arxiv: 2604.13123 · v2 · submitted 2026-04-13 · 💻 cs.LG · cs.AI

Recognition: 2 theorem links

· Lean Theorem

Spectral Entropy Collapse as a Phase Transition in Delayed Generalisation: An Interventional and Predictive Framework for Grokkin

Truong Xuan Khanh , Truong Quynh Hoa , Luu Duc Trung , Phan Thanh Duc

Authors on Pith no claims yet

Pith reviewed 2026-05-13 06:39 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords grokkingspectral entropyrepresentational collapsedelayed generalizationphase transitionmodular arithmeticFourier alignmentneural network representations

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The pith

Spectral entropy of neural representations collapses before generalization in grokking.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates grokking, the sudden shift from memorization to generalization in neural networks after extended training. It finds that the spectral entropy of the covariance matrix of learned representations decreases steadily and crosses a consistent task-specific threshold just prior to the rise in test accuracy. An intervention that mixes representations to slow this entropy reduction also postpones the generalization transition, even when controlling for parameter norms. The distance to this entropy threshold can predict the remaining training steps until grokking occurs with reasonable accuracy on unseen data. These patterns hold across multiple modular arithmetic problems and random initializations, pointing to a geometric signature of the transition.

Core claim

Across modular arithmetic tasks, spectral entropy of the representation covariance matrix decreases gradually during training and crosses a stable task-specific threshold before test accuracy rises. A representation-mixing intervention delays this collapse and thereby delays grokking. The entropy gap predicts remaining time to grokking. Entropy collapse couples strongly to the emergence of Fourier-aligned representations, indicating concentration into task-structured directions.

What carries the argument

Spectral entropy of the representation covariance matrix, which quantifies the spread of learned features across dimensions and tracks their concentration into task-relevant directions.

If this is right

Test accuracy rises after spectral entropy crosses its threshold.
Representation mixing that delays entropy collapse also delays grokking.
The entropy gap to threshold predicts time until generalization.
Similar entropy dynamics appear in non-abelian group composition tasks.
Entropy collapse does not produce grokking without suitable inductive bias in the model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Monitoring spectral entropy during training could provide an early indicator for when generalization will begin.
The approach might extend to predicting generalization in larger-scale models like transformers.
Techniques to control representation entropy could be developed to accelerate or delay generalization as needed.
Connections to spectral dynamics in feature learning suggest broader applicability beyond the studied tasks.

Load-bearing premise

The representation-mixing intervention affects only the timing of spectral entropy collapse without introducing other changes to the training dynamics or model biases.

What would settle it

Finding training runs where test accuracy improves substantially before the spectral entropy crosses the observed threshold, or where delaying the collapse does not correspondingly delay grokking.

Figures

Figures reproduced from arXiv: 2604.13123 by Luu Duc Trung, Phan Thanh Duc, Truong Quynh Hoa, Truong Xuan Khanh.

**Figure 1.** Figure 1: Entropy collapse precedes grokking. Mean ± 1.96 SE over 10 seeds. (A) Accuracy curves showing the classic grokking delay. (B) Normalised spectral entropy H˜(t) decreasing monotonically, crossing the threshold H˜ ∗ = 0.609 (dashed line) before test accuracy rises. (C) Parameter norm increasing then plateauing. Vertical dashed line marks mean Tgrok = 14,360 steps [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Figure 2: Stability of the threshold H˜ ∗ . Histogram of H˜(Tgrok) across 10 seeds. Mean H˜ ∗ = 0.609 (95% CI: [0.595,0.624]). 6 Causal Analysis Correlation between H˜ and Tgrok does not establish causality. We therefore conduct a do-calculus-style intervention [Pearl, 2000]: at every training step, we mix representations before computing the loss, z˜i = (1−α)zi +α zσ(i) , (3) where σ is a cyclic shift (a valid dera… view at source ↗

**Figure 3.** Figure 3: Causal intervention delays grokking. Mean ± 1.96 SE over 10 seeds. (A) Test accuracy: intervention (orange) and norm-controlled (green) conditions generalise later than baseline (blue). (B) Entropy: mixing prevents H˜ from collapsing below H˜ ∗ . (C) Norm: norm-controlled condition matches baseline, confirming that the delay is attributable to entropy, not norm. ∆Tgrok = +5,020 steps (p = 0.044, d = 0.70) … view at source ↗

**Figure 4.** Figure 4: Left: Each point is one evaluation step from one seed, coloured by test accuracy. Pearson ρ(∥θ∥2,H˜) = −0.248 (p = 4×10−23). Right: Power-law fit to 1,428 data points across 10 seeds. R 2 = 0.543 (95% CI: [0.513,0.573]), γ = 1.65 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: Entropy collapse is consistent across modular arithmetic tasks. Each panel shows mean ± 1.96 SE test accuracy (orange, left axis) and H˜ (blue dashed, right axis) for 5 seeds. H˜ ∗ : 0.605 (add), 0.589 (mul), 0.589 (sub), range 2.7%. (median relative error 21%), implying a 95% predictive interval of roughly ±6,000 steps; predictions should be interpreted as probabilistic estimates rather than point forecas… view at source ↗

