{"id":"e53158d5-f137-46db-94a8-7bb5bd756df5","arxiv_id":"2607.06290","paper_version":1,"verdict":"ACCEPT","confidence":"UNKNOWN","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Finite-width executions of Netsor tensor programs converge to their infinite-width Gaussian-process limits in Wasserstein distance at rate O(1/√n) per hidden width, covering weight-sharing architectures including RNNs and attention.","lead":"The paper proves that random neural networks with shared weights (including RNNs and attention layers) converge to Gaussian processes at a rate of 1/√width in Wasserstein distance. This gives engineers and theorists explicit finite-width error bounds for architectures where prior quantitative results did not apply.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"Non-degeneracy reduction for basic Netsor relies on lifting infinite-width linear identities to finite-width, but the argument has a gap when the identity depends on random (not deterministic) coefficients.","rationale":"The reader correctly identified the non-degeneracy assumption for Netsor K (Theorem 2.4) as the weakest point. My analysis confirms this: the basic Netsor non-degeneracy reduction (Proposition 3.11) is sound because the coefficients αᵢ are deterministic and the open-set argument works for jointly non-degenerate Gaussians. However, for Netsor K, Remark 2.5 explicitly acknowledges that the reduction fails because scalar kernel variables are deterministic at infinite width but random at finite width, breaking the lifting argument. This means Theorem 2.4's coverage of attention architectures is conditional on a property that (a) must be verified case-by-case, (b) could fail for practical configurations, and (c) cannot be reduced away. The main theorem (Theorem 1.2) for basic Netsor is well-supported: the line-by-line induction correctly handles the circularity with the kernel LLN, the structure lemmas are clean, and the non-degeneracy reduction is valid. The O(1/√n) rate is honest. The Lipschitz requirement is stated, not hidden. The numerical experiments are preliminary but transparent about the sampling noise floor. The paper makes a solid contribution: quantitative Wasserstein bounds for tensor programs with weight sharing, extending prior feed-forward-only results. The Netsor K extension is honestly stated with its limitation. The verdict of ACCEPT is appropriate: the central claim (Theorem 1.2) is well-proven, and Theorem 2.4 is a conditional extension that is transparently flagged as such. No verdict change is needed, but the reader's identification of the non-degeneracy assumption as the weakest point is correct and important for anyone applying Theorem 2.4 to attention architectures.","tokens_in":33295,"tokens_out":1126,"duration_ms":806192,"concrete_test":"Construct a minimal Netsor K program for a single-head attention layer (Algorithm 5) with u=2 tokens and width n, and compute the joint covariance of all G-vars (q_1, q_2, k_1, k_2, v_1, v_2) in the infinite-width execution. Check whether this covariance matrix is full-rank (non-degenerate). If it is degenerate for any input configuration, Theorem 2.4 does not apply to standard attention. If it is always non-degenerate, the concern about attention coverage is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The non-degeneracy reduction (Lemma 3.9 / Proposition 3.11) is load-bearing for Theorem 1.2's proof: it ensures Gram matrices are invertible at the infinite-width point, so the maps involving pseudoinverses and matrix square roots are differentiable there (§4.2, 'Covariance stability'). The key step is Proposition 3.11(ii)⇒(iii): if h̄ lies in the L²-span of previous parents {h̄⁽ⁱ⁾}, then there exist deterministic αᵢ such that ḡ = Σαᵢḡ⁽ⁱ⁾ a.s. (infinite-width), and the claim is that the same linear relation g = Σαᵢg⁽ⁱ⁾ holds in finite-width execution too. The proof argues: since the previous G-vars are non-degenerate Gaussians and Φ(ḡ⁽ʲ⁾)=0 a.s. with Φ continuous, the zero set is open and hence all of R^k', so Φ(g⁽ʲ⁾)=0 a.s. in finite-width. This works when the coefficients αᵢ are deterministic constants. However, the αᵢ arise from the L²-span relation h̄ = Σαᵢh̄⁽ⁱ⁾, and in programs with weight sharing (the paper's main contribution), the parents h⁽ⁱ⁾ can themselves depend on previously generated G-vars from the same A-var W. The coefficients αᵢ = (H_W^{(ℓ-1),*} H_W^{(ℓ-1)})⁺ (h̄ * H_W^{(ℓ-1)}) are computed from infinite-width Gram data, which are deterministic. So the αᵢ are indeed deterministic. The concern narrows: the argument that Φ's zero set being open requires non-degeneracy of ALL previous G-vars jointly, not just marginally. In a program with weight sharing, joint non-degeneracy of G-vars using the same W requires that the parents h⁽ⁱ⁾ be linearly independent in L², which is assumed by the sequential reduction. This appears sound