Paper deep dive
Convergent Evolution: How Different Language Models Learn Similar Number Representations
Deqing Fu, Tianyi Zhou, Mikhail Belkin, Vatsal Sharan, Robin Jia
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 97%
Last extracted: 4/26/2026, 6:50:28 PM
Summary
The paper investigates the 'convergent evolution' of number representations in language models, distinguishing between 'spectral convergence' (the presence of periodic Fourier spikes at periods T=2, 5, 10) and 'geometric convergence' (the ability to linearly classify numbers mod-T). The authors prove that while Fourier spikes are a universal feature across diverse architectures (Transformers, Linear RNNs, LSTMs, and classical embeddings) and even raw token distributions, they are necessary but not sufficient for geometric separability. Through controlled experiments, they demonstrate that geometric convergence depends on the alignment of data signals (text-number co-occurrence and cross-number interaction), architecture (Transformers and Linear RNNs succeed while LSTMs fail), and the tokenizer (multi-token addition tasks drive circular representations).
Entities (8)
Relation Signals (6)
Transformer → achieves → Geometric Convergence
confidence 100% · Transformers and linear RNNs achieve strong probing while LSTMs trained on the same data remain at chance
Transformer → exhibits → Spectral Convergence
confidence 100% · Transformers... all learn features that have period-T spikes in the Fourier domain
LSTM → exhibits → Spectral Convergence
confidence 100% · LSTMs... all learn features that have period-T spikes in the Fourier domain
LSTM → failstoachieve → Geometric Convergence
confidence 100% · LSTMs trained on identical data develop more prominent Fourier spikes but achieve chance-level probing
Fourier Spikes → isnecessaryfor → Geometric Convergence
confidence 100% · we prove that Fourier domain sparsity is necessary but not sufficient for mod-T geometric separability
Text-Number Co-occurrence → drives → Geometric Convergence
confidence 90% · they can learn them from complementary co-occurrence signals in general language data, including text-number co-occurrence
Cypher Suggestions (0)
No Cypher suggestions yet.
Abstract
Abstract:Language models trained on natural text learn to represent numbers using periodic features with dominant periods at $T=2, 5, 10$. In this paper, we identify a two-tiered hierarchy of these features: while Transformers, Linear RNNs, LSTMs, and classical word embeddings trained in different ways all learn features that have period-$T$ spikes in the Fourier domain, only some learn geometrically separable features that can be used to linearly classify a number mod-$T$. To explain this incongruity, we prove that Fourier domain sparsity is necessary but not sufficient for mod-$T$ geometric separability. Empirically, we investigate when model training yields geometrically separable features, finding that the data, architecture, optimizer, and tokenizer all play key roles. In particular, we identify two different routes through which models can acquire geometrically separable features: they can learn them from complementary co-occurrence signals in general language data, including text-number co-occurrence and cross-number interaction, or from multi-token (but not single-token) addition problems. Overall, our results highlight the phenomenon of convergent evolution in feature learning: A diverse range of models learn similar features from different training signals.
Tags
Links
- Source: https://arxiv.org/abs/2604.20817v1
- Canonical: https://arxiv.org/abs/2604.20817v1
Trouble viewing inline? Open PDF directly →
Full Text
77,470 characters extracted from source content.