**Figure 6.** Figure 6: Entropy collapse in S5 permutation composition (non-abelian). (A) Mean ± 1.96 SE for 10 seeds; all grokked. (B) H˜ ∗ across group structures: modular arithmetic (Z/pZ, blue) versus S5 (orange). H˜ ∗ = 0.655 for S5 vs 0.594 for modular, consistent with higher output complexity (120 vs 97 classes). These results show that entropy collapse is a consistent signature across both abelian and non-abelian group st… view at source ↗

**Figure 7.** Figure 7: Entropy-based grokking forecaster (representative seed). (A) Purple region: interval between first accurate prediction and grokking. (B) Online prediction converging within ±20% band. (C) Prediction error over time [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗

**Figure 8.** Figure 8: Prediction accuracy across all 10 seeds. (D) Final prediction error per seed; all below 20%. (E) Advance warning lead times (mean 12,370, min 8,800 steps). An MLP and a 1-layer Transformer are trained on the same task (p = 41). Both memorise the training set within 500 steps. The MLP’s H˜ collapses from 0.76 to 0.15 — well below H˜ ∗ — yet test accuracy remains near zero for the full 80,000 steps ( [PITH_… view at source ↗

**Figure 9.** Figure 9: Entropy collapse without grokking in an MLP. Top (MLP, p = 41): (A) Train accuracy reaches 1.0; test accuracy stays near zero for 80,000 steps. (B) H˜ drops well below H˜ ∗ — collapse occurs but grokking does not. (C) Fourier alignment remains near zero (peak = 0.052). Bottom (1-layer Transformer): (D) Grokking at step 1,600. (E) H˜ crosses H˜ ∗ and generalisation follows. Entropy collapse alone does not g… view at source ↗

**Figure 10.** Figure 10: Probe robustness. (A) Nearly identical H˜ trajectories from training-set and test-set probes. (B) Absolute difference stays below 0.02 at all steps. (C) Per-seed H˜ ∗ values are highly correlated (r = 0.998). C Representation Mixing Intervention Details Given a mini-batch of representations {zi} B i=1 , the mixing operation is z˜i = (1−α)zi +α zi+1 mod B, (4) where α = 0.1. This cyclic shift is a valid de… view at source ↗

read the original abstract

Grokking - the delayed transition from memorisation to generalisation in neural networks - remains poorly understood. We study this phenomenon through the geometry of learned representations and identify a consistent empirical signature preceding generalisation: collapse of the spectral entropy of the representation covariance matrix. Across modular arithmetic tasks and multiple random seeds, spectral entropy decreases gradually during training and crosses a stable task-specific threshold before test accuracy rises. A representation-mixing intervention that delays this collapse also delays grokking, including under norm-matched controls, indicating that the effect is not explained by parameter norm alone. We further show that the entropy gap predicts the remaining time until grokking with useful out-of-sample accuracy. To probe the structure underlying this transition, we introduce a Fourier-alignment observable for cyclic-group tasks. Entropy collapse is strongly coupled to the emergence of Fourier-aligned representations, suggesting that spectral entropy tracks concentration of the representation into task-structured directions rather than generic compression alone. The same qualitative dynamics appear in non-abelian group composition tasks, while MLP controls show that entropy collapse by itself is insufficient for grokking in the absence of appropriate inductive bias. Taken together, the results support a view of grokking as a representational phase transition with an observable geometric signature. We discuss the scope and limitations of this interpretation, connections to recent feature-learning and spectral-dynamics work, and directions for testing whether similar transitions appear in larger-scale learning systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Spectral entropy collapse precedes grokking with an intervenable signature, but the mixing intervention may not isolate entropy from Fourier effects.

read the letter

The paper's central observation is that spectral entropy of the representation covariance matrix falls gradually and hits a task-specific threshold before test accuracy jumps in these grokking setups. They back this with an intervention that mixes representations to delay the collapse and thereby delay grokking, plus a predictor that uses the entropy gap to forecast remaining time to generalization. They do a few things right. The pattern holds across multiple seeds and both abelian and non-abelian group tasks. Linking the entropy drop to Fourier alignment gives a structural story rather than pure phenomenology. The MLP controls correctly show that entropy collapse alone does not produce grokking without the right bias. The prediction step is out-of-sample, which adds some credibility. The main soft spot is the intervention. Representation mixing on these cyclic tasks is likely to affect the emergence of Fourier-aligned directions directly, since the paper already ties entropy collapse to that alignment. If the mixing perturbs the projection onto those modes or the curvature, the delay in grokking could stem from that rather than the entropy threshold per se. They use norm-matched controls, but that does not rule out changes in alignment dynamics. Threshold selection also looks post-hoc and task-specific, which limits how predictive the framework really is without more validation. This is for people tracking geometric signatures in neural training dynamics. A reader working on grokking or phase transitions will pick up a concrete observable and a way to intervene on it. The work shows clear thinking on the empirical side and honest engagement with limits, so it deserves a serious referee even if the causal isolation is not fully settled. I recommend sending it for peer review, with the expectation that referees will press on the intervention's side effects and the generality of the thresholds.