for basic Netsor. The real gap is in Theorem 2.4 (Netsor K): Remark 2.5 explicitly states the reduction fails because 'a scalar kernel variable is deterministic in infinite-width but random at finite width, so a linear relation that holds identically in the infinite-width execution need not hold identically in the finite-width execution.' This means Theorem 2.4's non-degeneracy assumption is not merely a checkable condition but a非","agreement_with_reader":"agree"},"referee_report":{"model":"glm-5.2","summary":"This paper establishes quantitative Wasserstein-distance convergence rates (of order sum of inverse square roots of hidden widths) between finite-width and infinite-width executions of Netsor tensor programs. The main result (Theorem 1.2) covers architectures with weight sharing (RNNs, multi-input evaluation) that prior feed-forward quantitative CLTs could not handle, via a line-by-line inductive coupling argument. An extension to Netsor K programs (Theorem 2.4) covers attention-type layers under a non-degeneracy assumption on G-vars. The proof uses structure lemmas for conditional Gaussian laws, a non-degeneracy reduction, and a kernel LLN (Corollary 1.3) that serves both as a consequence and an inductive ingredient.","tokens_in":33596,"tokens_out":1693,"duration_ms":473243,"significance":"The paper makes a genuine contribution by extending quantitative Gaussian-process convergence bounds beyond feed-forward architectures to the tensor-program framework, which naturally accommodates weight sharing. The line-by-line induction correctly handles the apparent circularity between Theorem 1.2 and Corollary 1.3 by using the theorem only for truncated programs at each inductive step. The structure lemmas (3.5, 3.6) cleanly characterize conditional laws in both finite and infinite-width executions. The Netsor K extension and the explicit acknowledgment of the non-degeneracy limitation (Remark 2.5) are commendable. The numerical experiments in Section 5, while not theorem-level for non-MLP architectures, provide reasonable empirical support.","major_comments":[{"comment":"Proposition 3.11, proof of (ii)=>(iii): The argument that Phi's zero set being open requires non-degeneracy of ALL previous G-vars jointly, not just marginally. The proof states: 'Since the (g_bar^(j)) are non-degenerate, the set {z : Phi(z) != 0} has null Lebesgue measure, and by continuity of Phi it is an open set, hence empty.' This step requires that the joint law of (g_bar^(j))_{j=1,...,k'} be non-degenerate (i.e., supported on all of R^{k'}). The non-degeneracy assumption in Lemma 3.9 is stated conditionally ('conditionally upon F_{l-1}'), but the proof of Proposition 3.11(ii)=>(iii) appears to use marginal non-degeneracy of the joint vector. In programs with weight sharing, the parents h^(i) can depend on previously generated G-vars from the same A-var W, so joint non-degeneracy of the G-vars is not automatic from conditional non-degeneracy at each step. The authors should clarify","section":null},{"comment":"Proposition 3.11, proof of (ii)=>(iii): The lifting from infinite-width to finite-width linear identities is the load-bearing step for the non-degeneracy reduction (Lemma 3.9), which in turn is needed for differentiability of the maps involving pseudoinverses and matrix square roots in Section 4.2 ('Covariance stability'). The proof argues that since Phi(g_bar^(j)) = 0 a.s. (infinite-width) and the zero set is open, Phi vanishes identically, hence Phi(g^(j)) = 0 a.s. in finite-width too. This works when the coefficients alpha_i are deterministic constants, which they are (computed from infinite-width Gram data). However, the argument that the zero set {z : Phi(z) != 0} is open AND has null Lebesgue measure requires that the joint distribution of (g_bar^(j)) has full support. The proof should explicitly verify that the non-degeneracy hypothesis (conditional non-degeneracy at each MatMul) ","section":null},{"comment":"Theorem 2.4 (Netsor K extension): The non-degeneracy assumption is stated as 'its G-vars are jointly non-degenerate Gaussians in the infinite-width execution.' Remark 2.5 honestly acknowledges that the reduction does not lift because 'a scalar kernel variable is deterministic in infinite-width but random at finite width, so a linear relation that holds identically in the infinite-width execution need not hold identically in the finite-width execution.' This means Theorem 2.4's coverage of attention architectures is conditional on a property that must be verified case-by-case. The paper should provide at least one concrete example (e.g., the single-head attention of Program 5) where this non-degeneracy is verified, or clarify that the attention example is illustrative rather than covered by the theorem. As stated, it is unclear whether any practical attention configuration satisfies