{"id":"24317d94-c6a9-44c4-b1d9-718f85c0d449","arxiv_id":"2607.07678","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":1,"one_line_summary":"RoPE frequency usage is determined by a data-induced dependency width W, with the optimal frequency scaling as π/W, explaining both learned spectra and the success of position interpolation.","lead":"This paper shows that the specific frequencies a transformer learns to use from Rotary Position Embeddings (RoPE) are determined by the distance structure of dependencies in the training data. It provides a mathematical rule for this matching and uses it to explain when position interpolation techniques successfully extend context windows.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The geometric utility U(θ;K) is proven correct but never connected to the training objective; the link between 'maximizes positional contrast' and 'gradient descent concentrates energy here' is assumed, not derived.","rationale":"The reader correctly identified the central gap: the utility function U(θ;K) is a geometric measure of positional contrast, and its connection to what gradient descent actually learns is assumed rather than proven. I agree this is the most load-bearing concern. The theorem itself is correct — the proof in Appendix B is straightforward and sound. The issue is entirely in the bridge from theory to practice. The empirical evidence is suggestive but insufficient: the key quantitative fit (c=3.02 vs π) lacks error bars and uses a minimal model on a task uniquely suited to the theory (pure positional retrieval with no competing content cues). The iGSM results are only qualitative. The PI/self-similarity framework (Section 4, Theorem 2) is elegant and the arithmetic counterexample (Figure 4C) is a nice falsification, but the natural-language self-similarity evidence (Figure 4B) uses tokenization-scale variation as a proxy for positional dilation, which is indirect. Despite these gaps, the paper's core insight — that RoPE frequency usage reflects data dependency structure, and that PI works by dilating positional fields — is valuable and the theoretical framework is clean. The CONDITIONAL verdict with HIGH confidence is appropriate: the contribution is real but the theory-to-practice bridge needs the additional evidence I described. My concern does not change the verdict because the reader already identified this as the key weakness and calibrated the verdict accordingly.","tokens_in":38498,"tokens_out":3048,"duration_ms":241091,"concrete_test":"Run the block-structured experiment (Section D.1) with ≥5 random seeds per B value and report 95% confidence intervals on the fitted constant c. Additionally, plot the full energy spectrum {E_m} for each B (not just the geometric mean θ_eff) and verify the spectrum actually peaks near θ = π/B rather than being broad or multi-modal. Then, as a critical control, run a variant where the target token is identifiable by content (e.g., a unique marker token) in addition to position, and check whether energy still concentrates at π/B. If the content-cue variant shifts energy away from π/B, the theory's scope is narrower than claimed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1 correctly shows that U(θ;K) = ∫(1−cos(θr)) dPK(r) is strictly increasing on the admissible set A_W = {θ : θW ≤ π}, yielding θ⋆ = π/W. This is mathematically sound. The load-bearing gap is that U is a purely geometric quantity — it depends only on the kernel K and the frequency θ, with no dependence on model weights, loss function, or learning dynamics. The paper's central empirical claim is that learned frequency energy E_m = E[a²_{ij,m} + b²_{ij,m}] concentrates near θ⋆. But nothing in the theory establishes that minimizing cross-entropy loss drives E_m toward the frequency that maximizes U. The coefficients a_{ij,θ}, b_{ij,θ} are content-dependent learned projections; the model could in principle achieve low loss using a frequency with low U by compensating with large weight norms, or by relying on content-based attention rather than positional contrast. The softmax normalization in attention creates inter-frequency competition that the single-frequency utility does not capture. The block-structured experiment (Section 2, Figure 2C) provides the strongest evidence: c=3.02 vs π≈3.14. But this is a single fit over 7 data points with no error bars, on a 2-layer single-head model, on a task where positional retrieval is the only solution strategy (no content cues compete). The iGSM experiment (Figure 3) provides only qualitative support. Without either a learning theorem showing gradient descent converges to the predicted frequency, or larger-scale experiments with statistical rigor across diverse task structures, the bridge from geometric optimality to learned behavior remains an untested assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"glm-5.2","summary":"This paper proposes a data-centered theory of RoPE frequency selection in transformers. The central theoretical result (Theorem 