{"id":"97a5ff21-2e63-4407-bcf2-109eaa26fb11","arxiv_id":"2511.06063","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"A product of linear and Wendland kernels is proposed so that the Koopman operator's spectrum lies in the unit disk exactly when a conjugate linear system is stable.","lead":"The paper builds a new function space for Koopman operators by multiplying a linear kernel with a radial kernel, aiming to make the operator's eigenvalues reveal whether an equilibrium is stable. It claims this gives a data-driven stability certificate, but the proof requires a global conjugacy to a linear system and the error bounds do not support the spectral conclusion as written.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 13's certificate uses Lemma 10's H→L2 error as an H→H perturbation in Lemma 11; no such norm transfer is shown, so the main stability claim is unproven.","rationale":"The reader's weakest assumption identifies the same load-bearing gap: Lemma 10 supplies only an H→L² error bound, while the spectral perturbation argument in Lemma 11 and the subsequent comparison of T=A*A and T̂=Â* require an H→H bound on A−Â. This is not a minor technicality; the two norms are not equivalent on the compact domain, and the missing implication can fail in the worst case. Because Theorem 13 is the central stability certificate, this gap alone invalidates the main claim as stated.\n\nAdditional problems strengthen the rejection: Theorem 8's spectral equality appears to ignore the possibility that a non-surjective f produces a zero eigenvalue from functions vanishing on f(X), and Lemma 12's inference of contractivity from diagonalizability in a non-orthogonal basis is not generally valid. However, the H→L² versus H→H mismatch is the most direct obstruction to the stated proof.\n\nThe numerical experiments are illustrative but do not provide independent support for the theorem: the paper supplies no code or machine-checked verification, and Appendix B.1 shows that the method is inconclusive when the domain contains a limit cycle, which is consistent with the theoretical gap rather than a demonstration of the certificate.","tokens_in":13820,"tokens_out":9414,"duration_ms":88659,"concrete_test":"Analytical check: take d=1, X=[0,1], and H as the product-kernel RKHS in (7). Let ψ_N be a unit vector in W^{s,2}(0,1) with high-frequency oscillation, e.g. ψ_N(x) = c_N sin(Nπx) normalized in W^{s,2}, and set φ_N(x) = x ψ_N(x) ∈ H. Define the rank-one perturbation E_N f = ⟨φ_N, f⟩_H φ_N. Show that ∥E_N∥_{H→H} = 1 while ∥E_N∥_{H→L²} = ∥φ_N∥_{L²} → 0 as N→∞. This disproves the implicit implication 'small H→L² error ⇒ small H→H error' used in Theorem 13. If desired, also compute ∥A−Â∥_{H→H} directly in the Section 5 Van der Pol experiment and compare with the Lemma 10 bound.","verdict_should_be":"REJECT","load_bearing_attack":"The load-bearing step is in Section 4.3. The paper writes\n\n∥T−T̂∥ ≤ ∥Â−A∥² + 2∥A∥·∥Â−A∥ = (ε_{n,δ} + 2∥A∥)ε_{n,δ}\n\nand feeds this into Lemma 11, which is a spectral perturbation statement for self-adjoint operators on H. But Lemma 10 bounds only the H→L²(X) operator norm:\n\n∥Â−A∥_{H→L²} ≤ √μ_{r+1}∥A∥_{H→H} + c_r ε.\n\nThe H→H norm is required for the displayed inequality and for Lemma 11. On a compact domain the H norm dominates L², not conversely; an operator can be O(1/N) in H→L² and have H→H norm 1—for example, the rank-one operator E = ⟨φ_N,·⟩_H φ_N, where φ_N is a unit vector in H that is a high-frequency oscillation. Hence Lemma 10's ε_{n,δ} cannot be substituted for the H→H spectral perturbation error. Even if EDMD errors in practice are benign, the proof of Theorem 13 has a missing premise, and the claimed probabilistic stability certificate is not established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a reproducing kernel Hilbert space (RKHS) construction—the product of a linear kernel and a Wendland radial kernel—on which the Koopman operator of a nonlinear discrete-time system is well-defined and bounded. The authors claim that, under a homeomorphic conjugacy to a linear system, the Koopman spectrum is the semigroup generated by the eigenvalues of the linearization, so that asymptotic stability of the equilibrium is characterized by the spectrum lying in the open unit disk. They then combine a kernel EDMD error bound with spectral perturbation estimates to argue that the finite-rank empirical Koopman operator can provide a probabilistic stability certificate. The theoretical claims are supported by proofs in an appendix and by numerical experiments on a Van der Pol oscillator.","tokens_in":14205,"tokens_out":8610,"duration_ms":78330,"significance":"If the central claims were fully established, the paper would contribute a concrete kernel design that makes the RKHS 'aware' of an equilibrium point, thereby connecting Koopman spectra to stability. The product-kernel idea is novel and potentially useful