Expand or collapse full text
April 22, 2026 Convergent Evolution: How Different Language Models Learn Similar Number Representations Deqing Fu σ Tianyi Zhou σ Mikhail Belkin ψ Vatsal Sharan σ Robin Jia σ σ University of Southern California ψ UC San Diego deqingfu,tzhou029,vsharan,robinjia@usc.edu, mbelkin@ucsd.edu Abstract Language models trained on natural text learn to represent numbers using periodic features with dominant periods atT =2, 5, 10. In this paper, we identify a two-tiered hierarchy of these features: while Transformers, Linear RNNs, LSTMs, and classical word embeddings trained in different ways all learn features that have period-Tspikes in the Fourier domain, only some learn geometrically separable features that can be used to linearly classify a number mod-T. To explain this incongruity, we prove that Fourier domain sparsity is necessary but not sufficient for mod-Tgeometric separability. Empirically, we investigate when model training yields geometrically sepa- rable features, finding that the data, architecture, optimizer, and tokenizer all play key roles. In particular, we identify two different routes through which models can acquire geometrically separable features: they can learn them from complementary co-occurrence signals in general language data, including text-number co-occurrence and cross-number interaction, or from multi-token (but not single-token) addition problems. Overall, our results highlight the phenomenon of convergent evolution in feature learning: A di- verse range of models learn similar features from different training signals. Models: https://hf.co/collections/deqing/convergent-evolution 1 Introduction Language models trained on natural language develop periodic representations for number tokens. For many Transformer-based language models, Zhou et al. (2024) show that the embeddings of integer tokens have consistent spikes in the Fourier domain at periodsT = 2, 5, and 10. Such periodic features have also been well-documented in models’ intermediate representations (Levy & Geva, 2025) and in the model mechanisms that implement addition (Zhou et al., 2024; Kantamneni & Tegmark, 2025). Engels et al. (2025) even find analogous periodic structures for other cyclical concepts such as days of the week and months of the year. These findings have been broadly interpreted as evidence that language models learn structured numerical representations through next-token prediction. In this paper, we first demonstrate that this phenomenon is far more general than previously recognized. Figure 1 shows that the sameT =2, 5, 10 spikes appear not only in Transformers (Vaswani et al., 2017) of varying scale (GPT-2 (Radford et al., 2019), GPT-OSS (OpenAI, 2025), Llama-3 (Meta, 2024), Llama-4 (Meta, 2025), and DeepSeek-V3 (DeepSeek, 2025)), but also in non-Transformer LLMs (Mamba (Gu & Dao, 2024), Falcon-Mamba (Zuo et al., 2024), xLSTM (Beck et al., 2024), Kimi-Linear (Kimi, 2025)) and classical word embeddings (GloVe (Pennington et al., 2014) and FastText (Bojanowski et al., 2017)). Even the raw token frequency distribution of numbers in the training corpus, with no model at all, exhibits the same periodic spectrum (see Figure 2). We view this universality as a case of convergent evolution: different systems independently develop the same representation because they share the same constraints from training data and tokenization. In biology, convergent evolution refers to the independent emergence of similar traits in unrelated organisms facing shared environmental pressures, such as the independent evolution of eyes in vertebrates and cephalopods (McGhee, 2011). Fourier features in number embeddings are analogous: a shared trait that arises from shared constraints on the learning process. arXiv:2604.20817v1 [cs.CL] 22 Apr 2026 Convergent Evolution2 GPT-2-1.5B GPT-OSS-120B Llama-3.1-8B Llama-4-Scout-109B DeepSeek-V3-671B Transformer LLMs Mamba-2.8B Falcon-Mamba-7B xLSTM-7B Kimi-Linear-48B Non-Transformer LLMs 0.00.10.20.30.40.5 Fourier Frequency GloVe-Dolma FastText-WikiNews Word Embeddings Fourier Spectrum on Number Embeddings of Pretrained Models 1234 Norm. Magnitude Transformer (Muon) Transformer (AdamW) Gated DeltaNet Mamba-2 LSTM Word2Vec PPMI 9-digit Addition 3-digit Addition Token Distribution Transformer Linear RNNs Gated RNNs Autoregressive Language