{"id":"e120b86a-2f6b-4616-936d-43fb3043b139","arxiv_id":"2508.12647","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A neural-operator diffusion model gives more stable Hessian-diagonal estimates, improving score-based causal ordering and enabling LLM-guided causal reasoning.","lead":"This paper proposes SciNO, a diffusion model built on neural operators that estimates the second derivatives of data densities more stably, and uses it to improve causal ordering algorithms. It also shows how to combine these estimates with LLM priors to guide causal reasoning without fine-tuning.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The load-bearing gap is that denoising score matching controls only first-order score error; the §3.1 Hilbert-space completeness argument does not imply Hessian accuracy, so SciNO's central claim lacks a transfer guarantee.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing gap: score-matching training provides no direct control over Hessian accuracy, and the completeness argument in Section 3.1 does not bridge that gap. I agree that this is the central theoretical weakness. The paper does have independent empirical support: Figure 2 directly measures Hessian-diagonal residuals, the ablations in Section A.5.1 compare LTE against PE and additive variants, and the results are internally consistent with the claim that the architecture helps. My concern is not that the empirical results are fabricated, but that the stated theoretical justification is invalid, and the central claim is phrased as a stability guarantee. The abstract/full-text mismatch (CSG in the metadata versus SciNO in the full text) is a separate submission-integrity issue that also warrants attention, but it does not change the internal assessment of the SciNO argument. Given the empirical evidence, the appropriate verdict remains conditional: the paper should provide code, error bars, and a corrected or weakened theoretical claim connecting score error to Hessian error. No change to the reader's CONDITIONAL verdict is needed.","tokens_in":27868,"tokens_out":4202,"duration_ms":50427,"concrete_test":"On a 2D SEM with a known closed-form density, compute the ground-truth Hessian diagonal by automatic differentiation. Train both DiffAN with MLP and DiffAN with SciNO under the identical denoising score-matching objective, and at each checkpoint record both the score-matching loss and the mean absolute error of the Hessian diagonal. Check whether Hessian error tracks score loss and whether SciNO's Hessian-error advantage persists when the two models are compared at matched values of the score-matching loss. If the advantage disappears at matched score loss, or if Hessian error does not decrease as score loss decreases, the claimed first-order-to-second-order transfer is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that SciNO 'stably approximate[s] the Hessian diagonal of the data distribution' and thereby improves ordering-based causal discovery. The training objective, however, is denoising score matching, which penalizes only first-order score error. Section 3.1 and Theorem A.2 establish that neural operators can approximate a target score function in a Sobolev/Hölder norm, and therefore also its derivatives, but this is a purely existential approximation statement about a fixed target. It says nothing about the sequence of models produced by a specific stochastic optimizer. The sentence 'the completeness of Hilbert space implies that the distance between ... will close to zero as we train score models accurately' is a non-sequitur: completeness guarantees that Cauchy sequences converge in the ambient space, not that the training trajectory is Cauchy or that its limit minimizes the Hessian error. The gap is load-bearing because the leaf-node criteria (3) and (4) depend on the Hessian diagonal of the log-density, and equations (5) and (6) use second derivatives of the trained score model. Without a bound connecting score error to Hessian error, the theoretical justification does not support the headline claim, leaving only the empirical boxplots in Figure 2 to carry that weight.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"As submitted, the front matter (title and abstract) describes a different paper: 'Cognitive Structure Generation: From Educational Priors to Policy Optimization,' about pretraining a Cognitive Structure Diffusion Probabilistic Model and optimizing it with reinforcement learning. The body, however, is a causal discovery paper proposing SciNO (Score-informed Neural Operator), a functional diffusion model that aims to stably approximate the Hessian diagonal of log-densities and thereby improve ordering-based causal discovery under additive noise models. The body's central claim is that SciNO reduces order divergence by 42.7% on synthetic graphs and 31.5% on real-world datasets compared to DiffAN, and that a probabilistic control method combining SciNO evidence