for Koopman-based control. The paper is honest about strong assumptions (global conjugacy to a linear map, finite hitting time, etc.) and includes a useful appendix discussion of why radial-only kernels cannot provide certificates. However, the main stability-certificate theorem rests on several unproven or incorrect technical steps. The paper does not provide code or machine-checked proofs, and the numerical experiments are illustrative rather than predictive, as they use data whose stability properties are already known. The result, as written, does not meet the standard for a rigorous stability certificate.","major_comments":[{"comment":"Lemma 10 bounds only the operator norm from H to L^2(X): ||Â−A||_{H→L²} ≤ ε_{n,δ}. The subsequent displayed inequality ||T−T̂|| ≤ (ε_{n,δ}+2||A||)ε_{n,δ} and the application of Lemma 11 require an H→H bound on ||Â−A||. On a compact domain the H norm dominates L², not conversely; an operator can be small in H→L² while having H→H norm of order 1 (e.g., a rank-one operator built from a highly oscillatory unit vector in H). The paper never establishes the needed H→H error bound, so the chain leading to Theorem 13 is broken.","section":"Section 4.3, before Lemma 11"},{"comment":"The proof infers that ||Ah|| ≤ max_{λ∈σ(A)} |λ| · ||h|| from the fact that A is diagonal with respect to the monomial basis. This is only valid if the basis is orthonormal (or A is normal). The monomials are not orthogonal in the H inner product, and the proof gives no normality property. Consequently, the spectral radius of A*A is not shown to be |λ₁(F)|², and the discrete-spectrum/contractivity claim needed for Lemma 11 is unsupported. This is a load-bearing gap for the stability certificate.","section":"Lemma 12 and Appendix A.6"},{"comment":"The stated equality σ(A)=⟨σ(F)⟩ is not correct as written. When F is stable with |λ_i|<1, finite products λ_1^{α_1}...λ_d^{α_d} accumulate at 0, so 0 lies in the closure of this set and hence in the spectrum of A (as a limit point), even though 0∉⟨σ(F)⟩. The proof claims (λI−A)^{-1} exists for all λ∉⟨σ(F)⟩; for λ=0 this is false because the eigenvalues accumulate at 0. The correct statement would be σ(A)=closure(⟨σ(F)⟩), which still gives σ(A)⊂D for stable F, but the theorem as stated is false.","section":"Theorem 8 and Appendix A.4"}],"minor_comments":[{"comment":"The phrase 'ifwell-defined' should be 'if well-defined'.","section":"Abstract"},{"comment":"The notation 'κ =' and 'κ −' is visually confusing; consider renaming the kernels (e.g., κ_lin and κ_rad).","section":"Section 3.1"},{"comment":"The numerical section reports only qualitative trajectory and spectrum plots. Quantitative metrics (e.g., prediction error, convergence of the spectrum with n and r) would strengthen the verification. The statement that the spectrum 'overflows the unit circle' for μ=+1 is explicitly dependent on the sampling domain avoiding the limit cycle; this limitation is only discussed in Appendix B and should be flagged in the main text.","section":"Section 5"},{"comment":"The control-theoretic discussion is speculative and contains no theorems; it would be clearer to present it as future work rather than as a contribution.","section":"Section 6"},{"comment":"The claim that 'such a certification did not exist with existing RKHS constructions' is too strong. Existing weighted-space or contractive Koopman constructions can provide stability certificates under different assumptions.","section":"Conclusion"}],"recommendation":"reject","confidential_remarks":"The product-kernel construction is interesting, but the central theorem chain has multiple load-bearing gaps—especially the missing H→H error bound and the invalid use of a non-orthogonal diagonal basis. These are not merely presentation issues; they affect the main claim of providing a stability certificate. The paper would require substantial new analysis to repair, and in its current form I cannot recommend acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The product kernel idea is the real contribution here. Writing H as sums of coordinate functions times Sobolev functions is clean, and Theorem 6 gives a plausible invariance condition under f_i in Phi^s. That part is worth keeping. The numerical study is honest too: the authors explicitly show that when the domain contains the limit cycle, the spectrum is inconclusive, which is more candor than most papers manage.\n\nThe soft spots are not minor, though. Theorem 8 states σ(A) = ⟨σ(F)⟩, but this is false when f is not surjective. If f(X) is a proper subset of X, there are nonzero functions in H supported outside f(X) that are in the kernel of A, so 0 is in σ(A), while 0 is never in the semigroup generated by σ(F). The conjugation by ψ to a linear contraction F automatically makes F non-surjective on a bounded Z, so this is not an edge case; it is precisely the asymptotically stable regime the paper targets. The density argument over polynomials does not rescue this, because the inverse on the polynomial subspace does not have to extend to an inverse on the whole space.