear inclusion $x(m+1)\\in\\mathcal A x(m)$.","On any wedge, the generalized and joint spectral subradii coincide, and a discrete inclusion is selectably stable exactly when this common subradius is below 1."],"supporting_citations":[{"why":"Introduced the joint spectral radius for pairs of matrices, the quantity whose nonlinear analogue is studied here.","marker":"[23]"},{"why":"Introduced the generalized spectral radius and conjectured its equality with the joint spectral radius for matrices.","marker":"[6]"},{"why":"Proved the matrix Berger–Wang equality for bounded sets, the result the paper generalizes to order-preserving homogeneous maps.","marker":"[2]"},{"why":"Established an earlier nonlinear joint/generalized spectral radius equality under an eigenvector-in-interior assumption, which the present theorem relaxes for polyhedral cones.","marker":"[7]"},{"why":"Supplies the cone facts used throughout: normality, condition G, parts of polyhedral cones, and existence of eigenvectors for cone maps.","marker":"[14]"},{"why":"Gives limiting formulas for the cone spectral radius and guarantees eigenvectors with eigenvalue $r(f)$, used to define and compute $r(\\mathcal A)$.","marker":"[17]"},{"why":"Proves continuity of the cone spectral radius, used in the Hausdorff-continuity theorem for the joint spectral radius.","marker":"[13]"},{"why":"Proved equality of joint and generalized spectral subradii for matrices, which Section 6 extends to homogeneous maps on wedges.","marker":"[5]"},{"why":"Introduced spectral subradii and their role in stability of discrete linear inclusions, the motivation for the selectable-stability criterion.","marker":"[12]"}],"fun_headline_variants":["Berger-Wang formula holds for cone maps","Cone maps unify joint and generalized radii","Spectral radii equal for cone families","Equicontinuous maps on cones: Berger-Wang"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is equicontinuity of the whole family $\\mathcal A$: without it a bounded family can contain maps with unbounded one-sided slopes on arbitrarily small scales, and Example 2.7 shows that then the two spectral radii can differ.","fun_headline_variants_meta":{"raw":{"variants":["Berger-Wang formula holds for cone maps","Cone maps unify joint and generalized radii","Spectral radii equal for cone families","Equicontinuous maps on cones: Berger-Wang"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000338,"raw_usage":{"total_tokens":1860,"prompt_tokens":929,"completion_tokens":931,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":545,"completion_tokens_details":{"reasoning_tokens":872}},"tokens_in":545,"tokens_out":931,"duration_ms":8638,"temperature":1.0,"reasoning_tokens":872,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:36:57.983351+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"To test the theorem's necessity, compute the two radii for Example 2.7: $\\mathcal A=\\{f_\\lambda\\}_{0<\\lambda<1}$ on $\\mathbb R^2_{\\ge0}$ with $f_\\lambda(x_1,x_2)=(x_1^\\lambda x_2^{1-\\lambda},\\tfrac12 x_2)$ gives $r(\\mathcal A)=\\tfrac12$ but $\\hat r(\\mathcal A)=1$. This family is bounded but not equicontinuous — the slope in the $x_2$-direction blows up as $x_1\\to0$ — so it isolates exactly the hypothesis in Theorem 2.1 that must fail for the Berger–Wang equality to break. A genuine counterexample to the theorem would have to be a bounded equicontinuous family on a polyhedral cone with unequal radii, and checking equicontinuity is the gate.","supporting_citations":[{"cited_title":"Rota and G","cited_arxiv_id":null,"evidence_quote":"Introduced the joint spectral radius for pairs of matrices, the quantity whose nonlinear analogue is studied here."},{"cited_title":"Daubechies and J","cited_arxiv_id":null,"evidence_quote":"Introduced the generalized spectral radius and conjectured its equality with the joint spectral radius for matrices."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proved the matrix Berger–Wang equality for bounded sets, the result the paper generalizes to order-preserving homogeneous maps."},{"cited_title":"Nonlinear Joint Spectral Radius","cited_arxiv_id":"2507.11314","evidence_quote":"Established an earlier nonlinear joint/generalized spectral radius equality under an eigenvector-in-interior assumption, which the present theorem relaxes for polyhedral cones."},{"cited_title":"Lemmens and R","cited_arxiv_id":null,"evidence_quote":"Supplies the cone facts used throughout: normality, condition G, parts of polyhedral cones, and existence of eigenvectors for cone maps."},{"cited_title":"Mallet-Paret and R","cited_arxiv_id":null,"evidence_quote":"Gives limiting formulas for the cone spectral radius and guarantees eigenvectors with eigenvalue $r(f)$, used to define and compute $r(\\mathcal A)$."},{"cited_title":"Lemmens and R","cited_arxiv_id":null,"evidence_quote":"Proves continuity of the cone spectral radius, used in the Hausdorff-continuity theorem for the joint spectral radius."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proved equality of joint and generalized spectral subradii for matrices, which Section 6 extends to homogeneous maps on wedges."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced spectral subradii and their role in stability of discrete linear inclusions, the motivation for the selectable-stability criterion."}],"review_version":2}