with autoregressive model priors improves LLM-based causal ordering without fine-tuning or prompt engineering. The main theoretical support is Theorem 3.1, which asserts that neural operators can approximate the target score function in a Sobolev/Hölder norm, and the main empirical support consists of synthetic and real-world experiments comparing SciNO with DiffAN, CaPS, and other baselines, supplemented by ablations of the proposed architectural components.","tokens_in":28021,"tokens_out":3991,"duration_ms":44131,"significance":"If the empirical results hold, the paper makes a useful contribution: stable estimation of second-order score derivatives is genuinely important for score-matching-based causal ordering, and the proposed neural-operator architecture with learnable time encoding shows consistent improvements on high-dimensional graphs while reducing memory footprint. The paper also benefits from a fairly complete set of ablations (PE vs. LTE, additive vs. multiplicative LTE, DiffAN with LTE), from a clear statement of the underlying ANM assumptions, and from explicit runtime and memory comparisons against kernel-based methods. The probabilistic control method is a reasonable and potentially practical way to combine data-driven statistics with LLM priors. However, the theoretical justification for the central claim contains a logical gap, and the mismatch between the submitted abstract/body is a substantial consistency problem that must be addressed before the paper can be considered publishable.","major_comments":[{"comment":"The theoretical support for the central claim is incomplete. The paper states that because neural operators can approximate the target score function in a Sobolev/Hölder norm, the trained score model's second-order derivatives approximate the Hessian diagonal, and it justifies training convergence by saying that 'the completeness of Hilbert space implies that the distance between ... will close to zero as we train score models accurately.' This is a non-sequitur: completeness guarantees that Cauchy sequences converge in the ambient space, but it does not imply that the stochastic optimizer's trajectory is Cauchy, nor that the limit has small Hessian error. Since the leaf-node criteria (3) and (4) and the residue approximation (6) use second derivatives of the trained model, this gap is load-bearing for the headline claim. The empirical evidence in Figure 2 and Table 2 may support the claim, but the theoretical statements in Section 3.1 and the proof of Theorem 3.1 should be revised to either provide a score-error-to-Hessian-error bound under the actual training objective or explicitly reframe the claim as empirical rather than theoretical.","section":"Section 3.1 and Theorem 3.1"},{"comment":"The title and abstract submitted for review describe Cognitive Structure Generation (CSDPM with reinforcement learning), but the entire body is about SciNO for causal discovery. None of the abstract's claims—cognitive structure assessment, pretraining a diffusion model over student structures, or hierarchical reward optimization—appears anywhere in the full text, and none of the body's contributions (SciNO, Hessian diagonal approximation, LLM control) is mentioned in the abstract. This is a fundamental internal inconsistency in the manuscript as submitted and must be corrected before the paper can be evaluated as a coherent contribution.","section":"Title and Abstract"}],"minor_comments":[{"comment":"There is a typo in the theorem statement: 'compect subset' should be 'compact subset.'","section":"Theorem 3.1"},{"comment":"The caption says 'Comparion,' which should be 'Comparison.'","section":"Figure 3 caption"},{"comment":"The sentence beginning 'SciNO elaboratesCausal Relationship' appears garbled and should be rewritten, as it is not a readable English sentence.","section":"Section 3.2"},{"comment":"The term 'deciduous score' is unusual and likely a typo or mistranslation; I assume 'residual score' or 'residue score' is intended, but the term should be defined or corrected for clarity.","section":"Section 2, DiffAN paragraph"},{"comment":"The paper reports large percentage improvements but does not provide confidence intervals or significance tests for the order-divergence reductions across the 10 runs; adding such statistics would make the comparison more convincing.","section":"Section 4.3 and Table 4"}],"recommendation":"major_revision","confidential_remarks":"The abstract/body mismatch is highly unusual and should be checked against the authors' original submission. If the mismatch is genuine, it is a serious editorial issue; if it is an artifact of the review pipeline, then this report should be read only with respect to the body text, in which case the main remaining concern is the theoretical gap in Section 3.1."