Models Word Embeddings Transformer Trained on Addition Learned Models Counting Only Both Spectral & Geometric ConvergenceSpectral Convergence Only Spectral & Partial Geometric Convergence Neither Figure 1: Universality of Fourier Features and Convergent Evolution. (Left) Fourier spectrum of number embeddings across three architecture families: Transformer LLMs, non-Transformer LLMs, and classical word embeddings. Each row shows the median- normalized magnitude at each Fourier frequency. All models exhibit consistent spikes at frequencies of periodT =2, 5, 10, etc. (Right) Convergent Evolution of various models studied in this paper. It shows two types of convergence: spectral convergence where models learn Fourier spikes, and geometric convergence where models learn modular probes. But do these Fourier spikes indicate that models have learned functional numerical struc- ture? We identify a two-tiered hierarchy of periodic features: only some systems with Fourier spikes cleanly encode modular arithmetic properties in their embeddings. By this we mean that the residue classn mod Tis linearly decodable from the embeddinge(n). Period-Tfeatures naturally group numbers by their value modT, and the linear representa- tion hypothesis (Park et al., 2023) conjectures that such structure should be accessible via linear probes. We call the emergence of Fourier spikes spectral convergence and the emergence of linearly separable mod-Tclasses geometric convergence. Spectral convergence appears in almost every system we examine, but geometric convergence does not: Transformers and linear RNNs trained on 10 billion tokens develop embeddings where mod-Tclasses are linearly separable, while LSTMs trained on identical data develop more prominent Fourier spikes but achieve chance-level probing. Understanding what separates these two levels of convergence is the central question of this paper. We summarize our contributions below. Fourier spikes are universal but probing is not. We show that Fourier spikes atT =2, 5, 10 appear in every system we examine, spanning Transformer and non-Transformer LLMs, classical word embeddings, and even the raw token frequency distribution (Figure 1). We demonstrate both theoretically (Theorem 1) and empirically (Figure 2) that Fourier spikes are necessary but not sufficient for mod-Tprobing, and explain how models with similar spectra can have vastly different probing accuracy (§3). Geometric convergence requires data, architecture, and optimizer to align. Through controlled experiments on 300M-parameter models trained on identical data, we isolate three factors that jointly determine whether mod-Tclasses become linearly separable in the embeddings. Our experimental methodology can be viewed as a form of structure attribution. Analogous to how influence functions (Koh & Liang, 2017) or Shapley values (Ghorbani & Zou, 2019) attribute model predictions to individual training examples, our controlled perturbations attribute the emergence of learned representations to specific structural properties of the data distribution. We find that geometric convergence depends on several complementary data signals: perturbations that progressively remove text- number co-occurrence, cross-number interaction, or context length each degrade probing, while Fourier spikes persist across all conditions (§4.1). The architecture plays a critical role: Transformers and linear RNNs achieve strong probing while LSTMs trained on the same data remain at chance (§4.2). Regardless of the optimizer, both Transformers and linear RNNs learn the same Fourier spectrum but different probing performance. 3Fu, Zhou, Belkin, Sharan, & Jia Convergent evolution takes a different form under arithmetic task pressure. In mod- els trained on addition from scratch, the tokenizer determines whether Fourier structure emerges (§5). Multi-token addition requires computing each output digit as a sum modulo 1000, forcing the model to solve modular subproblems that produce circular representa- tions. Single-token addition admits multiple strategies, and the resulting representations randomly depend on the optimizer. 