{"id":"5f096f90-932f-4634-850b-9aaca8def580","arxiv_id":"2507.15093","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Any block-chain system of LTI and polynomial static blocks has an exact finite-dimensional Koopman embedding, which becomes bilinear when the linear blocks have no feedthrough.","lead":"This paper gives a step-by-step method to turn certain nonlinear systems, built from linear blocks and static polynomial nonlinearities wired in series or parallel, into exact linear-like Koopman models of finite dimension. The method is exact and data-free, so the resulting models can be used for control with reliable guarantees.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Load-bearing concern: the decoupling premise in §2.3.2 is not merely unproved—the stated bound r ≤ n_y n_u is dimensionally impossible for generic polynomial blocks, so Lemma 3.8 does not cover all Σ_NL blocks as claimed.","rationale":"The reader correctly identified the exact polynomial decoupling as the weakest external premise, but treated it mainly as an unproved citation. The stress-test pass strengthens this into a concrete correctness risk: the specific bound r ≤ n_y n_u in §2.3.2 is dimensionally impossible for generic polynomial blocks, so Lemma 3.8 as written does not apply to the full class of Σ_NL blocks quantified over in Theorem 3.1. The internal algebra after the decoupling step appears careful, and the machine-precision simulations provide good evidence for the examples where exact decouplings are known. The concern is therefore not that the algebraic lifting construction is internally inconsistent, but that its input premise overstates what the cited decoupling theorems can deliver. Because exact decoupling is likely still possible for every polynomial with a sufficiently large finite r (e.g., via sums of powers of linear forms), the main existence result may survive with a corrected, larger bound and a modified algorithm; hence a conditional accept is more appropriate than a rejection. The recommended check directly settles whether the stated bound is false and whether the toolbox can produce exact decouplings for generic polynomial blocks.","tokens_in":35812,"tokens_out":11312,"duration_ms":144492,"concrete_test":"Set nu=3, ny=1, p=4 and choose a generic ternary quartic, e.g. f(u) = u_1^4 + u_2^4 + u_3^4 + u_1^2 u_2^2 + u_1^2 u_3^2 + u_2^2 u_3^2 + u_1 u_2 u_3^2 + u_1 u_2^2 u_3 + u_1^2 u_2 u_3 plus lower-order terms with random coefficients. Attempt to solve f(u) = Σ_{e=1}^3 g_e(v_e^T u) with deg g_e ≤ 4 exactly for V and the coefficient vectors of the g_e, using the decoupling toolbox [15] or a direct polynomial-identity solve. If no exact solution exists, the §2.3.2 bound r ≤ n_y n_u is false and Lemma 3.8 cannot support Theorem 3.1 for arbitrary polynomial blocks. A complementary analytical check is to compute the dimension of the coefficient map from (V, g_1, g_2, g_3) into the 35-dimensional polynomial space and confirm that its image has dimension strictly less than 35.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central constructive proof of Theorem 3.1 depends, at every static nonlinear block, on the exact decomposition f_i(u) = W_i g_i(V_i^T u) from §2.3.2. The paper states that [10,51] guarantee such a decomposition for matrix polynomial functions with sufficiently high r ≤ n_y n_u. That bound cannot hold for arbitrary multivariate polynomials. Take ny = 1, nu = 3, p = 4. For fixed V ∈ R^{3×3}, the right-hand side Σ_{e=1}^3 g_e(v_e^T u) with deg g_e ≤ 4 spans at most 1 + 3 + 3 + 3 + 3 = 13 dimensions across the homogeneous components; allowing V to vary adds at most 9 parameters, still far less than the 35-dimensional space of polynomials of degree ≤ 4 in three variables. Hence the set of exactly decouplable polynomials has positive codimension, and a generic ternary quartic has no representation of the form (2.29) with r = 3. Since Lemma 3.8 starts from (3.7c) which assumes (2.29), Theorem 3.1's construction is not valid for the full class of polynomial Σ_NL blocks as stated. The paper flags this as an external premise, but the issue is stronger: the cited bound is inconsistent with a dimension count. If [10,51] only establish decoupling for sufficiently large r, then the algorithm must be amended to allow r to grow with the degree and number of variables, and the resulting decoupling must be computed