Referee Report

2 major / 2 minor

Summary. The paper claims that grokking on modular arithmetic and group composition tasks is preceded by a gradual collapse in the spectral entropy of the learned representation covariance matrix, which crosses a stable task-specific threshold before test accuracy rises. A representation-mixing intervention delays this collapse (and grokking) under norm-matched controls; an entropy-gap predictor forecasts remaining time to grokking; and entropy collapse is coupled to the emergence of Fourier-aligned representations on cyclic tasks, while MLP controls show that entropy collapse alone is insufficient without appropriate inductive bias. The results are presented as evidence for grokking as a representational phase transition with an observable geometric signature.

Significance. If the central empirical pattern and intervention hold, the work supplies a concrete, measurable precursor to delayed generalization that is predictive out-of-sample and intervenable, thereby linking representation geometry to the memorization-to-generalization transition. The coupling to Fourier alignment and the contrast with MLP controls further situate the finding within spectral and feature-learning accounts of grokking, offering a falsifiable geometric lens that could be tested in larger models.

major comments (2)

[interventional experiments] The representation-mixing intervention (described in the interventional experiments) is claimed to isolate the effect of entropy collapse, yet the manuscript does not demonstrate that the mixing leaves the projection onto task-relevant Fourier modes or the curvature along those directions unchanged until the entropy threshold is crossed. Because the paper itself reports strong coupling between entropy collapse and Fourier alignment, any perturbation to alignment dynamics would confound the causal attribution of the observed delay in grokking to the entropy threshold alone.
[predictive framework] The claim that the entropy gap predicts remaining time to grokking with 'useful out-of-sample accuracy' lacks reported statistical tests, confidence intervals, or details on whether the task-specific thresholds were chosen post-hoc versus pre-specified. Without these, the predictive utility and the assertion of a stable threshold cannot be fully evaluated from the presented results.

minor comments (2)

[methods] Notation for the spectral entropy (eigenvalue-based or otherwise) and the precise definition of the representation covariance matrix should be stated explicitly in the methods section to allow direct reproduction.
[figures] Figure captions for the entropy trajectories and intervention results should include the exact number of seeds, the norm-matching procedure, and any post-selection criteria for the displayed runs.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which identify key areas for strengthening the causal interpretation of the interventional results and the statistical grounding of the predictive claims. We address each point below and will incorporate revisions that clarify these aspects while preserving the manuscript's core empirical findings.

read point-by-point responses

Referee: The representation-mixing intervention (described in the interventional experiments) is claimed to isolate the effect of entropy collapse, yet the manuscript does not demonstrate that the mixing leaves the projection onto task-relevant Fourier modes or the curvature along those directions unchanged until the entropy threshold is crossed. Because the paper itself reports strong coupling between entropy collapse and Fourier alignment, any perturbation to alignment dynamics would confound the causal attribution of the observed delay in grokking to the entropy threshold alone.

Authors: We appreciate the referee highlighting this potential confound given the reported coupling. The representation-mixing procedure was constructed to perturb covariance structure while preserving per-sample norms and without explicit targeting of Fourier directions; however, the current manuscript does not include post-intervention verification of Fourier-mode projections or curvature. In revision we will add these diagnostics, showing that alignment trajectories under mixing remain statistically indistinguishable from controls until the entropy threshold is reached. This addition, together with the existing norm-matched controls, will better isolate the entropy effect. We note that the observed delay in grokking is consistent across multiple tasks and seeds, supporting the interpretation even if full causal isolation requires the proposed checks. revision: yes
Referee: The claim that the entropy gap predicts remaining time to grokking with 'useful out-of-sample accuracy' lacks reported statistical tests, confidence intervals, or details on whether the task-specific thresholds were chosen post-hoc versus pre-specified. Without these, the predictive utility and the assertion of a stable threshold cannot be fully evaluated from the presented results.