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nTwo things you should know before reading this one. First, the arXiv listing says it is about Cognitive Structure Generation, but the full text is actually a different paper: Score-informed Neural Operator for Enhancing Ordering-based Causal Discovery. I reviewed the full text, which is what matters, but the mismatch needs to be fixed before anything else. Second, the paper's central empirical claim is probably right, but the theoretical justification in Section 3.1 is not.\n\nWhat is genuinely new: replacing the MLP score model in DiffAN with a neural-operator functional diffusion model, plus a learnable time encoding and a real/imaginary spectral decomposition, gives substantially better Hessian-diagonal estimates and lower order divergence, especially as the number of variables grows. The 42.7%/31.5% average reductions over DiffAN are consistent across the reported boxplots, tables, and ablations. The probabilistic control of autoregressive LLMs is also a real contribution: multiplying LLM leaf-node priors by SciNO's variance-based evidence reduces order divergence by 64% on average over uncontrolled models, and it does so without fine-tuning or prompt engineering. The memory-efficiency arguments against kernel-based CaPS are quantitatively supported.\n\nWhere the soft spots are. The theory in Section 3.1 and Appendix A.2.2 proves an existence result: some neural operator can approximate the target score function and its derivatives in a Sobolev/Hölder norm. That is fine as an approximation-theoretic statement, but the text then says Hilbert-space completeness implies the distance between the trained model and the target closes to zero during training. That is a non-sequitur. Completeness says Cauchy sequences converge in the ambient space; it says nothing about the trajectory of stochastic gradient descent or about whether minimizing first-order score error drives second-order Hessian error down. The leaf-node criteria (3) and (4) depend on second derivatives, so the paper's headline claim lacks the advertised transfer guarantee. The empirical results can carry that weight, but the paper should say so honestly. Minor but real: no code or configuration files are provided, several tables are garbled to the point of unverifiability, and some headline numbers (e.g., the 31.5% average) cannot be fully recomputed from what is printed.\n\nMy read: the engineering is sound, the experiments are broad and mostly convincing, and the LLM-control section is a nice addition. The paper is not ready as-is, but it is not broken. It deserves a serious referee and a conditional accept after the authors fix the metadata, release code, and rewrite the theory section to claim only what the approximation theorem actually supports.\n\nFor a causal-discovery reading group, I would bring it. I would cite it if I worked on score-based ordering methods.","headline":"The submission is a solid empirical causal-discovery paper wrapped in a confusing wrapper: the arXiv metadata advertises a different paper, the theory section overclaims a convergence guarantee it does not prove, and the empirical results are strong enough to deserve a serious referee.","tokens_in":28652,"tokens_out":1381,"would_cite":true,"duration_ms":17517,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper argues that ordering-based causal discovery under additive noise models can be substantially improved by replacing the MLP score model with a function-space neural-operator diffusion model that stably approximates the Hessian…","keywords":["causal discovery","ordering-based causal discovery","score matching","Hessian diagonal","diffusion models","neural operators","additive noise models","large language models"],"falsifier":"Take a 2D nonlinear Gaussian SEM whose log-density and Hessian diagonal are known analytically, train SciNO and an MLP score model on the same samples, and compare both the score error and the Hessian-diagonal error on a held-out grid. The central claim fails if SciNO's second-derivative error is not lower than the MLP's at comparable first-order score error, or if the reported order-divergence gains disappear once Hessian accuracy is matched. A second check: rerun the LLM control experiments with randomized variable-name masks and no descriptions; if the evidence term no longer improves on the uncontrolled LLM, the semantic-versus-statistical contribution is misattributed.","tokens_in":27574,"feed_emoji":"🧩","tokens_out":9757,"duration_ms":92177,"temperature":0.7,"pith_summary":"The paper tries to establish that ordering-based causal discovery under additive noise models can be made substantially more accurate by replacing the usual multilayer-perceptron score model with a score model that lives in a function space and is built from neural operators. This model, SciNO, is designed to approximate not just the score but its Jacobian diagonal—the Hessian diagonal of the log-density—which is the statistic the leaf-node criteria in SCORE, DiffAN, and CaPS rely on. Reported results show SciNO reduces order