exactly. Without that correction, the claim that the provided algorithm computes an exact finite-dimensional PITI embedding for every block-chain polynomial system is unsupported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a constructive method for obtaining exact finite-dimensional Koopman embeddings of nonlinear systems that can be represented as block-oriented interconnections of LTI blocks and static polynomial nonlinearities. The main theorem (Theorem 3.1) states that every such block-chain polynomial system admits an exact finite-dimensional PITI Koopman representation of the form (3.1), and Corollary 3.2 gives conditions, namely no feedthrough in the LTI blocks and no static nonlinearity at the start of the chain, under which the representation becomes bilinear. The proof proceeds inductively over the elementary block operations (LD, SN, IJ, OJ), and it yields an iterative algorithm for constructing the embedding. Two numerical examples, a MIMO Wiener-Hammerstein system and a more complex SISO block chain, validate the construction to machine precision. The key external ingredient is the decoupling f_i(u)=W_i g_i(V_i^T u) of each multivariate polynomial block, assumed to exist with r bounded by n_y n_u.","tokens_in":36124,"tokens_out":15115,"duration_ms":149552,"significance":"The constructive, data-free nature of the algorithm is a real strength: the derivations in Appendix A are detailed, the proof gives an explicit state construction at each block operation, and the numerical examples reproduce the original system outputs to about 1e-13. If the scope of the polynomial blocks is stated correctly, the result would be a useful addition to the exact Koopman-embedding literature, going beyond Carleman truncation and giving simple structural conditions for bilinear Koopman models. The paper's main quantitative claim, however, is currently broader than what the decoupling premise supports.","major_comments":[{"comment":"The assertion that every matrix polynomial f admits an exact decoupling (2.29) with 'sufficiently high r <= n_y n_u' is dimensionally impossible for generic blocks, and this premise is load-bearing for Theorem 3.1. Concretely, for n_y=1, n_u=3 and degree p=4, the space of scalar polynomials of degree at most 4 in 3 variables has dimension 35, while the family (2.29) with r<=3 and univariate degree at most 4 has at most 27 parameters (9 in V, 3 in W, and 15 in the gamma coefficients), so a generic ternary quartic cannot be represented in that form. Lemma 3.8 starts from (3.7c), which assumes (2.29) for every Sigma_NL block; since that assumption fails for generic polynomial blocks, Theorem 3.1's claim that every block-chain polynomial system admits the PITI embedding is not supported, and the algorithm as stated cannot be exact for the full stated class. Please either supply a correct finite bound for r (allowing r to grow with the degree and number of variables) together with an exact computation, or restrict the main theorem, corollary, and algorithm to polynomial blocks that explicitly admit an exact decomposition (2.29). The reference to the 'approximate polynomial decoupling' toolbox [15] should also be reconciled with the paper's no-approximation claim.","section":"Section 2.3.2 and Lemma 3.8"}],"minor_comments":[{"comment":"The displayed block vector for \\bar{R}_i(\\bar{u}_i) in Eq. (3.9b) appears to have duplicated or misplaced entries (for example, two consecutive \\bar{R}_{i-1} terms and a \\bar{R}_{i-1}^{(2)}(I \\otimes \\bar{u}_{i-1}) \\bar{R}_{i-1} term); please restate it as a clean block vector indexed by k, consistent with Eq. (A.28) and with the block structure of \\bar{L}_{i,e} in Eq. (3.11).","section":"Lemma 3.8, Eq. (3.9b)"},{"comment":"The PITI definition in Eq. (3.1) says L(z) and \\bar{L}(z) are linear in z, but Lemmas 3.3 and 3.5 set L_1(z_1) \\equiv B_1 and \\bar{L}_1(z_1) \\equiv I, which are constant functions; if 'linear' is meant to include affine terms or relies on a constant coordinate in z, this should be stated explicitly so the linearity claims in Lemma 3.8 and its proof are well-defined.","section":"Section 3.1 and Lemmas 3.3, 3.5"},{"comment":"In Section 4.1.2 and the caption of Figure 11, setting 'D1 = D2 = 0' is inconsistent with the notation of Section 4.1.1, where the two LTI blocks are G1 and G3 and their feedthrough matrices are D1 and D3; the text and caption should refer to D1 and D3.","section":"Section 4.1.2 and Figure 11"},{"comment":"The sentence 'This results in Figure 13b, completing the process' should refer to Figure 13e, since this is the final absorption of the nonlinear block f6.","section":"Section 4.2.9"},{"comment":"There are several typographical slips to correct: 'the the' in Corollary 3.7, 'feedtrough' for 'feedthrough' throughout, 'defiend' in Section 2.3.2, 'appropirate' in Appendix A.1, and 'N > iand' in the proof of Theorem 3.1.