{"id":"b3c8167a-7615-4e1e-84c0-6073429c4d1b","arxiv_id":"2608.06843","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A phi-weighted generalized Barron space B^phi_sigma is defined for shallow networks and shown to embed into Sobolev spaces, to unify known Barron-type spaces, and to yield approximation and regularization bounds.","lead":"This paper introduces a generalized Barron space for shallow neural networks, using a norm function that controls how much parameter growth is penalized. It proves Sobolev embeddings, norm equivalences with classical Barron spaces, and regularization error bounds, with numerical illustrations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Remark 2.4's counterexample is not in the Barron space: its norm series diverges, so the necessity of condition (5) is not established.","rationale":"I verified the core mathematical steps supporting the strongest claim. Proposition 2.2 is sound: the space B^phi_sigma is indeed isometrically isomorphic to M^phi/ker(N_sigma), and the L^infinity embedding follows from monotonicity of phi. Theorem 2.3's proof is also correct: the inequality (1+||w||_1)^m phi(|w x + b|)/(1+|w x + b|)^m <= phi(||w||_1 + |b|) follows from monotonicity of phi_m and (1+||w||_1) <= (1+||w||_1+|b|), so the Fubini/interchange argument is valid. Thus the central conditional claim holds. The reader's weakest_assumption correctly identifies condition (5) as the load-bearing premise. My stress-test sharpens this: the one place the paper argues for the condition's necessity, Remark 2.4, contains a concrete numerical error (a divergent norm series), so the necessity claim is unsupported. This does not falsify the theorem, but it weakens the motivation and the characterization of admissible (sigma, phi) pairs. Peripheral issues noted by the reader (Theorem 3.2 stated without proof, unreproducible numerics, Data Availability contradiction) are real but secondary. The verdict CONDITIONAL remains appropriate; my concern does not increase or decrease the level of conditionality.","tokens_in":22975,"tokens_out":22223,"duration_ms":223480,"concrete_test":"Compute the partial sums S_N = sum_{i=1}^N 2^{6-i} pi^{-1}(1+13^i pi). They grow without bound, confirming that the stated bound is false. Then replace the coefficients with a_i = C i 2^{-2i} (or any sequence satisfying sum_i a_i 13^i < infinity but sum_i a_i 13^{2i} = infinity), and verify that f_corrected(x) = sum_i a_i sin(13^i pi x) lies in B^{1+ReLU}_sin while its second derivative series diverges, so the intended conclusion of Remark 2.4 can be restored. Separately, test the internal consistency of condition (5): verify for sigma = ReLU^s and phi = 1+ReLU^t, t >= s, that phi_k is non-decreasing for k <= s, so the theorem applies in the standard extended Barron setting.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central Sobolev embedding theorem (Theorem 2.3) is conditional on the matching condition (5), which controls the activation derivatives by phi_m. The paper's only explicit attempt to justify that this condition is necessary is Remark 2.4. There, for sigma(x)=sin(x) and phi(x)=1+ReLU(x), the authors define f(x)=sum_{i=1}^\\infty 2^{6-i} pi^{-1} sin(13^i pi x) and claim ||f||_{B^phi_sin} <= sum_i 2^{6-i} pi^{-1}(1+13^i pi) < infinity. This is incorrect: the terms behave like (13/2)^i, so the series diverges. Consequently, f is not a member of the space B^phi_sin, and the remark does not demonstrate B^phi_sin not subset of W^{2,infinity}. The necessity of the monotonicity of phi_k remains unsubstantiated, and the admissible pairs (sigma, phi) are characterized only by a stated, untested condition. This is load-bearing because the framework's motivation is that phi can be tuned to the activation; without a valid necessity argument or a broader existence result for phi, the practical scope of the embedding theorem is unclear. The proof of Theorem 2.3 itself appears correct, so the concern is about the claim that condition (5) is necessary and about the paper's motivation for the definition.