\n\nThe second load-bearing gap is the one the stress-test note flags. Lemma 10 gives an error bound in the H→L2 operator norm. Theorem 13 feeds that same epsilon into an inequality for ∥T−T̂∥ that is an H→H statement, and then invokes Lemma 11, which is a spectral perturbation result for self-adjoint operators on H. There is no inequality ∥A−Â∥_{H→H} ≤ C ∥A−Â∥_{H→L2} on a compact domain; high-frequency rank-one operators make the H→H norm arbitrarily large while the H→L2 norm goes to zero. So the probabilistic certificate in Theorem 13 is not established.\n\nLemma 12 has its own problem: it infers contractivity of A from diagonalizability in a monomial basis. That eigenbasis is not orthonormal, and for a non-normal operator the norm can exceed the spectral radius arbitrarily. The inequality ∥Ah∥ ≤ max|λ|∥h∥ does not follow.\n\nThese are not cosmetic issues. They sit on the main claim. The construction itself is salvageable — with explicit surjectivity or a careful treatment of the 0 spectrum, and with an actual H→H error bound or a different spectral perturbation argument — but the paper as written overstates what it proves.\n\nWho should read it: researchers working on Koopman RKHS constructions and data-driven stability certificates. It deserves a serious peer review rather than a desk reject, because the kernel construction is new and the question is important. My recommendation is reject for the current version, but with a clear path to revision: fix the spectrum theorem, fix the norm mismatch, and weaken the claims accordingly.","headline":"The linear–radial product kernel is a genuinely new construction, but the stability certificate is unproven: Theorem 8 misses the spectrum at 0 for non-surjective f, and the error-bound argument uses an H→L2 norm where an H→H norm is required.","tokens_in":14607,"tokens_out":5533,"would_cite":false,"duration_ms":53787,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37M25","47B32","93D05","93C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"A learned Koopman operator can certify whether an equilibrium point is stable, using a linear–radial product kernel.","keywords":["Koopman operator","reproducing kernel Hilbert space","stability analysis","kernel EDMD","Wendland kernel","linear–radial product kernel","spectrum–stability relation","bifurcation detection"],"falsifier":"Find a smooth nonlinear system with an asymptotically stable equilibrium that satisfies the paper's smoothness and homeomorphism conditions, but where the learned Koopman operator (computed via kernel EDMD with the linear–radial product kernel, sufficient sample size, and appropriately chosen rank) has a largest eigenvalue magnitude ≥ 1−ε with high probability. Alternatively, exhibit a diagonalizable non-normal Koopman operator A (as in Lemma 12's setting) for which the derived bound on the spectral radius of A*A fails, invalidating the perturbation argument.","tokens_in":13661,"feed_emoji":"📈","tokens_out":1552,"duration_ms":14297,"temperature":0.7,"pith_summary":"This paper aims to close the gap between the Koopman operator's spectrum and the stability of an equilibrium point in nonlinear dynamical systems. It constructs a reproducing kernel Hilbert space (RKHS) using the product of a linear kernel and a Wendland radial kernel, so the space is both regular enough for the Koopman operator to be well-defined and sensitive to the equilibrium point at the origin. Under smoothness and homeomorphism conditions, the paper proves that if the equilibrium is asymptotically stable, the Koopman spectrum lies inside the unit circle; if the equilibrium becomes unstable through bifurcation, the spectrum escapes. It then shows that a finite-rank approximation learned from data, via kernel extended dynamic mode decomposition (EDMD), inherits this property with high probability, making the learned spectrum a stability certificate.","feed_headline":"Learned Koopman spectrum can certify stability","feed_subtitle":"A linear–radial kernel space makes the learned operator's spectrum reveal whether a nonlinear system's equilibrium is stable, with a probabi","key_machinery":"The central object is the linear–radial product kernel κ(x,x') = xᵀx' · ρ(|x−x'|), where xᵀx' is the linear kernel and ρ is the Wendland radial kernel. This product kernel defines an RKHS H whose members are finite sums of linear coordinate functions multiplied by Sobolev functions, making the space simultaneously regular enough for Koopman invariance and locally linear near the equilibrium. The spectral analysis then uses a homeomorphism to a linear system, reducing the Koopman spectrum to products of the Jacobian eigenvalues, and the stability certificate follows from a perturbation bound