the假设","section":null}],"minor_comments":[{"comment":"Section 1.1: 'A depth-M Multilayer Perceptron' — the hyphenation is inconsistent with later usage.","section":null},{"comment":"Equation (1.2): The notation W^{(ell)}_{ij} ~ iid N(0, n_ell^{-1}) should specify that i ranges over {1,...,n_{ell+1}} and j over {1,...,n_ell} for clarity.","section":null},{"comment":"Section 2.2, Example 2.2: The LayerNorm epsilon is mentioned but the specific value or range is not specified. Clarifying that epsilon > 0 is a structural constant would help.","section":null},{"comment":"Figure 6 caption: 'Schematic program graph for a single-head attention layer' — the figure shows dependencies for a fixed query token i and key/value token j, but the caption does not mention this restriction.","section":null},{"comment":"Section 4.2, MatMul case: The text states 'We first dispose of the case m_r in n_in' and then later repeats 'We first consider the case m_r in n_in.' This appears to be duplicated text from a draft revision.","section":null},{"comment":"Section 5.2: The reference to 'Theorem 1 in [FG15]' for the SW_1 empirical measure convergence rate should specify the exact statement or page, as the result in that paper is for W_p, not SW_1 specifically.","section":null},{"comment":"Table 1: The RNN slope (-0.579) is close to -0.5, and the text acknowledges slopes 'strictly below -1/2.' A brief discussion of why the RNN slope is notably shallower than the MLP slopes would be informative.","section":null},{"comment":"References: Several arXiv preprints are cited with future dates (2026). Ensure these are the correct final versions or update to published references where available.","section":null},{"comment":"Notation: The use of both 'hidd' subscript and 'hidd(r)' for truncated programs could be made more consistent; a brief remark defining n_{hidd(r)} explicitly as the hidden widths appearing in lines 1,...,r would help.","section":null}],"recommendation":"major_revision","confidential_remarks":"The core proof structure for basic Netsor (Theorem 1.2) appears sound, and the circularity concern raised in the stress-test is adequately addressed by the truncated-program induction. The main issue is whether the non-degeneracy reduction (Proposition 3.11) is fully rigorous for weight-sharing programs, which is the paper's central contribution. The skeptic's concern about joint non-degeneracy of G-vars in the lifting argument is legitimate and should be addressed explicitly. If the authors can verify that conditional non-degeneracy at each MatMul step implies joint non-degeneracy of the full G-var vector (or add this as an explicit hypothesis), the result stands. The Netsor K extension (Theorem 2.4) is more conditional; the authors are honest about this limitation, but providing at least one verified attention example would substantially strengthen the paper's scope claims."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful and constructive report. The referee identifies a genuine gap in the proof of Proposition 3.11 (ii)=>(iii) concerning the distinction between marginal and joint non-degeneracy of G-vars, and raises an important question about whether the non-degeneracy assumption in Theorem 2.4 is ever satisfied for practical attention architectures. We address both points below. The first requires a revision to clarify the proof; the second requires either a verification for a concrete example or a recharacterization of the attention example's status.","responses":[{"response":"The referee is correct that the proof of Proposition 3.11(ii)=>(iii) requires joint non-degeneracy of the vector (g_bar^(j))_{j=1,...,k'}, not merely marginal non-degeneracy of each component. The current wording 'Since the (g_bar^(j)) are non-degenerate' is ambiguous and should read 'Since the joint law of (g_bar^(j))_{j=1,...,k'} is non-degenerate.' We will revise the proof to explicitly state this requirement. Regarding the concern that joint non-degeneracy is not automatic from conditional non-degeneracy at each step: this is also correct in general. However, in the context of Lemma 3.9, the reduction proceeds sequentially and the non-degeneracy hypothesis of Lemma 3.9 is precisely that every MatMul line has non-degenerate conditional Gaussian output in the infinite-width execution. We will add a remark clarifying that this sequential conditional non-degeneracy, combined with the inductive structure of the reduction (where previously processed G-vars are already non-degenerate), ensures that the joint law of the relevant G-vars at each step is non-degenerate. If this implication does not hold in full generality for weight-sharing programs, we will state the joint non-degeneracy as an explicit hypothesis of Proposition 3.11 and verify it for the architectures considered in Section 2.1.","revision_made":"partial","referee_comment":"Proposition 3.11, proof of (ii)=>(iii): The argument that Phi's zero set being open requires non-degeneracy of ALL previous G-vars jointly, not just marginally. The proof states: 'Since the (g_bar^(j)) are non-degenerate, the set {z : Phi(z) != 0} has null Lebesgue measure, and by continuity of Phi it is an open set, hence empty.' This step requires that the joint law of (g_bar^(j))_{j=1,...,k'} be non-degenerate (i.e., supported on all of R^{k'}). The non-degeneracy assumption in Lemma 3.9 is stated conditionally ('conditionally upon F_{l-1}'), but the proof of Proposition 3.11(ii)=>(iii) appears to use marginal non-degeneracy of the joint vector. In programs with weight sharing, the parents h^(i) can depend on previously generated G-vars from the same A-var W, so joint non-degeneracy of the G-vars is not automatic from conditional non-degeneracy at each step."},{"response":"The referee correctly identifies that the lifting argument requires full support of the joint law of (g_bar^(j))_{j=1,...,k'}, which is equivalent to joint non-degeneracy (positive definiteness of the joint covariance matrix). The current proof does not explicitly verify this, and as noted in our response to the first comment, we will revise the proof to either (a) show that the sequential conditional non-degeneracy hypothesis of Lemma 3.9 implies joint non-degeneracy of the relevant G-vars at each step of the reduction, or (b) state joint non-degeneracy as an explicit hypothesis. We agree that the coefficients alpha_i are deterministic (computed from infinite-width Gram data), so the core lifting argument is sound once full support is established. We will add the missing verification and clarify the logical structure of the proof to make the dependency on joint non-degeneracy explicit.","revision_made":"partial","referee_comment":"Proposition 3.11, proof of (ii)=>(iii): The lifting from infinite-width to finite-width linear identities is the load-bearing step for the non-degeneracy reduction (Lemma 3.9), which in turn is needed for differentiability of the maps involving pseudoinverses and matrix square roots in Section 4.2 ('Covariance stability'). The proof argues that since Phi(g_bar^(j)) = 0 a.s. (infinite-width) and the zero set is open, Phi vanishes identically, hence Phi(g^(j)) = 0 a.s. in finite-width too. This works when the coefficients alpha_i are deterministic constants, which they are (computed from infinite-width Gram data). However, the argument that the zero set {z : Phi(z) != 0} is open AND has null Lebesgue measure requires that the joint distribution of (g_bar^(j)) has full support. The proof should explicitly verify that the non-degeneracy hypothesis (conditional non-degeneracy at each MatMul)"},{"response":"We agree that the paper should either verify the non-degeneracy assumption for at least one concrete attention configuration or explicitly state that the attention example is illustrative rather than covered by Theorem 2.4. Upon reflection, we believe the non-degeneracy assumption can be verified for the single-head attention of Program 5 under generic conditions on the input (specifically, when the input tokens are in general position so that the projected query/key vectors have a non-degenerate joint Gaussian law in the infinite-width execution). We will add a proposition or remark verifying this for Program 5 with appropriate hypotheses on the input. If the verification turns out to be more subtle than expected for the full attention layer (particularly due to the softmax nonlinearity creating dependencies), we will instead recharacterize the attention example as illustrative and state clearly that Theorem 2.4's coverage of attention is conditional on a case-by-case verification of non-degeneracy. Either way, the current ambiguity in the manuscript will be resolved.","revision_made":"partial","referee_comment":"Theorem 2.4 (Netsor K extension): The non-degeneracy assumption is stated as 'its G-vars are jointly non-degenerate Gaussians in the infinite-width execution.' Remark 2.5 honestly acknowledges that the reduction does not lift because 'a scalar kernel variable is deterministic in infinite-width but random at finite width, so a linear relation that holds identically in the infinite-width execution need not hold identically in the finite-width execution.' This means Theorem 2.4's coverage of attention architectures is conditional on a property that must be verified case-by-case. The paper should provide at least one concrete example (e.g., the single-head attention of Program 5) where this non-degeneracy is verified, or clarify that the attention example is illustrative rather than covered by the theorem. As stated, it is unclear whether any practical attention configuration satisfies the假设"}],"tokens_in":33250,"tokens_out":1492,"duration_ms":339512,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"The main thing to