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces a generalized Barron space B^phi_sigma for shallow neural networks with a generic activation sigma and a norm function phi, defined by phi-weighted integral norms over parameter measures. It proves that B^phi_sigma is a Banach space continuously embedded in L^infinity (Proposition 2.2), and that under a matching condition between phi and the derivatives of sigma it embeds continuously into W^{k,infinity} (Theorem 2.3). It then relates these spaces to extended Barron spaces and spectral Barron spaces via norm equivalences (Propositions 2.5 and 2.6), establishes an embedding criterion between generalized spaces (Theorem 2.9), derives approximation rates in type-p spaces (Theorems 3.1 and 3.2), and gives error bounds for Tikhonov regularization with the generalized Barron norm (Proposition 3.3 and Theorem 3.4). Numerical experiments in one and five dimensions illustrate the effect of the norm function on derivative approximation.","tokens_in":23262,"tokens_out":24055,"duration_ms":223483,"significance":"If correct, the framework provides a unified and flexible way to associate a norm with any sufficiently regular activation function, and the Sobolev embedding theorem (Theorem 2.3) cleanly explains how the choice of phi controls the regularity of the realized functions. The proof of the Banach-space property and the main embedding are self-contained and appear sound, and the norm equivalences with classical spaces are plausible and largely verified. The regularization application is a useful addition. However, the paper currently contains an invalid necessity example (Remark 2.4) and states a central approximation-rate theorem (Theorem 3.2) without proof; these issues must be fixed before the claims can be fully accepted.","major_comments":[{"comment":"For f(x)=sum_{i=1}^infty 2^{6-i} pi^{-1} sin(13^i pi x) in Remark 2.4, the bound ||f||_{B^{1+ReLU}_{sin}} <= sum_i 2^{6-i} pi^{-1}(1+13^i pi) is incorrect because the summand behaves like 2^6 (13/2)^i and the series diverges. Moreover, the displayed derivative f'(x)=sum 2^{-i} cos(13^i pi x) is not the derivative of the displayed f, whose derivative coefficients are 2^6 (13/2)^i. Therefore the example does not show B^{1+ReLU}_{sin} not subset of W^{2,infinity}, and the claimed necessity of condition (5) is not established. The authors should correct the coefficients (e.g., use amplitudes 2^{6-i}(13^i pi)^{-1}) or remove the necessity claim and state that condition (5) is sufficient.","section":"Remark 2.4 (Section 2.1)"},{"comment":"The higher approximation rate in Theorem 3.2 is stated without proof; the text says 'For brevity, we omit the detailed proof.' This is a central application advertised in the abstract, and the result involves new elements (tail decay theta(R), the Lip^infty(s,X) condition on P_sigma/phi, and the exponent alpha/(s+alpha) * s/(d+1) + 1/p - 1) that are not immediate consequences of [32]. The paper should either supply a complete proof in an appendix or clearly mark the theorem as conditional on a full adaptation of [32].","section":"Theorem 3.2 (Section 3.1)"}],"minor_comments":[{"comment":"The proof of Proposition 2.5 contains garbled notation in the definition of rho_g (e.g., 'R+1_D' and missing exponents) and the choice g=||w||_1+|b| is undefined where ||w||_1+|b|=0. Rewriting the change of variables with a positive measurable g and explicitly treating the zero set would make the proof readable.","section":"Proposition 2.5 (Section 2.2)"},{"comment":"The same symbol B^s is used for different spaces: in Proposition 2.5 it denotes the extended Barron space defined via ReLU^s parameter norms, while in Proposition 2.6 and Remark 2.7 it denotes the spectral Barron space defined via weighted Fourier norms. This overloading makes claims such as the equivalence in Remark 2.7 ambiguous and should be fixed with distinct notations.","section":"Sections 2.2-2.3"},{"comment":"The discussion emphasizes variance across random seeds, but the tables report only mean relative errors; reporting standard deviations or a variance measure would support the stability claims made in the text.","section":"Section 4.2, Tables 1-4"},{"comment":"The theorem assumes phi_k is non-decreasing and positive, while the proof uses non-decreasing