Authors: We agree that the predictive section would benefit from greater statistical transparency. Thresholds were identified from the stabilization point observed in pilot runs on a disjoint set of seeds and then applied to held-out data; they were not tuned on the evaluation set. In the revised manuscript we will report: bootstrap confidence intervals on out-of-sample prediction accuracy, Pearson correlation coefficients with associated p-values between entropy gap and time-to-grokking, and an explicit description of the pre-specification procedure. These additions will allow readers to evaluate both the stability of the thresholds and the practical utility of the predictor. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical measurements and external intervention remain independent of fitted inputs

full rationale

The paper reports direct computation of spectral entropy from the representation covariance matrix, its gradual decrease during training, and its crossing of a task-specific threshold before test accuracy increases. The representation-mixing intervention is introduced as an external manipulation (with norm-matched controls) that alters the timing of entropy collapse and thereby delays grokking. The entropy-gap predictor is evaluated on held-out timing data rather than being a re-expression of any fitted parameter. No equations, self-citations, or ansatzes are shown to reduce the central claims to their own inputs by construction; the derivation chain consists of observational and interventional evidence that does not collapse into self-definition or renaming of known results.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The framework rests on the assumption that spectral entropy of the covariance matrix meaningfully tracks concentration into task-structured directions rather than generic compression, and that the chosen intervention affects only this quantity.

free parameters (1)

task-specific entropy threshold
Stable threshold crossed before accuracy rise; appears chosen or fitted per task from the data.

axioms (1)

domain assumption Spectral entropy of representation covariance tracks concentration into task-structured directions
Invoked to interpret collapse as more than generic compression.

pith-pipeline@v0.9.0 · 5575 in / 1195 out tokens · 72921 ms · 2026-05-13T06:39:58.327843+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/EntropyCollapse or Cost.FunctionalEquation entropy collapse as first-order phase transition; J-cost uniqueness (washburn_uniqueness_aczel) echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

We propose that a useful diagnostic quantity is the normalised spectral entropy ˜H(t)=H(t)/log d of the representation covariance matrix, and present empirical evidence that grokking is associated with a collapse of ˜H below a task-specific threshold ˜H∗.
IndisputableMonolith/Foundation/BranchSelection or ArithmeticFromLogic J uniquely calibrated via higher-derivative or coupling-combiner forcing refines

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refines
Relation between the paper passage and the cited Recognition theorem.

entropy collapse is necessary but not sufficient for grokking... architectural inductive biases play a critical role

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages · 1 internal anchor

[1]

Neel Nanda, Lawrence Chan, Tom Lieberum, Jess Smith, and Jacob Steinhardt

URLhttps://arxiv.org/abs/2303.11873. Neel Nanda, Lawrence Chan, Tom Lieberum, Jess Smith, and Jacob Steinhardt. Progress measures for grokking via mechanistic interpretability. InThe Eleventh International Conference on Learning Representations, 2023. URL https://arxiv.org/abs/2301.05217. Catherine Olsson, Nelson Elhage, Neel Nanda, Nicholas Joseph, Nova ...

work page arXiv 2023
[2]

Vardan Papyan, XY Han, and David L Donoho

URL https://transformer-circuits.pub/2022/in-context-learning-and-induction-heads/ index.html. Vardan Papyan, XY Han, and David L Donoho. Prevalence of neural collapse during the terminal phase of deep learning training.Proceedings of the National Academy of Sciences, 117(40):24652–24663, 2020. Judea Pearl.Causality: Models, Reasoning, and Inference. Camb...

work page 2022
[3]

Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets

URLhttps://arxiv.org/abs/2201.02177. Marten Scheffer, Jordi Bascompte, William A Brock, Victor Brovkin, Stephen R Carpenter, Vasilis Dakos, Hermann Held, Egbert H Van Nes, Max Rietkerk, and George Sugihara. Early-warning signals for critical transitions.Nature, 461(7260):53–59, 2009. Yuandong Tian, Xinlei Chen, and Surya Ganguli. Understanding self-superv...

work page internal anchor Pith review Pith/arXiv arXiv 2009
[4]

Instrument training.Add a fixed probe set (N≥4d ) and compute ˜H every 200–500 steps using theSpectralEntropyMonitor class

work page
[5]

3.Activate the predictor.Once ˜H(t)< ˜H∗ +0.15, callpredict_grok_time()at each eval step

Identify ˜H∗ empirically.Run 3–5 seeds to completion, record ˜H at test accuracy ≥0.99 , and average to obtain task-specific ˜H∗. 3.Activate the predictor.Once ˜H(t)< ˜H∗ +0.15, callpredict_grok_time()at each eval step. 4.Apply early stopping.When the prediction stabilises, halt training

work page
[6]

Diagnose failures.If ˜H does not collapse below ˜H∗ after ≥30,000 steps, the configuration is unlikely to grok (Table 5). D.2 Diagnostic Guide D.3 Computational Overhead Computing ˜H requires one forward pass over the probe set ( N=512 , d=128 ) and an eigendecomposition of a 128×128 covariance matrix: approximately8 msper eval call, <0.05% of total train...

work page