between the true and learned operators combined with spectral perturbation theory for self-adjoint ope","core_discovery":"The central claim is that the spectrum of the Koopman operator defined on the RKHS specified by the product kernel κ = κ⁻κ= (linear kernel times Wendland radial kernel) directly reflects the stability of the equilibrium at the origin. Specifically, assuming a Φ^s-homeomorphism conjugating the dynamics to a linear map, the Koopman spectrum equals the multiplicative semigroup generated by the eigenvalues of the Jacobian at the origin. Consequently, asymptotic stability forces the spectrum into the open unit disk, while instability places at least one eigenvalue outside. For the learned finite-rank operator  from kernel EDMD, Theorem 13 states that if |λ₁(Â)| + (ε_{n,δ} + 2||A||)ε_{n,δ} < 1, t","pith_inferences":["The paper's homeomorphism assumption effectively restricts applicability to systems with a single global invariant structure; when an additional attractor like a limit cycle lies in the state space, the learned spectrum reflects the attractor rather than the equilibrium, as App. B shows. A natural extension would be to design kernels that encode multiple invariant structures so the spectrum disamb","The stability certificate requires choosing the Wendland kernel smoothness parameter k in concert with the dynamics' smoothness; a testable practical extension is an adaptive scheme that selects k from data and checks whether the certificate remains valid, potentially using the error bound's constants.","The proposed kernel Lyapunov equation (Eq. 12) is conjectural in the paper; a concrete next step would be to prove existence and uniqueness of its solution P and to bound the estimation error of the resulting Lyapunov function when  replaces A, which would turn the framework into a data-driven stability certificate with a computable Lyapunov function."],"forward_implications":["If the learned Koopman spectrum lies strictly inside the unit circle with the stated margin, one can certify asymptotic stability of the equilibrium without prior stability knowledge, purely from data.","A spectrum escaping the unit circle can serve as a data-driven bifurcation detector, flagging when the equilibrium has become unstable.","The spectrum–stability correspondence opens a route to Koopman-based control synthesis via kernel Lyapunov equations of the form V(f(x))−V(x) = −w(x), where V is a kernel quadratic form.","The framework naturally extends to continuous-time systems through Koopman semigroups, potentially giving Lyapunov certificates for nonlinear control systems.","The algebraic structure of the RKHS (locally linear, globally regular) may support learning Lyapunov functions that are locally quadratic but globally generic, bridging local stability analysis and global dynamics."],"fun_headline_variants":["Koopman spectrum in product kernel certifies stability","Linear-radial kernel confines Koopman spectrum for stable equilibria","Koopman operator spectrum reveals stability and bifurcation","Stability via Koopman spectrum with Wendland kernel","Product kernel enables Koopman spectral stability proof"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The stability certificate relies on the assumption that the dynamics are conjugated to a linear system by a suitable homeomorphism, and on an unproven step where an H→L2 error bound is used as if it were an H→H error bound to control the spectral perturbation of  relative to A.","fun_headline_variants_meta":{"raw":{"variants":["Koopman spectrum in product kernel certifies stability","Linear-radial kernel confines Koopman spectrum for stable equilibria","Koopman operator spectrum reveals stability and bifurcation","Stability via Koopman spectrum with Wendland kernel","Product kernel enables Koopman spectral stability proof"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000246,"raw_usage":{"total_tokens":1374,"prompt_tokens":743,"completion_tokens":631,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":487,"completion_tokens_details":{"reasoning_tokens":562}},"tokens_in":487,"tokens_out":631,"duration_ms":5646,"temperature":1.0,"reasoning_tokens":562,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T23:24:04.549473+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a smooth nonlinear system with an asymptotically stable equilibrium that satisfies the paper's smoothness and homeomorphism conditions, but where the learned Koopman operator (computed via kernel EDMD with the linear–radial product kernel, sufficient sample size, and appropriately chosen rank) has a largest eigenvalue magnitude ≥ 1−ε with high probability. Alternatively, exhibit a diagonalizable non-normal Koopman operator A (as in Lemma 12's setting) for which the derived bound on the spectral radius of A*A fails, invalidating the perturbation argument.","supporting_citations":[],"review_version":1}