divergence by 42.7% on synthetic graphs and by 31.5% on real-world graphs relative to DiffAN, and that it stays within memory limits where kernel-based estimators fail. The paper also argues that the same evidence can be combined with autoregressive-model probabilities to guide LLM causal reasoning without fine-tuning or prompt engineering.","feed_headline":"One score model cuts causal-order divergence by 42.7%","feed_subtitle":"A function-space diffusion model stabilizes the Hessian diagonal that leaf-node selection needs, and steers LLM reasoning without…","key_machinery":"The load-bearing object is SciNO, a functional diffusion model whose score network is a neural operator rather than an MLP: the score is treated as a map between function spaces, so derivatives can be learned as functions instead of being numerically differentiated from a pointwise network. Two design choices carry the argument: Fourier-layer outputs are split into real and imaginary parts in the spectral domain, and a Learnable Time Encoding is injected multiplicatively so the model can learn spatial and temporal derivatives jointly. The quantities that matter are the leaf-selection statistics—$\\mathrm{Var}(H_{ii})$ under SCORE's criterion and $\\mathbb{E}[H_{ii}]$ under CaPS's—where $H_{ii} = \\partial_i^2 \\log p(x)$ is the Hessian diagonal; SciNO approximates these through the deciduous-score formula (6) inherited from DiffAN. The paper justifies the approximation with a Sobolev-embedding argument showing neural operators can approximate the score in a norm that controls derivatives, together with pointwise approximation theorems for the target score function.","core_discovery":"The central claim is that stable second-derivative estimation is the unlock for ordering-based causal discovery, and that SciNO achieves it. Under the additive noise model, a leaf node is identifiable from the variance (SCORE) or expectation (CaPS) of the Hessian diagonal of the log-density; DiffAN avoids refitting by approximating these terms from a trained diffusion score model. SciNO replaces the MLP with a time-conditioned Fourier neural operator in a Hilbert space, adding a Learnable Time Encoding and a real/imaginary spectral decomposition so the model can track spatiotemporal derivatives. In experiments, DiffAN with SciNO cuts order divergence from 117.0 to 86.6 on 100-node random graphs, and CaPS with a probed SciNO scales to sample sizes where CaPS hits out-of-memory errors. The paper further claims that multiplying an autoregressive model's next-token probabilities by SciNO-based evidence—average-rank or confidence-interval statistics from deep ensembles—improves LLM causal ordering by up to 75% while reducing the number of LLM calls from quadratic to linear.","pith_inferences":["A testable extension would be to use SciNO's Hessian-diagonal estimates for other score-derivative tasks, such as sensitivity analysis, marginal likelihood approximations, or second-order optimization diagnostics; the paper does not evaluate these.","The rank-based and confidence-interval evidence terms are one way to fuse LLM priors with data statistics; the same Bayesian multiplication could be applied to other autoregressive models or to non-LLM sequential predictors, although the paper only demonstrates LLMs.","Because the identifiability guarantee rests on the additive noise model, an obvious stress test is to run the same pipeline on non-additive or discrete data; the paper flags this as a limitation, so a neutral reader should treat the ordering claims as scoped to continuous ANMs.","The correlation between MMD and order divergence under SciNO suggests generated samples could serve as a cheap diagnostic for ordering reliability, but this connection is an inference from the paper's Figure 3 rather than a stated result."],"forward_implications":["Swapping SciNO's score model into an existing ordering pipeline is enough to lower order divergence on graphs from 2 to 100 nodes, with the largest gains at high dimensionality.","Causal ordering stays usable when samples are large: the probing variant avoids CaPS's cubic kernel-matrix inversion and out-of-memory failures beyond 100,000 samples.","LLM-based causal reasoning can be controlled with data-derived evidence, reducing order divergence by up to 75% on a 107-node graph and needing only O(n) LLM calls per ordering.","The improvement is not just generative fit: order divergence tracks goodness-of-fit under SciNO, whereas the MLP baseline shows no such correlation.","Partial context is enough: when variable names are masked and only 10% of variables have descriptions, the combined control still reduces order divergence relative to the uncontrolled LLM."],"supporting_citations":[{"why":"DiffAN — the MLP score-model baseline SciNO replaces; supplies the deciduous-score Hessian approximation used for leaf nodes.","marker":"[35]"},{"why":"SCORE — defines the variance-of-Hessian-diagonal leaf criterion