1) shows that for a data-induced positional dependency kernel of width W, the optimal admissible RoPE frequency—defined as the one maximizing average positional contrast U(θ;K)—scales as θ⋆ = π/W. The authors then connect this to position interpolation (PI) for length generalization, showing via Theorem 2 that PI preserves frequency utility if and only if the test-time dependency structure is a self-similar dilation of the training-time structure. Empirical validation includes: (1) a controlled block-structured synthetic task where the effective frequency follows an inverse scaling law with block size (c=3.02 vs. π≈3.14), (2) iGSM experiments showing broader dependencies shift frequency usage lower, and (3) demonstrations that PI helps on self-similar natural language but fails on non-self-similar arithmetic tasks.","tokens_in":38820,"tokens_out":1589,"duration_ms":381441,"significance":"The paper addresses a well-motivated question: why do trained models use RoPE frequencies non-uniformly? The parameter-free prediction θ⋆ = π/W (Theorem 1) is a clean, falsifiable result, and the block-structured experiment (Section 5.1, Figure 2C) provides a direct test with c=3.02 close to π. The connection between PI and self-similarity (Theorem 2, Corollary 1) yields a principled explanation for when PI should help, supported by the arithmetic counterexample (Figure 4C) and the natural-language mutual information analysis (Figure 4B). The perplexity-vs-accuracy analysis for arithmetic (Section 5.2, Tables 1–2) is a useful practical finding. The framework is internally consistent and the predictions are testable, which are strengths.","major_comments":[{"comment":"§3.3, Theorem 1 and the link to learned weights: The utility function U(θ;K) = ∫(1−cos(θr)) dPK(r) is purely geometric—it depends only on the kernel K and frequency θ, with no dependence on model weights, loss function, or learning dynamics. The paper's central empirical claim is that learned frequency energy E_m = E[a²_{ij,m} + b²_{ij,m}] concentrates near θ⋆. However, nothing in the theory establishes that minimizing cross-entropy loss drives E_m toward the frequency that maximizes U. The coefficients a_{ij,θ}, b_{ij,θ} are content-dependent learned projections; the model could in principle achieve low loss using a frequency with low U by compensating with large weight norms or by relying on content-based attention. This gap between the idealized single-frequency utility and the actual learned multi-frequency attention mechanism is the load-bearing assumption. The paper should either (","section":null},{"comment":"§5.1, Figure 2C: The block-structured experiment provides the strongest quantitative evidence, but the fit c=3.02 vs. π≈3.14 is based on only 7 data points (B ∈ {32,...,2048}) with no error bars, on a 2-layer single-head model, on a task where positional retrieval is the only solution strategy (no content cues compete). The lack of error bars or confidence intervals makes it impossible to assess whether the agreement with π is statistically meaningful or coincidental. Adding bootstrap confidence intervals on c, or reporting results across multiple random seeds, would substantially strengthen this central claim.","section":null},{"comment":"§5.1, Figure 3B: The iGSM experiment provides only qualitative support—Figure 3B shows that log θ_eff decreases as ops increase, but there is no quantitative fit to the predicted 1/W scaling (unlike the block experiment). The dependency width W is not directly measured or controlled here; it is only qualitatively argued to be broader for larger ops. A quantitative comparison between the measured W (from the MI profiles in Figure 3A) and the predicted θ⋆ = π/W would make this a genuine test rather than a qualitative trend.","section":null},{"comment":"§4.2, Theorem 2 and Definition 4: The self-similarity condition P_{αW} = (S_α)_# P_W is a strong structural assumption. The natural-language evidence (Figure 4B) uses tokenization granularity as a proxy for positional dilation, but it is unclear whether changing BPE vocabulary size truly corresponds to the dilation map S_α(r) = αr in the sense of Definition 4. The paper should discuss whether this proxy is a valid approximation of the formal self-similarity condition, or whether it is merely suggestive.","section":null}],"minor_comments":[{"comment":"§2: The definition of θ_eff = exp(Σ_m ω_m log(θ_m)) is introduced in the block experiment paragraph but the notation ω_m is only defined later. Consider defining it at first use.","section":null},{"comment":"Figure 2C: The y-axis label 'log-mean effective frequency' could be clarified—is it log(θ_eff) or θ_eff on a log scale? The text uses both formulations in different places.","section":null},{"comment":"§3.2: The admissible set A_W is defined as {θ > 0 : θW ≤ π}, but the field F(θ) = π/θ is defined as the half-period. It would help to explicitly note that admissibility requires F(θ) ≥ W, i.e., the field covers the dependency width, to make the connection