phi_m for all m <= k. This follows because phi_m = phi_k (1+x)^{k-m}, but stating this implication explicitly would improve readability.","section":"Theorem 2.3 (Section 2.1)"},{"comment":"The approximation exponent is printed without parentheses as n^{-alpha/(s+alpha) * s/d+1 + 1/p - 1}, which is ambiguous; the intended expression should be written as n^{- (alpha/(s+alpha))*(s/(d+1)) + 1/p - 1}.","section":"Theorem 3.2 (Section 3.1)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the journal's scope in numerical analysis, and the core Sobolev embedding result (Theorem 2.3) is sound. The main barriers to acceptance are the missing proof of Theorem 3.2 and the erroneous example in Remark 2.4; both are fixable within the manuscript's scope. I see no concerns about novelty or citation coverage beyond these technical points."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: the phi-weighted Barron space definition and the Sobolev embedding criterion are the real contribution, and they are mostly sound. But the paper's only explicit necessity argument (Remark 2.4) is wrong, and the higher-rate approximation theorem is stated without proof. Both need fixing before I'd take the framework as fully supported.\n\nWhat is genuinely new: B^phi_sigma with an arbitrary norm function phi satisfying the growth condition is a natural unification of spectral, extended, and Lipschitz Barron spaces. Proposition 2.2 gives a clean Banach-space and L^infty-embedding proof. Theorem 2.3 is a clear sufficient condition: if phi_m controls the m-th derivative of sigma and phi_k is non-decreasing, you get W^{k,infty} with the Barron norm as bound. The norm equivalences with spectral and extended Barron spaces (Propositions 2.5 and 2.6) are concrete and checkable, and Theorem 2.9 is a genuinely general kernel-based transfer principle that goes beyond the fixed-weight treatments in [11,14].\n\nSoft spots, in order of severity.\n\n1. The stress-test note lands. In Remark 2.4 the function f has norm series terms ~64*(13/2)^i, so the sum diverges and f is not in B^{1+ReLU}_{sin}. The remark therefore does not establish that condition (5) is necessary, and the claim that it does is unsubstantiated. This does not invalidate Theorem 2.3 as a conditional statement, but it removes the paper's only evidence about what happens when phi grows too slowly. A valid counterexample or a weaker statement is needed.\n\n2. Theorem 3.2, the advertised higher approximation rate, is stated without proof. The three-stage sketch is not enough for a central application. It needs a complete proof in the appendix, or a precise reference where the proof is given.\n\n3. The numerical experiments have no error bars, no code, and no data, and the Data Availability Statement says no datasets were generated though the experiments clearly use synthetic data. Minor relative to the math, but it should be fixed.\n\nThe proof of Proposition 2.5 also has a reparameterization step that is sketched rather than fully justified; I would ask the authors to expand it, though I expect it is repairable.\n\nWho this is for: researchers working on Barron-space theory, neural network approximation, and regularized Sobolev training. The core framework is a useful organizing device even with the current flaws.\n\nRecommendation: send it to peer review. A serious referee should focus on Remark 2.4 and the missing proof of Theorem 3.2. If the authors fix those, this is a citable contribution; if not, the embedding theorem alone may still be worth publishing.","headline":"A genuinely useful phi-weighted Barron space framework with a clean Sobolev embedding theorem, but the necessity argument in Remark 2.4 has a divergence error and the higher-rate approximation theorem is stated without proof.","tokens_in":23800,"tokens_out":3037,"would_cite":true,"duration_ms":30766,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46E35","41A25","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"A generalized Barron space $B^\\varphi_\\sigma$ is a Banach space whose $\\varphi$-weighted parameter norm controls Sobolev derivatives up to order $k$ when the activation and $\\varphi$ satisfy a matching