know: this paper proves an O(1/√n) Wasserstein bound between finite-width and infinite-width executions of Netsor programs, including programs with weight sharing (RNNs, multi-input evaluation). That is genuinely new — prior quantitative CLTs [BT22; Tre23; Fav+25] covered feed-forward architectures with independent weights per layer, and Yang's tensor program master theorem was qualitative only. The extension to weight-sharing architectures is the real contribution here, and it is earned. The proof is a clean line-by-line induction with an explicit coupling (same Gaussian noise for finite and infinite conditional Gaussians), and the apparent circularity between Theorem 1.2 and the kernel LLN (Corollary 1.3) is correctly handled by applying the theorem only to truncated programs at each induction step. The structure lemmas (3.5, 3.6) are well-organized and the non-degeneracy reduction for basic Netsor (Proposition 3.11) is clean: the key lifting argument works because the coefficients αᵢ are deterministic (computed from infinite-width Gram data) and the G-vars are jointly non-degenerate Gaussians, so the continuity argument for propagating infinite-width linear identities to finite-width is valid. The stress-test concern about this step does not actually land on reading the proof — it narrows to the Netsor K case, which the authors already flag themselves. The Lipschitz requirement is a stated hypothesis, not a hidden assumption. The O(1/√n) rate is weaker than known feed-forward rates, but the architectural generality justifies it. Numerical experiments are preliminary but honestly interpreted — they acknowledge the sampling noise floor and don't overclaim. The one real soft spot is Theorem 2.4 (Netsor K extension covering attention). The non-degeneracy assumption on G-vars is not merely a checkable condition — Remark 2.5 honestly acknowledges that the reduction used for basic Netsor fails because scalar kernel variables are deterministic at infinite width but random at finite width. This means Theorem 2.4's coverage of attention architectures is conditional on a property that could fail for certain weight-sharing patterns and must be verified case-by-case. This is a genuine limitation, not a fatal one: the basic Netsor result stands on its own, and the Netsor K theorem is a reasonable first step toward attention. The paper is for researchers in infinite-width neural network theory who need quantitative finite-width guarantees beyond feed-forward architectures. It deserves a serious referee — the core proof is detailed and correct, and the Netsor K limitation is appropriately scoped rather than papered over.","headline":"Solid quantitative CLT for tensor programs with weight sharing; the basic Netsor proof is sound, the Netsor K extension has an honest but real limitation.","tokens_in":34246,"tokens_out":1433,"would_cite":true,"duration_ms":100893,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Finite-width neural nets converge to Gaussian limits at rate 1/√n","keywords":["tensor programs","Gaussian process limit","Wasserstein distance","neural network width","central limit theorem","attention mechanism","weight sharing","Netsor programs"],"falsifier":"Construct an attention-based Netsor K program whose G-vars are degenerate in the infinite-width execution, demonstrating that the non-degeneracy assumption is not automatically satisfied and that the convergence bound cannot be applied without verifying it.","tokens_in":33462,"feed_emoji":"🎯","tokens_out":1068,"duration_ms":131624,"temperature":0.7,"pith_summary":"This paper proves that randomly initialized neural networks of essentially any architecture — feed-forward, recurrent, and attention-based — approach their infinite-width Gaussian-process limits at an explicit, quantifiable rate. The central object is the Netsor program, a formal language that expresses a neural network's forward pass as a sequence of matrix multiplications and nonlinearities. The authors show that for any such program with Lipschitz activation functions, the Wasserstein distance between the finite-width network's outputs and the idealized infinite-width Gaussian process is bounded by a constant times the sum of 1/√m over all hidden layer widths m. The proof works by constructing an explicit line-by-line coupling: at each matrix multiplication step, both the finite and infinite executions are conditionally Gaussian, and the authors couple them using shared random noise, bounding the error through a law of large numbers for empirical kernels. A key extension to Netsor K programs adds scalar kernel variables and parametric nonlinearities, covering attention mechanisms like softmax and layer normalization, though under a non-degeneracy assumption on the Gaussian variables that must hold in the infinite-width limit.","feed_headline":"Finite-width neural nets converge to