2 Related Work Fourier Features. Fourier features have been used for many years in computer vision as edge and orientation detectors (Olshausen & Field, 1997; Olah et al., 2020; Fiquet & Simoncelli, 2023). The original Transformer applies sinusoidal position encodings (Vaswani et al., 2017), and several works have found that explicitly injecting high-frequency components into inputs helps with spatial and numerical tasks (Tancik et al., 2020; He et al., 2023; Hua et al., 2024). Recently, this structure is also found to emerge without being designed in. Transformers trained on modular addition embed numbers on a circle and rotating to compute the answer (Nanda et al., 2023; Zhong et al., 2023; Gromov, 2023). The same holds in pretrained LLMs that number token embeddings break into Fourier components, and there are recognizable addition circuits in the attention and MLP layers (Zhou et al., 2024; Kantamneni & Tegmark, 2025; Levy & Geva, 2025). Zhou et al. (2025) show that hard-coding these Fourier features improves arithmetic learning. These studies document spectral structure but do not test whether it implies geometric separability — a distinction our work shows is critical. In this paper, we further study the question these papers leave open, where the structure comes from in the first place. Mechanistic Interpretability. Mechanistic interpretability aims to reverse-engineer the representations and algorithms learned by language models. The linear representation hypothesis (Park et al., 2023) conjectures that high-level concepts, if learned, should be linearly decodable from model representations. Probing (Orgad et al., 2025; Kossen et al., 2024) is the standard tool for this. However, number representations are not linearly encoded (Nanda et al., 2023; Zhong et al., 2023; Gromov, 2023). Karkada et al. (2026) also find circular representations for days of the week, which suggests this is a fairly general solution to any problem with rotational symmetry. Allen-Zhu (2025) use controlled synthetic pretraining to isolate which capabilities emerge from which architectural and data choices. Separately, Huh et al. (2024) argue that representations across models and modalities are converging toward a shared statistical model of reality, measured via global kernel alignment. We ask a different question: whether models that converge on the similar representation of a specific concept have learned the same functional structure, and show that they can diverge fundamentally. In this paper, we vary tokenization, architecture, optimizer, and task to understand the convergent evolution of number representations. 3 Problem Setup and Preliminary Analysis We study the token embeddings of numbers 0 throughN−1 in language models, where N =1000 corresponds to the set of numbers that receive single-token representations in the Llama-3 tokenizer (Meta, 2024). Lete(n)∈ R d denote the token embedding of numbern. To detect periodic structure in these embeddings, following Zhou et al. (2024), we compute the discrete Fourier transform along the token index: F ν = 1 √ N N−1 ∑ n=0 e(n) e −2πiνn ∈ C d ,ν = k N , k = 0, . . . , N− 1. The power at frequencyνis∥F ν ∥ 2 = ∑ d j=1 |F (j) ν | 2 . A Fourier spike at periodTrefers to a visible peak in∥F 1/T ∥ 2 relative to neighboring frequencies. For a periodTdividingN, define the mod-Tresidue classesC r = n : n ≡ r (mod T)forr =0,. . .,T−1, each of size |C r | = N/T. To evaluate whether the embeddings encode modular arithmetic at periodT, we train a linear probe (T-class logistic regression) to predict n mod T from e(n). A natural question is whether the presence of a Fourier spike at periodTguarantees good mod-Tprobing accuracy. As we will show in Figure 2, the answer is strikingly no: an Convergent Evolution4 0.00.10.20.30.40.5 0 4 8 T=2 T=5 T=10 Transformer Fourier Magnitude Spectrum 2345678910 0 50 100 96 -4 35 64 11 -9 10 -5 85 Cohen's of Linear Probe (%) 123456789 0 15 30 max = 23.314 Fourier 10 = 33.8 cond(S W ) = 122.1 Eigenvalues of S 1 W S B (T=10) 0.00.10.20.30.40.5 0 4 8 T=2 T=5 T=10 Gated DeltaNet 2345678910 0 50 100 95 -5 37 55 11 -7 14 -5 78 123456789 0 15 30 max = 19.963 Fourier 10 = 24.6 cond(S W ) = 126.3 0.00.10.20.30.40.5 0 4 8 T=2 T=5 T=10 LSTM 2345678910 0 50 100 9 0 0 3 0 -1 0 -1 2 123456789 0 15 30 max = 0.290 Fourier 10 = 337.4 cond(S W ) = 8569.0 0.00.10.20.30.40.5 Fourier Frequency 0 4 8 T=2 T=5 T=10 Token Dist. 2345678910 Modulus T 0 50 100 4 0 1 4 1 -1 1 0 3 123456789 Eigenvalue Rank i 0 15 30 max = 0.046 Fourier 10 = 94.0 Figure 2: A “Spiky” Fourier