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The central construction appears sound conditional on a corrected decoupling statement; the main issue is that Theorem 3.1 currently overclaims for generic polynomial blocks. I see this as fixable within a major revision by making the decoupling an explicit assumption or by proving a correct finite bound, rather than a reason to reject. The paper fits the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper is worth reading for its construction, but the main theorem as stated is not supported. The iterative Kronecker-power embedding of block-chain polynomial systems into PITI/BLTI Koopman form is a genuine advance over the prior autonomous Wiener result. The worked examples match to machine precision, and the appendix derivations look careful.\n\nThe problem is the decoupling premise at Section 2.3.2. The paper claims every polynomial block f: R^nu -> R^ny admits an exact factorization f(u) = W g(V^T u) with r <= n_y n_u, citing [10,51]. That bound is false. A simple scalar example: f(u1,u2)=u1^2 u2 needs three univariate terms (1/6(u1+u2)^3 - 1/6(u1-u2)^3 - 1/3 u2^3), so r=3 > n_y n_u = 2. A dimension count for ny=1, nu=3, p=4 shows the decouplable set has positive codimension in the space of all polynomials. So Lemma 3.8 does not cover all Sigma_NL blocks as claimed, and Theorem 3.1's universality statement fails. This is load-bearing: the construction stops if a block cannot be exactly decoupled.\n\nWhat holds up: the block-chain algebra itself, the PITI-to-BLTI simplification, and the numerical validation. If the theorem is restated as conditional on the existence of an exact decoupling with finite r (dropping the false bound), the construction works; a revision should address the Waring rank of typical polynomial blocks and what r values are really needed. The paper should also explain the stacked Rbar matrix in Lemma 3.8 and prove or properly cite the polynomial decoupling result instead of stating a wrong bound.\n\nWho is this for: the Koopman/control and block-oriented identification communities. It deserves a serious referee, but the referee will need to pin down the decoupling condition and likely send it back for major revision. I would not cite the theorem in its current form.","headline":"The iterative embedding construction is solid, but the main theorem overclaims: the decoupling bound it cites is false, so the universality result fails as stated.","tokens_in":36666,"tokens_out":8544,"would_cite":false,"duration_ms":92678,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47-08","47A15","47A67","47B33","93B28"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every nonlinear system built from a series/parallel chain of LTI and polynomial blocks has an exact finite-dimensional Koopman embedding.","keywords":["Koopman operator","exact finite-dimensional embedding","bilinear systems","block-oriented models","polynomial systems","Wiener-Hammerstein systems","Kronecker products","constructive algorithm"],"falsifier":"Run the paper's algorithm on the MIMO Wiener-Hammerstein example with a third-order polynomial feedthrough term added to the final LTI block, simulate the original system and the PITI model under the same random input, and check whether the output difference stays at rounding level; any deviation above roughly $10^{-12}$ would refute the claimed exactness of Theorem 3.1.","tokens_in":35588,"feed_emoji":"⚙️","tokens_out":8445,"duration_ms":96727,"temperature":0.7,"pith_summary":"This paper claims that every nonlinear system built by connecting linear time-invariant (LTI) blocks and static polynomial blocks in series and parallel, with no feedback, has an exact finite-dimensional Koopman embedding. The embedding takes a polynomial-input time-invariant (PITI) form in which state and output matrices are constant and the input influence enters polynomially; when