and order divergence used in most experiments.","marker":"[33]"},{"why":"CaPS — defines the expectation-based leaf criterion; the probing variant of SciNO is evaluated against it.","marker":"[45]"},{"why":"HDM — provides the Hilbert-space diffusion formalism and the time-conditioned Fourier neural operator SciNO modifies.","marker":"[26]"},{"why":"Neural operators — the function-space map family used to model scores and their derivatives.","marker":"[16]"},{"why":"FNO — contributes spectral convolution and the universal approximation theorems cited in the pointwise approximation proof.","marker":"[24]"},{"why":"Stein gradient estimator — the expensive kernel estimator that SCORE/CaPS use and that SciNO's probing avoids.","marker":"[23]"},{"why":"Causal-order framing for LLM experts — motivates the probabilistic control of autoregressive causal ordering.","marker":"[42]"},{"why":"Deep ensembles — the mechanism for obtaining multiple Hessian-variance samples used in the rank and confidence-interval evidence terms.","marker":"[19]"},{"why":"DDPM — supplies the denoising score-matching objective and time-conditioning convention that the functional model adapts.","marker":"[13]"}],"fun_headline_variants":["AI generates students' cognitive structures via diffusion and RL","New framework generates cognitive structure for student modeling","Diffusion model plus RL yields interpretable cognitive structure","CSG: generating cognitive structure from educational priors"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that training a score model with a first-order denoising score-matching objective also makes its second derivatives approximate the true Hessian diagonal; the Hilbert-space completeness argument in the paper says a good approximator exists in principle, not that the optimizer finds one.","fun_headline_variants_meta":{"raw":{"variants":["AI generates students' cognitive structures via diffusion and RL","New framework generates cognitive structure for student modeling","Diffusion model plus RL yields interpretable cognitive structure","CSG: generating cognitive structure from educational priors"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000308,"raw_usage":{"total_tokens":1742,"prompt_tokens":911,"completion_tokens":831,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":527,"completion_tokens_details":{"reasoning_tokens":770}},"tokens_in":527,"tokens_out":831,"duration_ms":8524,"temperature":1.0,"reasoning_tokens":770,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:20:02.826832+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a 2D nonlinear Gaussian SEM whose log-density and Hessian diagonal are known analytically, train SciNO and an MLP score model on the same samples, and compare both the score error and the Hessian-diagonal error on a held-out grid. The central claim fails if SciNO's second-derivative error is not lower than the MLP's at comparable first-order score error, or if the reported order-divergence gains disappear once Hessian accuracy is matched. A second check: rerun the LLM control experiments with randomized variable-name masks and no descriptions; if the evidence term no longer improves on the uncontrolled LLM, the semantic-versus-statistical contribution is misattributed.","supporting_citations":[{"cited_title":"Diffusion Models for Causal Discovery via Topological Ordering","cited_arxiv_id":null,"evidence_quote":"DiffAN — the MLP score-model baseline SciNO replaces; supplies the deciduous-score Hessian approximation used for leaf nodes."},{"cited_title":"Score Matching Enables Causal Discovery of Nonlinear Additive Noise Models","cited_arxiv_id":null,"evidence_quote":"SCORE — defines the variance-of-Hessian-diagonal leaf criterion and order divergence used in most experiments."},{"cited_title":"Ordering-Based Causal Discovery for Linear and Nonlinear Relations","cited_arxiv_id":null,"evidence_quote":"CaPS — defines the expectation-based leaf criterion; the probing variant of SciNO is evaluated against it."},{"cited_title":"Score-based Generative Modeling through Stochastic Evolution Equations in Hilbert Spaces","cited_arxiv_id":null,"evidence_quote":"HDM — provides the Hilbert-space diffusion formalism and the time-conditioned Fourier neural operator SciNO modifies."},{"cited_title":"Fourier Neural Operator for Parametric Partial Differential Equations","cited_arxiv_id":null,"evidence_quote":"FNO — contributes spectral convolution and the universal approximation theorems cited in the pointwise approximation proof."},{"cited_title":"Gradient Estimators for Implicit Models","cited_arxiv_id":null,"evidence_quote":"Stein gradient estimator — the expensive kernel estimator that SCORE/CaPS use and that SciNO's probing avoids."},{"cited_title":"Balasubramanian, and Amit Sharma","cited_arxiv_id":null,"evidence_quote":"Causal-order framing for LLM experts — motivates the probabilistic control of autoregressive causal ordering."}],"review_version":1}