Spectrum Does Not Imply Good Feature Learning. (Left) Token embeddings of Transformer, Gated DeltaNet and LSTM, and even simply the Number Token Distribution Frequency exhibit distinct Fourier spikes atT =2, 5, and 10. (Middle) Linear probes reveal only the Transformer and Gated DeltaNet learned functional modular arithmetic with high Cohen’sκ, while others remain at chance. (Right) Theorem 1 explains this discrepancy through the internal noise structure of the T = 10 embeddings. LSTM trained on the same data as a Transformer develops larger Fourier power atT =10 yet achieves chance-level probing. The following result shows how this is possible and demonstrates that the presence of a Fourier spike is a necessary but not a sufficient condition for learning modular probes. Theorem 1. Given embeddingse(n) N−1 n=0 and residue classesC r T−1 r=0 defined above, let the class meansμ r , grand meanμ, between-class scatter matrix S B , and within-class scatter matrix S W be μ r = 1 |C r | ∑ n∈C r e(n),μ = 1 N N−1 ∑ n=0 e(n), S B = 1 T T−1 ∑ r=0 (μ r −μ)(μ r −μ) ⊤ ,S W = 1 N T−1 ∑ r=0 ∑ n∈C r (e(n)−μ r )(e(n)−μ r ) ⊤ . LetΦ T = ∑ T−1 ℓ=1 ∥F ℓ/T ∥ 2 be the total power at harmonics of periodT, andH T = 0, 1/T, 2/T, . . . , (T−1)/T be the set of harmonic frequencies. (i) IfΦ T = 0, then S B = 0 and no linear probe can classify n mod T above chance. (i)For anyT ≥2,C >0, andε >0, there existNdivisible byTand embeddingse(n) satisfyingΦ T > Cyet noT-class linear classifier achieves accuracy above 1/T + ε, i.e., no more than ε above random guessing. 5Fu, Zhou, Belkin, Sharan, & Jia 01020 e(n) A=5, B=0.2, T =1250, best linear acc=100.0% N=25 Points. Period T=5 e(n) = A (n mod T) + Bn/T 0 5 10 15 20 1 6 11 16 21 2 7 12 17 22 3 8 13 18 23 4 9 14 19 24 mod 5 01234 0.00.10.20.30.40.5 frequency 0 200 400 600 800 1000 F 2 T=5 0204060 e(n) A=5, B=11, T =1250, best linear acc=36.0% 0 1 2 5 3 6 4 7 10 8 11 9 12 15 13 16 14 17 20 18 21 19 22 23 24 0.00.10.20.30.40.5 frequency 0 200 400 600 800 1000 F 2 T=5 Figure 3: Examples for the Proof of Theorem 1 Part (i) in § A.3. In the proof, every number n ∈ 0,. . .,N−1has a unique decompositionn = r + mTwith residuer = n mod T ∈ 0,. . .,T−1and block indexm = ⌊n/T⌋ ∈ 0,. . .,K−1, whereK = N/T. We set the embeddinge(n) = Ar + Bm, so thatAcontrols the between-class scatterS B (and hence Φ T = N· S B ) whileBcontrols only the within-class scatterS W , with neither parameter affecting the other. As illustrated in the figure, fixingAproduces a persistent Fourier spike at periodTregardless ofB: whenBis small the residue classes cluster into linearly separable groups, but asBgrows the embeddings interleave across classes so that the best linear classification accuracy goes closer to random guessing 1/T. Theorem 1 establishes thatΦ T >0 is necessary but not sufficient. A natural follow-up is: what quantitatively determines whether a spikyΦ T translates into geometric separability? Remark 2. The gap between two parts of Theorem 1 can be made more precise through the lens of Fisher’s Linear Discriminant Analysis (Fisher, 1936). Assumed≥ T−1 andS W is invertible. Assumed > T−1 andS W is invertible. While a linear probe forTclasses separates data across a (T−1)-dimensional subspace, the absolute ceiling on its performance is dictated by its principal axis of separation: the single directionvthat maximizes the ratio of between-class to within-class variance, given by the Rayleigh quotient v ⊤ S B v v ⊤ S W v . The maximum achievable separability along this optimal axis is exactly the largest generalized eigenvalue of the scatter matrices,λ max (S −1 W S B ) . We show (see § A.4) that this maximum discriminant satisfies: 1 (T− 1)· cond(S W ) · Φ T N· λ min (S W ) ≤ λ max (S −1 W S B ) ≤ Φ T N· λ min (S W ) . The Fourier powerΦ T only guarantees a large total variance among the class means, which drives Φ T = Tr(S B )/Nin the numerator. It ensures the class centers are dispersed in space. However, actual linear separability depends on how this signal aligns with the within-class scatter,S W . A large condition number,cond(S W ) = λ max (S W )/λ min (S W ), drastically lowers the minimum bound. If the periodic signal aligns with the dimensions of maximum within-class noise, the massiveΦ T is entirely swallowed by within-class variance. This drivesλ max toward zero and forces