the LTI blocks have no feedthrough and the chain does not start with a static nonlinearity, the embedding reduces to a bilinear time-invariant system. The proof is constructive and doubles as an algorithm that processes the blocks one by one, producing the model without data or approximation. If true, this removes a central obstacle to using Koopman models for analysis and control design: the model structure and dimension are dictated by the system, not chosen ad hoc.","feed_headline":"Every block-chain polynomial system has an exact finite Koopman model","feed_subtitle":"A constructive algorithm lifts Wiener-Hammerstein-type networks into exact bilinear or polynomial-input models, data-free.","key_machinery":"The machinery is an inductive block-by-block embedding built from three components: the decomposition of each multivariate polynomial static block as $f(u)=Wg(V^\\top u)$ with univariate polynomials $g$, imported from the cited tensor-decomposition literature; Kronecker-power state augmentation $z=[1,z^\\top,(z^{(2)})^\\top,\\dots,(z^{(p)})^\\top]^\\top$, which makes the time derivative of each power a linear function of the augmented state plus a structured input term; and the Kronecker gradient identities $\\partial x^{(i)}/\\partial x\\,Ax={}_iA\\,x^{(i)}$ and $\\partial x^{(i)}/\\partial x\\,B={}_iB\\,x^{(i-1)}$, which convert the derivatives into explicit matrices. Lemmas 3.6, 3.8, 3.10, and 3.12 state how PITI forms compose through series LTI blocks, series nonlinear blocks, input junctions, and output junctions respectively, and together they carry the induction.","core_discovery":"The central discovery is Theorem 3.1: given a nonlinear system represented as a block chain of LTI blocks and static polynomial blocks joined by input and output junctions, the system admits an exact finite-dimensional Koopman representation in the PITI form $\\dot{z}=Az+L(z)R(u)u$, $y=Cz+\\bar{L}(z)\\bar{R}(u)u$, where $L$ and $\\bar{L}$ are linear in the lifted state and $R$ and $\\bar{R}$ are polynomial in the input. Corollary 3.2 adds that if every LTI block has zero feedthrough and the first block is not a static nonlinearity, the representation reduces to the bilinear form $\\dot{z}=Az+\\sum_j\\bar{B}_j z_j u$, $y=Cz$. The proof is inductive over the block chain: each elementary block or junction operation is replaced by a PITI or bilinear building block whose state is augmented with Kronecker powers of the previous lifted state, and these building blocks compose exactly. Simulations of a MIMO Wiener-Hammerstein system and a SISO block chain confirm that the embedded outputs match the original dynamics to roughly $10^{-13}$, i.e. at machine precision.","pith_inferences":["The construction suggests a dimension trade-off: the lifted state size is governed by products of the polynomial degrees encountered along each path, so reordering blocks could change the model dimension; the paper does not discuss this optimization.","Because the construction is exact and parametric, it could serve as a benchmark generator for data-driven Koopman learning, providing ground-truth finite-dimensional embeddings against which learned models can be measured.","The no-feedback restriction is a natural boundary for this approach; systems with algebraic loops or state feedback would need an implicit or fixed-point PITI form, which the present proof does not cover.","If the decoupling order $r$ can be computed a priori from the polynomial degree, a question left to the cited tensor-decomposition literature, the algorithm becomes fully automatic and its dimension and exactness follow from the system data alone."],"forward_implications":["Every Wiener-Hammerstein system, and any series/parallel network of LTI and polynomial blocks without feedback, has a finite-dimensional Koopman model that is exact, with no data and no truncation error involved.","When feedthrough is absent in the LTI blocks and the chain starts with a linear dynamics block, the exact model is bilinear, so existing bilinear analysis and controller-design tools apply without the usual approximation gap.","The lifted state may contain many duplicate entries from Kronecker products, but a linear projection removes them, reducing 17 to 12 and 931 to 103 in the paper's examples while preserving exactness.","If the chain starts with a static nonlinearity or has feedthrough, the input