immediate.","section":null},{"comment":"Appendix D.1: Training for only 200 iterations seems very short. Were the models converged? A note on convergence checking would be helpful.","section":null},{"comment":"Figure 4A: The figure is quite dense and the text in the panels is small. Consider enlarging or splitting into subfigures for readability.","section":null},{"comment":"§5.2: The arithmetic experiment uses Llama-2-7B with context length 4096, but Table 1 reports results for context windows up to 16384. It would be useful to clarify whether the 8192 and 16384 results use PI with α=2 and α=4 respectively.","section":null},{"comment":"References: Several citations are to non-peer-reviewed sources (Reddit posts [6,10], GitHub pull requests [5]). While these are relevant to the RoPE scaling literature, the authors should verify accuracy of these citations.","section":null}],"recommendation":"major_revision","confidential_remarks":"The core theoretical contribution (Theorem 1) is mathematically correct but the gap between the geometric utility function and the learning dynamics is the main concern. The paper would benefit from either (a) a formal or informal argument for why gradient descent on cross-entropy should concentrate energy at the U-maximizing frequency, or (b) acknowledgment that the theory is a normative principle (what frequency *should* be useful) rather than a descriptive one (what frequency *will be learned*), with a discussion of when the two coincide. The block experiment is the strongest evidence but needs error bars. If these issues are addressed, the paper is suitable for publication. The self-similarity framework for PI is novel and interesting even if the formal connection is not fully rigorous."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful and constructive review. The referee raises four major points: (1) the gap between the geometric utility U(θ;K) and the claim that cross-entropy minimization drives learned frequency energy E_m toward θ⋆; (2) the absence of error bars or multiple-seed confidence intervals for the c=3.02 fit in Figure 2C; (3) the lack of a quantitative θ⋆=π/W fit in the iGSM experiment (Figure 3B); and (4) whether BPE vocabulary variation is a valid proxy for the formal self-similarity condition in Definition 4. We agree that points (1), (2), and (3) identify genuine gaps between the theory and the empirical evidence, and we will revise the manuscript to address them. For point (4), we agree the discussion should be expanded but argue the proxy is appropriate as a first-order approximation. Details follow.","responses":[{"response":"The referee correctly identifies the central gap between Theorem 1 and the empirical claims. The theorem proves that θ⋆=π/W maximizes the average positional contrast U(θ;K), which is a geometric quantity. It does not prove that gradient descent on cross-entropy loss drives the learned energy E_m toward θ⋆. We agree this is a load-bearing assumption that the current manuscript does not adequately justify. In the revision, we will add a new subsection (Section 3.4) that explicitly states this as a linking assumption and provides the following supporting argument. The key insight is that in the block-structured task (Section 5.1), positional retrieval is the only solution strategy: the observed tokens are latent value plus i.i.d. Gaussian noise, and the target is a token at a fixed offset. Content-based attention cannot solve this task because the content at the target offset is uncorrelated with the query content across different inputs. The model must use positional information to identify the correct offset. In this setting, the attention score contribution from frequency θ_m is s_{ij}^{(m)} = a_{ij,m}cos(θ_m r) + b_{ij,m}sin(θ_m r), and the model must allocate energy to frequencies that provide unambiguous positional contrast over the dependency width W. A frequency with low U provides poor positional separation over the relevant distances, so compensating with large weight norms would amplify noise rather than signal. This is why the controlled experiment is designed to eliminate content cues: it isolates the positional mechanism that Theorem 1 analyzes. We acknowledge that for natural language, where content-based attention can partially substitute for positional information, the linking assumption is less tightly controlled. We will state this limitation explicitly.","revision_made":"yes","referee_comment":"§3.3, Theorem 1 and the link to learned weights: U(θ;K) is purely geometric with no dependence on model weights, loss, or learning dynamics. Nothing establishes that minimizing cross-entropy drives E_m toward the frequency maximizing U. The model could achieve low loss using a low-U frequency by compensating with large weight norms or content-based attention. This gap is the load-bearing assumption."