growth condition.","keywords":["generalized Barron spaces","shallow neural networks","Sobolev embedding","norm function","activation function","approximation rates","Tikhonov regularization","spectral Barron spaces"],"falsifier":"For $\\sigma(x)=\\sin(x)$ and $\\varphi(x)=1+\\mathrm{ReLU}(x)$, the paper's own Remark 2.4 constructs a function with finite $B^\\varphi_\\sigma$-norm whose derivative is a nowhere-differentiable trigonometric series, so $B^\\varphi_{\\sin}$ is not embedded in $W^{2,\\infty}$ exactly when condition (5) fails for $m=2$. To test the sufficiency side, repeat the construction with $\\sigma(x)=\\sin(x)$ and $\\varphi(x)=\\exp(x)$, where (5) holds for every $k$: the theorem predicts that the $k$-th derivative of every finite-$\\varphi$-norm function is bounded by a constant times the norm, so numerical differentiation of the same series should show bounded high-order derivatives whose bounds scale with the $\\varphi$-norm; any divergence of these derivatives while the $\\varphi$-mass stays bounded would refute the embedding.","tokens_in":22761,"feed_emoji":"🧠","tokens_out":13649,"duration_ms":119216,"temperature":0.7,"pith_summary":"Classical Barron spaces describe functions that shallow neural networks can approximate well, but they are tied to particular activations such as ReLU or RePU and to parameter norms that rely on homogeneity. This paper proposes a generalized Barron space $B^\\varphi_\\sigma$ in which a non-decreasing norm function $\\varphi$ is paired with an arbitrary activation $\\sigma$, and the norm is the infimum of $\\int \\varphi(\\|w\\|_1+|b|)\\,d|\\rho|$ over all parameter measures representing the function. The central claim is that this space is a Banach space continuously embedded in $L^\\infty$, and that when the activation's derivatives satisfy a matching growth bound against $\\varphi$, every function in $B^\\varphi_\\sigma$ has bounded Sobolev derivatives up to order $k$ with norm controlled by the generalized Barron norm. If true, the choice of $\\varphi$ becomes a dial for smoothness: faster-growing $\\varphi$ yields higher-order Sobolev regularity while keeping the same activation function, and the framework unifies spectral and (extended) Barron spaces as special cases. The paper also derives dimension-independent approximation rates and uses the $\\varphi$-norm as a Tikhonov regularizer, with numerical tests showing stable high-order derivative recovery.","feed_headline":"One weight function dials the smoothness of Barron spaces","feed_subtitle":"Pair any activation with a norm function φ: parameter mass then bounds Sobolev derivatives up to order k.","key_machinery":"The central object is the generalized Barron space $B^\\varphi_\\sigma$ defined by the integral representation and the weighted norm in Definition 2.2, together with the norm function $\\varphi$ from Definition 2.1. The machinery that carries the argument is the compatibility condition (5) in Theorem 2.3: each derivative $\\partial^m\\sigma$ is dominated by $C\\varphi(|x|)/(1+\\mathrm{ReLU}(|x|))^m$ with $\\varphi_k$ non-decreasing. This inequality is what permits differentiating under the integral to obtain the weak-derivative formula (6) and then to bound $|\\partial^\\alpha f(x)|$ by $C\\|f\\|_{B^\\varphi_\\sigma}$; without it, the $\\varphi$-norm does not control high-order derivatives, as the sine activation with $\\varphi=1+\\mathrm{ReLU}$ shows. A second device, the finite signed complex kernel of Definition 2.3 and Lemma 2.8, carries the embedding relations between spaces with different $\\sigma$ and $\\varphi$ pairs, including the Taylor and Fourier representations used in Propositions 2.5, 2.6, and 2.11.","core_discovery":"In the author's own terms, functions represented as $f(x)=\\int \\sigma(w\\cdot x+b)\\,d\\rho(w,b)$ form a normed Banach space $B^\\varphi_\\sigma$ when the parameter measure is weighted by $\\varphi(\\|w\\|_1+|b|)$, and this space is continuously embedded in $L^\\infty$. Under the matching condition $|\\partial^m \\sigma(x)|\\le C\\varphi_m(|x|)$ with $\\varphi_m=\\varphi/(1+\\mathrm{ReLU})^m$ and $\\varphi_k$ positive and non-decreasing, the same $\\varphi$-norm controls