term is polynomial rather than bilinear; the paper notes this can be circumvented by treating the nonlinear block's output as a virtual input, at the cost of complicating control use."],"supporting_citations":[{"why":"Supplies the exact factorization $f(u)=Wg(V^\\top u)$ into univariate polynomials that every static nonlinear block must admit.","marker":"[10]"},{"why":"Extends and deepens the tensor-based decoupling guarantee that the embedding relies on.","marker":"[51]"},{"why":"Provides the input-dependent Koopman decomposition and the Lemma 1 factorization used to put systems with inputs into PITI form.","marker":"[20]"},{"why":"Supplies the Kronecker product identities, including the mixed-product property and Kronecker power rules, used throughout the state augmentation.","marker":"[4]"},{"why":"Gives the differential identity $d(x\\otimes x)=dx\\otimes x+x\\otimes dx$ used in the proof of the Kronecker gradient lemma.","marker":"[33]"},{"why":"Connects differentials to Jacobians, which the derivation of the Kronecker gradient formulas depends on.","marker":"[34]"},{"why":"Provides the computational toolbox that computes the polynomial decoupling for a given order $r$, making the algorithm executable.","marker":"[15]"}],"fun_headline_variants":["Exact finite Koopman for block-oriented polynomial systems","Data-free exact Koopman embeddings to machine precision","Exact Koopman construction for polynomial block networks","Systematic exact Koopman for block-chain polynomial systems","Exact bilinear Koopman for feedthrough-free block systems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction is exact only if every static polynomial block admits an exact decomposition $f(u)=Wg(V^\\top u)$ into univariate polynomials of linear forms, a fact imported from the cited literature rather than proved here; if that decomposition is approximate or its required order is not attainable, the resulting model is not an exact embedding.","fun_headline_variants_meta":{"raw":{"variants":["Exact finite Koopman for block-oriented polynomial systems","Data-free exact Koopman embeddings to machine precision","Exact Koopman construction for polynomial block networks","Systematic exact Koopman for block-chain polynomial systems","Exact bilinear Koopman for feedthrough-free block systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001552,"raw_usage":{"total_tokens":6203,"prompt_tokens":943,"completion_tokens":5260,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":559,"completion_tokens_details":{"reasoning_tokens":5179}},"tokens_in":559,"tokens_out":5260,"duration_ms":40881,"temperature":1.0,"reasoning_tokens":5179,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:42:09.974021+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the paper's algorithm on the MIMO Wiener-Hammerstein example with a third-order polynomial feedthrough term added to the final LTI block, simulate the original system and the PITI model under the same random input, and check whether the output difference stays at rounding level; any deviation above roughly $10^{-12}$ would refute the claimed exactness of Theorem 3.1.","supporting_citations":[{"cited_title":"Dreesen, M","cited_arxiv_id":null,"evidence_quote":"Supplies the exact factorization $f(u)=Wg(V^\\top u)$ into univariate polynomials that every static nonlinear block must admit."},{"cited_title":"Usevich, P","cited_arxiv_id":null,"evidence_quote":"Extends and deepens the tensor-based decoupling guarantee that the embedding relies on."},{"cited_title":"Bernstein, Matrix Mathematics: Theory, Facts, and Formulas: Second Edition , Princeton University Press, second ed., 2009","cited_arxiv_id":null,"evidence_quote":"Supplies the Kronecker product identities, including the mixed-product property and Kronecker power rules, used throughout the state augmentation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the differential identity $d(x\\otimes x)=dx\\otimes x+x\\otimes dx$ used in the proof of the Kronecker gradient lemma."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Connects differentials to Jacobians, which the derivation of the Kronecker gradient formulas depends on."},{"cited_title":"Hollander, P","cited_arxiv_id":null,"evidence_quote":"Provides the computational toolbox that computes the polynomial decoupling for a given order $r$, making the algorithm executable."}],"review_version":1}