},{"response":"The referee is correct that the absence of error bars or confidence intervals weakens the quantitative claim. We will address this in the revision by reporting results across at least 5 random seeds for each block size B, computing bootstrap confidence intervals on the fitted constant c, and including these in an updated Figure 2C. We will also report the standard error of the fit. We note that the current experiment was designed as a controlled proof-of-concept: the 2-layer single-head architecture and the single-solution-strategy task were chosen precisely to isolate the positional mechanism. However, the referee's point that statistical significance cannot be assessed without error bars is well taken, and we will provide the necessary statistical evidence.","revision_made":"yes","referee_comment":"§5.1, Figure 2C: The fit c=3.02 vs. π≈3.14 is based on only 7 data points with no error bars, on a 2-layer single-head model, on a task where positional retrieval is the only solution strategy. Lack of error bars or confidence intervals makes it impossible to assess whether agreement with π is statistically meaningful."},{"response":"The referee correctly notes that the iGSM experiment provides only qualitative support. We agree that a quantitative comparison between the measured dependency width W and the predicted θ⋆=π/W would substantially strengthen the evidence. In the revision, we will add this quantitative analysis. Specifically, we will measure W from the mutual-information CDFs in Figure 3A by computing the ρ-field width W_ρ (Definition 2 in Section 3.1) for a fixed ρ (e.g., ρ=0.9) for each iGSM variant. We will then compare the measured effective frequency θ_eff against the predicted π/W_ρ for each of the three iGSM configurations. This will transform Figure 3B from a qualitative trend into a quantitative test of the scaling law. We acknowledge that the iGSM dependency profiles are not compactly supported, so the comparison will use the ρ-field width rather than the exact W from Theorem 1, and we will discuss this distinction explicitly.","revision_made":"yes","referee_comment":"§5.1, Figure 3B: The iGSM experiment provides only qualitative support—log θ_eff decreases as ops increase, but there is no quantitative fit to 1/W scaling. The dependency width W is not directly measured or controlled; it is only qualitatively argued to be broader for larger ops. A quantitative comparison between measured W and predicted θ⋆=π/W would make this a genuine test."},{"response":"We agree that the relationship between BPE vocabulary variation and the formal self-similarity condition deserves more discussion. In the revision, we will add a paragraph in Section 5.2 that addresses this point. The argument is as follows. Changing the BPE vocabulary size changes the number of tokens per unit of text. If the underlying linguistic dependency structure (measured in characters or words) is approximately scale-invariant, as suggested by prior work on fractal patterns in language (Alabdulmohsin et al., 2024), then the dependency profile measured in token distances should approximately dilate when the tokenization granularity changes. Specifically, if a vocabulary change maps k characters to one token instead of αk characters to one token, then a dependency at character-distance d appears at token-distance d/k in the finer tokenization and d/(αk) in the coarser one, corresponding to a dilation by factor α. The mutual-information profiles in Figure 4B show approximate stability under this transformation, which is consistent with—but does not formally prove—self-similarity in the sense of Definition 4. We will explicitly state that the BPE experiment provides suggestive but not definitive evidence, and that a rigorous verification of self-similarity would require controlled data where the dilation factor is known exactly, which is an interesting direction for future work.","revision_made":"partial","referee_comment":"§4.2, Theorem 2 and Definition 4: The self-similarity condition is a strong structural assumption. The natural-language evidence (Figure 4B) uses tokenization granularity as a proxy for positional dilation, but it is unclear whether changing BPE vocabulary size truly corresponds to the dilation map S_α(r)=αr in the sense of Definition 4. The paper should discuss whether this proxy is a valid approximation or merely suggestive."}],"tokens_in":38689,"tokens_out":1598,"duration_ms":275257,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"The main thing to know: this paper derives a parameter-free scaling law saying that the optimal RoPE frequency for a data-induced dependency profile of width W is θ* = π/W, and connects position interpolation to self-similarity of dependency structures. Both ideas are new and both are cleanly stated. The math is correct and simple — Theorem 1 is a short monotonicity argument, Theorem 2 is a change of variables. Neither is deep, but both are the right level of abstraction for the question being asked. The block-structured experiment is a genuinely controlled test: vary block length B, measure effective frequency, fit c/B, get c=3.02 against a theoretical prediction of π. One