the entire (T− 1)-dimensional subspace to collapse into overlapping classes. The proof of Theorem 1 is given in Appendix A, where part (i) follows naturally from Fourier identities in Lemma 4 which shows thatTr(S B ) =Φ T /N, and part (i) constructs examples (see Figure 3) so thatΦ T can be arbitrarily large but residue classes are not linearly separable. We now ask two empirical questions: how common are Fourier spikes in practice, and do they actually predict mod-T probe accuracy? Throughout this paper, unless stated otherwise, we train models with around 300 million parameters on 10B tokens from FineWeb-Edu (Lozhkov et al., 2024) using the Llama-3 tokenizer, which assigns single tokens to each integer from 0 to 999 such thatN =1000. We include architecture details in the Appendix B.1. We evaluate number representations Convergent Evolution6 using two complementary measurements: (1) the Fourier magnitude spectrum, computed as the power∥F ν ∥ 2 normalized by the median across frequenciesν, and (2) the mod-T probe accuracy, measured by Cohen’sκof a classifier trained to predictn mod Tfrom the token embeddinge(n). Cohen’sκadjusts for the baseline accuracy of random guessing (1/Tfor balanced classes), so thatκ =0 corresponds to chance andκ =100% to perfect classification regardless ofT. We report accuracy-based results in the appendix. We use three probe types: linear (logistic regression), MLP, and RFM kernel (Radhakrishnan et al., 2024); unless noted, we report linear probe results, with the others in § B.5. We report probe performance averaged over 30 runs: 3 random seeds, each with 10-fold cross-validation. The first measurement detects periodic structure while the second tests whether that structure supports mod-T classification. Fourier spikes are universal. Figure 1 shows that every pretrained model we examine, spanning Transformer LLMs, non-Transformer LLMs, and classical word embeddings, exhibits peaks atν =1/10, 1/5, 1/2. Spectral convergence is universal as long as natural languages are used for training. This holds across architectures across fundamentally different learning algorithms and across models never explicitly trained on numerical tasks. Fourier spikes do not imply modular arithmetic. Figure 2 presents a controlled comparison under our training setup: a Transformer, a Gated DeltaNet (Yang et al., 2025), an LSTM (Hochreiter & Schmidhuber, 1997) (all around 300M parameters), and the raw number token frequency distribution from the training corpus, where each numbernis represented by its scalar corpus frequencyp n via counting rather than a learned embedding. The left column shows that all four produce qualitatively similar Fourier spikes at periodsT =2, 5, 10. The middle column shows mod-Tprobe accuracy: the Transformer and Gated DeltaNet achieve κ =96 and 95 atT =2, andκ =85 and 78 atT =10, while the LSTM and the token distribution remain at chance across all moduli. The right column explains this gap through Theorem 1. The LSTM has larger Fourier powerΦ 10 than the Transformer, yet its Fisher discriminantλ max (S −1 W S B )is two orders of magnitude smaller. The difference lies in the condition numbercond(S W ): the LSTM’s within-class scatter is highly anisotropic, so the periodic signal is buried under within-class variance and the classes overlap despite visible Fourier spikes. Several recent studies have inspected Fourier spectra and concluded that models have learned modular structure (Nanda et al., 2023; Zhou et al., 2024). Our analysis shows why this inference is unreliable: a visible spike at periodTguaranteesTr(S B ) >0, but probe accuracy depends on the eigenspectrum ofS −1 W S B , which the Fourier power spectrum alone does not determine. We call this arrangement of embeddings into linearly separable mod-Tclasses Geometric Convergence. Unlike spectral convergence, geometric convergence is selective: only certain combinations of data, architecture, and optimizer produce it. Both are instances of con- vergent evolution: different systems arriving at similar representations because of shared constraints. The next sections identify what drives each. 4 Convergent Evolution in Language Model Pretraining Spectral convergence requires only the periodic frequency distribution of number tokens, but geometric convergence is selective. We now ask which constraints must be present for geometric convergence to emerge. Through controlled experiments that vary one factor at a time, we identify three: the data signal the model receives (§4.1), the architecture, and the optimizer (§4.2). All experiments use 300M-parameter models trained on 10B tokens from FineWeb-Edu (Lozhkov et al., 2024) with the Llama-3 tokenizer, unless stated otherwise. 