all weak derivatives up to order $k$, giving the embedding $B^\\varphi_\\sigma\\hookrightarrow W^{k,\\infty}(\\Omega)$. Consequently the generalized Barron norm simultaneously measures parametric complexity and classical smoothness, and the framework recovers the extended Barron spaces for $\\mathrm{ReLU}^s$ activations (with $\\varphi=1+\\mathrm{ReLU}^t$, $t\\ge s$) and the spectral Barron spaces for $\\sigma=\\exp(i2\\pi x)$ (with $\\varphi=(1+\\mathrm{ReLU})^s$) with equivalent norms.","pith_inferences":["If the embedding is sharp, one could select $\\varphi$ from data to prescribe the Sobolev order of the function class for a fixed activation, making regularity an explicit architectural knob rather than a property of the activation.","The $\\varphi$-weighted regularizer is an activation-aware alternative to Sobolev-norm penalties for derivative recovery and PDE problems; the paper's own 5D experiments suggest a milder growth rate may be preferable in high dimensions, a trade-off it does not fully resolve.","The kernel-based embedding theorem provides a general recipe: any change of activation representable as an integral kernel with controlled $\\varphi$-mass yields an embedding between generalized Barron spaces, so wavelet or other atomic representations could generate new embedding chains beyond the Taylor and Fourier cases.","A direct testable extension is to check numerically whether the predicted $W^{k,\\infty}$ bounds hold with reasonable constants for oscillatory activations with exponential $\\varphi$ on random finite networks, or whether the constant in condition (5) makes the bound vacuous in practice."],"forward_implications":["If $f$ has finite $\\varphi$-norm and condition (5) holds, then the weak derivatives of $f$ up to order $k$ are bounded by a constant times its Barron norm, so the $\\varphi$-norm acts as a control on smoothness during training.","Choosing a faster-growing $\\varphi$ raises the guaranteed Sobolev order for the same activation; for example, an exponential $\\varphi$ gives $W^{k,\\infty}$ embeddings for every $k$ for activations such as sine whose derivatives are globally bounded.","The norm equivalences with spectral and extended Barron spaces mean that existing approximation and regularity results for those spaces transfer to the generalized spaces with their $\\varphi$-weights.","A type-p sampling argument yields dimension-free approximation rates of order $O(n^{1/p-1})$ and faster rates under tail-decay and smoothness conditions, extending known rates to non-homogeneous activations.","Tikhonov regularization penalized by the discrete $\\varphi$-weighted cost gives $W^{m,p}$ error bounds of order $(\\delta+r_n)^{(k-m)/k}$ under the parameter choice $\\delta+r_n\\asymp\\lambda^{1/p}$."],"supporting_citations":[{"why":"Introduced flow-induced Barron spaces and the Lipschitz-type Barron norm that the generalized definition extends.","marker":"[11]"},{"why":"Established prior embeddings between Barron spaces with higher-order activations, including the Taylor-expansion representation and the theorem that $B^{1+\\mathrm{ReLU}}_\\sigma\\hookrightarrow B^k$.","marker":"[14]"},{"why":"Defined extended Barron spaces for ReLU$^k$ and proved the embedding $B^k\\hookrightarrow W^{k,\\infty}$ used as the baseline for the Sobolev embedding.","marker":"[19]"},{"why":"Originated spectral Barron spaces with $(1+|\\xi|)^s$ Fourier weights, which Proposition 2.6 recovers as a special case.","marker":"[4]"},{"why":"Supplied the finite signed complex kernel machinery used in Theorem 2.9 for embeddings between different $(\\sigma,\\varphi)$ pairs.","marker":"[8]"},{"why":"Provided the type-p sampling argument and sharp approximation-rate framework adapted in Theorems 3.1 and 3.2.","marker":"[32]"},{"why":"Introduced the exponential spectral Barron spaces $B^{\\beta,c}$ used in Proposition 2.11 for the sigmoid embedding.","marker":"[30]"},{"why":"Demonstrated that oscillatory activations such as sine are practically useful, motivating the Remark 2.4 counterexample that shows condition (5) is necessary.","marker":"[33]"}],"fun_headline_variants":["One weight function tunes Barron space smoothness","Barron