free parameter, tested against an independent constant. That's a real test, not a curve-fit. The self-similarity operationalization for natural language (mutual information profiles across tokenization granularities) is clever, and the arithmetic counterexample — where PI fails because dependencies don't dilate — is a good negative result that sharpens the theory. The perplexity-vs-accuracy observation on arithmetic is a useful practical point. Now the soft spot, which is real and load-bearing. The utility function U(θ;K) is purely geometric — it depends only on the kernel K and the frequency θ. The paper's empirical claim is that learned frequency energy concentrates near θ*. But nothing in the theory establishes that minimizing cross-entropy loss drives energy toward the frequency that maximizes U. The model could compensate with weight norms, rely on content-based attention, or exploit softmax competition across frequencies in ways the single-frequency utility doesn't capture. The paper assumes this bridge rather than deriving it. The block experiment partially mitigates this — the task is designed so positional retrieval is the only viable strategy, which removes some alternative explanations — but it's a 2-layer single-head model with 7 data points and no error bars on the fit. The iGSM experiment is qualitative only. This is a paper for people who think about positional encodings, data-dependent inductive biases, and context extension. The framework is useful even without the learning theorem, because it gives a principled vocabulary (field, resolution, admissibility, self-similarity) for reasoning about when and why frequency scaling works. A serious referee should push hard on the theory-to-practice gap — ideally asking for either a learning theorem or larger-scale experiments with statistical rigor across diverse task structures. But the core insight is sound and worth engaging with. I'd send it to review.","headline":"Clean scaling law θ* ≍ 1/W linking RoPE frequency to data dependency width; the gap between geometric optimality and learned behavior is the main soft spot.","tokens_in":39276,"tokens_out":1083,"would_cite":true,"duration_ms":124541,"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":"RoPE Frequencies Match Data's Dependency Width, Paper Shows","keywords":["Rotary Position Embeddings","RoPE","position interpolation","length generalization","transformer","positional encoding","frequency selection","self-similarity"],"falsifier":"Train transformers on datasets with controlled dependency widths W and measure whether the learned RoPE frequency spectrum concentrates at θ ≈ π/W. If the spectrum systematically deviates from the 1/W scaling — for instance, if it depends on sequence length, vocabulary size, or model width rather than W alone — the frequency-matching principle is incomplete.","tokens_in":38617,"feed_emoji":"📡","tokens_out":1332,"duration_ms":130672,"temperature":0.7,"pith_summary":"This paper argues that the RoPE (Rotary Position Embedding) frequencies a trained transformer learns to use are determined by the relative-distance structure of its training data. The central object is a data-induced positional dependency kernel K(r), which describes how strongly a prediction at one position depends on information at relative distance r. The characteristic width W of this kernel sets the optimal RoPE frequency via a field-resolution tradeoff: a frequency θ provides unambiguous positional contrast only within its field π/θ, and among frequencies whose field covers the dependency width, the highest (sharpest) one is optimal. This yields the frequency-matching principle θ⋆ = π/W. The paper validates this on synthetic block-structured data (where the fitted constant c ≈ 3.02 is close to the theoretical π) and on iGSM math problems (where broader reasoning chains shift usage toward lower frequencies). The same tradeoff is then applied to position interpolation (PI), the technique of rescaling RoPE frequencies by θ → θ/α to extend a model from context length L to αL. PI expands each frequency's field by α but coarsens its resolution by the same factor. The paper proves that PI preserves frequency utility if and only if the test-time dependency structure is a self-similar dilation of the training-time structure (Theorem 2). Under this self-similarity, the optimal training-scale frequency maps exactly to the optimal longer-context frequency. The paper shows natural language exhibits approximate self-similarity across tokenization scales, explaining why PI works for language models, while arithmetic tasks lack this structure and PI degrades accuracy even when perplexity appears acceptable.","feed_headline":"Optimal RoPE Frequency Scales as 1/W, Theory and Experiments Show","feed_subtitle":"A data-centered theory predicts that RoPE frequency usage matches the dependency width of training data, and explains when position interpol","key_machinery":"Positional dependency kernel K(r); field-resolution tradeoff (field = π/θ, resolution = arccos(1−τ)/θ); admissible frequency set A_W = {θ : θW ≤ π}; self-similarity condition P_{αW} = (S_α)_# P_W; RoPE frequency energy spectrum E_m","core_discovery":"The paper's central result is the frequency-matching principle (Theorem 1): for a positional dependency kernel of width W, the optimal admissible RoPE frequency is θ⋆ = π/W, where admissibility requires the frequency's unambiguous field π/θ to cover W. This is proven by showing that the average positional contrast U(θ;K) = ∫(1−cos(θr)) dP_K(r) is strictly increasing on the admissible set (0, π/W], so the maximum is attained at the boundary. The second key result (Theorem 2) shows that under self-similarity — defined as the test-time dependency measure being the pushforward of the training-time measure under dilation by α — position interpolation exactly preserves frequency utility: U(θ/α, αW","pith_inferences":["The theory could be extended to predict optimal per-layer or per-head frequency allocation: different heads or layers might specialize to different dependency widths present in the data, producing a mixture of optimal frequencies rather than a single one.","The self-similarity framework suggests a quantitative metric for how much PI will help on a given dataset: measure the deviation from exact self-similarity (e.g., the L1 distance between normalized dependency profiles at different scales) and correlate with PI performance degradation.","If the frequency-matching principle holds during training dynamics, one could monitor the learned RoPE spectrum during training to infer the effective dependency width the model has discovered, providing a data-independent probe of what the model has learned about positional structure.","The framework could inform RoPE frequency grid design: instead of the standard geometric spacing θ_m = base^{-2(m-1)/d}, one could space frequencies to match the expected distribution of dependency widths in the target domain."],"forward_implications":["If the frequency-matching principle is correct, one could diagnose a model's learned positional inductive bias by measuring the dependency width of its training data and checking whether the learned RoPE spectrum matches the predicted 1/W scaling.","The self-similarity condition for PI gives a concrete diagnostic: before applying PI to a new domain, measure whether that domain's dependency profiles are approximately dilation-invariant across scales. If not, PI should not be expected to help.","The result that perplexity can overstate long-context ability on non-self-similar tasks (arithmetic) suggests that long-context benchmarks should include tasks requiring precise positional resolution, not just next-token likelihood.","The mixture-of-scales view of natural language (KNL as a mixture over widths W) predicts that models trained on data with richer multi-scale dependencies should exhibit broader RoPE frequency bands, which is testable by comparing spectra across domains with different dependency profiles."],"fun_headline_variants":["RoPE Frequency Matches Training Dependency Width W, Theory Shows","Data Dependency Width Determines Optimal RoPE Frequency Usage","Position Interpolation Works When Dependencies Self-Similar Across Scales","Frequency-Matching Principle Links RoPE Usage to Training Data Structure","RoPE Frequency Selection Follows 1/W Scaling From Data-Induced Dependencies"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The theory assumes that the usefulness of a RoPE frequency can be captured by a purely geometric measure of positional contrast — how well the phase separates distances — while ignoring the content-dependent projection coefficients and softmax normalization in actual attention. The gap between this idealized single-frequency utility and the multi-frequency learned attention mechanism is not bridged by a learning theorem showing that training actually converges to thepredicted","fun_headline_variants_meta":{"raw":{"variants":["RoPE Frequency Matches Training Dependency Width W, Theory Shows","Data Dependency Width Determines Optimal RoPE Frequency Usage","Position Interpolation Works When Dependencies Self-Similar Across Scales","Frequency-Matching Principle Links RoPE Usage to Training Data Structure","RoPE Frequency Selection Follows 1/W Scaling From Data-Induced Dependencies"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":719,"prompt_tokens":649,"completion_tokens":70,"prompt_tokens_details":null},"tokens_in":649,"tokens_out":70,"duration_ms":22250,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T02:32:00.986868+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"Train transformers on datasets with controlled dependency widths W and measure whether the learned RoPE frequency spectrum concentrates at θ ≈ π/W. If the spectrum systematically deviates from the 1/W scaling — for instance, if it depends on sequence length, vocabulary size, or model width rather than W alone — the frequency-matching principle is incomplete.","supporting_citations":[],"review_version":1}