Gaussian limits at rate 1/√n","feed_subtitle":"Explicit error bounds cover feed-forward, recurrent, and attention architectures — not just simple MLPs.","key_machinery":"Netsor programs (a formal algebraic language for neural network forward passes); line-by-line inductive coupling via shared Gaussian noise; generalized Schur complements and Greville's formula for conditional Gaussian laws; non-degeneracy reduction to eliminate redundant matrix multiplication lines; Netsor K extension adding scalar kernel variables for attention mechanisms","core_discovery":"The paper's central result is a quantitative central limit theorem for tensor programs: the Wasserstein-p distance between a finite-width network's execution and its infinite-width Gaussian-process limit is bounded by c × Σ 1/√m (summed over hidden widths), where c depends only on structural constants. This extends prior quantitative CLTs — which covered only feed-forward networks with independent weights per layer — to architectures with weight sharing (RNNs, multi-input evaluation) and, via the Netsor K extension, to attention layers. The mechanism is a line-by-line inductive coupling that exploits the conditional Gaussian structure of matrix multiplication outputs in both finite and无限-宽度实","pith_inferences":["The non-degeneracy assumption for Netsor K programs suggests that certain attention configurations — particularly those with specific weight-sharing patterns or degenerate input geometries — may exhibit convergence to non-Gaussian limits or slower rates, which would be architecturally significant.","The gap between the theoretical O(1/√n) bound and the numerically observed slopes (ranging from -0.58 to -0.92) hints that the true convergence rate may be architecture-dependent and potentially sharper than the theorem guarantees, especially for feed-forward networks where O(1/n) rates are known.","Since the constant c depends on structural parameters like the number of program lines L, the bound may become loose for very deep networks, suggesting a regime where depth and width interact nontrivially in determining Gaussian approximation quality."],"forward_implications":["Practitioners can use the explicit 1/√m bound to estimate how wide a network must be for the Gaussian-process approximation to be accurate within a given tolerance, for architectures beyond simple MLPs.","The framework provides a principled way to certify when the NNGP or NTK approximation is valid for recurrent and attention-based architectures at finite width, rather than only asymptotically.","The line-by-line coupling technique could be extended to backward passes and gradient computations, potentially yielding quantitative convergence rates for NTK-based training dynamics in architectures with weight sharing.","The Netsor K extension with its non-degeneracy assumption identifies a precise mathematical condition that determines which attention configurations admit clean Gaussian-process limits and which do not."],"fun_headline_variants":["Infinite-width GP limits of tensor programs get explicit convergence rates","Wasserstein bounds: finite nets within O(1/√m) of their GP limits","Quantitative CLT for tensor programs covers RNNs and attention, not just MLPs","Tensor program GP convergence: explicit O(1/√width) error bounds, any architecture","Weight sharing doesn't break the 1/√width GP limit, with explicit bounds"],"cache_read_input_tokens":0,"weakest_assumption_plain":"For the extension to attention architectures (Netsor K programs, Theorem 2.4), the authors require that the Gaussian variables produced by matrix multiplications are jointly non-degenerate in the infinite-width limit. This property must be verified case-by-case and could fail for certain weight-sharing patterns or input configurations; if it fails, the theorem does not apply.","fun_headline_variants_meta":{"raw":{"variants":["Infinite-width GP limits of tensor programs get explicit convergence rates","Wasserstein bounds: finite nets within O(1/√m) of their GP limits","Quantitative CLT for tensor programs covers RNNs and attention, not just MLPs","Tensor program GP convergence: explicit O(1/√width) error bounds, any architecture","Weight sharing doesn't break the 1/√width GP limit, with explicit bounds"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":529,"prompt_tokens":420,"completion_tokens":109,"prompt_tokens_details":null},"tokens_in":420,"tokens_out":109,"duration_ms":30034,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T10:43:18.628050+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"Construct an attention-based Netsor K program whose G-vars are degenerate in the infinite-width execution, demonstrating that the non-degeneracy assumption is not automatically satisfied and that the convergence bound cannot be applied without verifying it.","supporting_citations":[],"review_version":1}