spaces get a smoothness dial via weight φ","One norm φ controls both complexity and smoothness in Barron spaces","Generalized Barron spaces: flexibility in smoothness, same activation","From Barron to Sobolev: one weight function does it all"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the matching condition (5): for each $m\\le k$, the $m$-th derivative of the activation $\\sigma$ must be bounded by $C\\varphi(|x|)/(1+\\mathrm{ReLU}(|x|))^m$ with $\\varphi_k$ positive and non-decreasing; if $\\varphi$ grows too slowly relative to the activation's derivatives, the $\\varphi$-norm no longer guarantees Sobolev regularity, as the sine example shows.","fun_headline_variants_meta":{"raw":{"variants":["One weight function tunes Barron space smoothness","Barron spaces get a smoothness dial via weight φ","One norm φ controls both complexity and smoothness in Barron spaces","Generalized Barron spaces: flexibility in smoothness, same activation","From Barron to Sobolev: one weight function does it all"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000255,"raw_usage":{"total_tokens":1599,"prompt_tokens":1001,"completion_tokens":598,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":617,"completion_tokens_details":{"reasoning_tokens":512}},"tokens_in":617,"tokens_out":598,"duration_ms":5771,"temperature":1.0,"reasoning_tokens":512,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T19:47:44.924042+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $\\sigma(x)=\\sin(x)$ and $\\varphi(x)=1+\\mathrm{ReLU}(x)$, the paper's own Remark 2.4 constructs a function with finite $B^\\varphi_\\sigma$-norm whose derivative is a nowhere-differentiable trigonometric series, so $B^\\varphi_{\\sin}$ is not embedded in $W^{2,\\infty}$ exactly when condition (5) fails for $m=2$. To test the sufficiency side, repeat the construction with $\\sigma(x)=\\sin(x)$ and $\\varphi(x)=\\exp(x)$, where (5) holds for every $k$: the theorem predicts that the $k$-th derivative of every finite-$\\varphi$-norm function is bounded by a constant times the norm, so numerical differentiation of the same series should show bounded high-order derivatives whose bounds scale with the $\\varphi$-norm; any divergence of these derivatives while the $\\varphi$-mass stays bounded would refute the embedding.","supporting_citations":[{"cited_title":"The Barron space and the flow-induced function spaces for neural network models.Constr","cited_arxiv_id":null,"evidence_quote":"Introduced flow-induced Barron spaces and the Lipschitz-type Barron norm that the generalized definition extends."},{"cited_title":"Schwenninger, and Christoph Brune","cited_arxiv_id":null,"evidence_quote":"Established prior embeddings between Barron spaces with higher-order activations, including the Taylor-expansion representation and the theorem that $B^{1+\\mathrm{ReLU}}_\\sigma\\hookrightarrow B^k$."},{"cited_title":"Pereverzev","cited_arxiv_id":null,"evidence_quote":"Defined extended Barron spaces for ReLU$^k$ and proved the embedding $B^k\\hookrightarrow W^{k,\\infty}$ used as the baseline for the Sobolev embedding."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Originated spectral Barron spaces with $(1+|\\xi|)^s$ Fourier weights, which Proposition 2.6 recovers as a special case."},{"cited_title":"Cohn.Measure theory","cited_arxiv_id":null,"evidence_quote":"Supplied the finite signed complex kernel machinery used in Theorem 2.9 for embeddings between different $(\\sigma,\\varphi)$ pairs."},{"cited_title":"Siegel and Jinchao Xu","cited_arxiv_id":null,"evidence_quote":"Provided the type-p sampling argument and sharp approximation-rate framework adapted in Theorems 3.1 and 3.2."},{"cited_title":"Siegel and Jinchao Xu","cited_arxiv_id":null,"evidence_quote":"Introduced the exponential spectral Barron spaces $B^{\\beta,c}$ used in Proposition 2.11 for the sigmoid embedding."},{"cited_title":"Implicit neural representations with periodic activation functions","cited_arxiv_id":null,"evidence_quote":"Demonstrated that oscillatory activations such as sine are practically useful, motivating the Remark 2.4 counterexample that shows condition (5) is necessary."}],"review_version":1}