4.1 Structural Attribution to Data To isolate the environmental pressures driving convergence, we fix the architecture (300M Transformer) and optimizer (Muon, Jordan et al., 2024) and vary only the training data. We apply controlled perturbations (Table 1) that each remove a specific type of co-occurrence while leaving others intact, attributing the emergence of spectral and geometric convergence to specific structural properties of the data. 7Fu, Zhou, Belkin, Sharan, & Jia Table 1: Data perturbation used in §4.1. All models on the 10B tokens from FineWeb-Edu. ConfigurationPerturbationStructure of Data Removed Original− Isolate-kEach sequence contains at mostknumbers token via packing. We use k = 1, 2, and 8. Interaction across numbers. k =1 indicates no interaction. ContextLength-ℓSequences split into windows ofℓ. We use ℓ = 2, 4, 8, and 64 Role of broad context Swap NumbersNumber token sequence replaced by that of another sequence, keeping number n-gram Number↔ text association Unigram ReplaceEvery number token resampled i.i.d. from marginal distribution to replace the original Co-occurrence v.s. frequency Original Transformer LM (Muon) 96.1-4.434.763.510.9-9.09.9-5.285.4 Context Length 64 Context Length 8 Context Length 4 Context Length 2 Reducing Context Length 93.5-7.131.446.611.7-7.810.7-6.372.0 88.7-4.129.834.910.1-7.310.0-3.751.7 87.3-7.227.734.78.1-5.610.2-4.147.4 71.7-6.326.229.77.6-3.912.1-1.040.6 Isolate (k=8) Isolate (k=2) Isolate (k=1) Reducing Cross-Number Interaction 94.1-4.432.257.29.4-8.711.9-6.477.2 89.6-4.731.636.111.2-5.912.1-5.353.0 85.9-0.430.134.314.1-4.711.8-2.445.0 0.00.10.20.30.40.5 Fourier Frequency Swap Numbers Unigram Replace Number Replacement mod 2mod 3mod 4mod 5mod 6mod 7mod 8mod 9mod 10 70.7-10.220.622.85.2-6.38.7-5.228.8 -5.9-0.7-1.81.2-2.20.6-1.00.21.2 1.0 1.5 2.0 3.0 4.0 Norm. Magnitude 0 20 40 60 80 100 Cohen's (%) Fourier Magnitude Spectrum Cohen's of Linear Probe (%) Figure 4: Spectral convergence is universal but geometric convergence depends on the data signal. (Left) Fourier spectra of Transformer embeddings trained under data per- turbations in Table 1. All perturbations produce similar spikes to the original at periods T =2, 5, 10, includingUnigram Replace, which destroys all co-occurrence structure among number tokens. (Right) Cohen’sκof linear probes for mod-Tclassification tells a different story:Original,Isolate-8, andContext Length 64achieve strong probing, but with shorter context length or fewer numbers within each sequence (e.g.Isolate-2) is much weaker at T = 5 and 10. Swap Numbers drops substantially, and Unigram Replace falls to chance. Spectral convergence requires only token frequencies. All perturbations produce nearly identical Fourier spectra (Figure 4, left), includingUnigram Replace, which destroys all co-occurrence structure by independently resampling every number token from its marginal distribution. As predicted by the universality observed in §3, the periodic frequency distribution of number tokens is sufficient to produce Fourier spikes, and no co-occurrence information is needed. Geometric convergence draws on several complementary data signals. The right panel of Figure 4 reveals that geometric convergence degrades gradually as different types of co-occurrence information are removed. The probing power is measured with Cohen’s κfor balanced classes, the metric that removes random guessing from accuracy forT- way classifications defined asκ = (Accuracy− 1 T )/(1− 1 T ). Whenκ =0, the probe is at chance andκ =100% is perfect classification regardless ofT.Swap Numbers, which preserves numbern-gram statistics but destroys the association between specific numbers and their text contexts, drops probing fromκ =85.4 to 28.8 atT =10, making text-number co-occurrence an important signal. But it is not the only one. Longer context provides a second signal. AtContext Length 2, where each token sees only one neighbor, mod-10 probing already reachesκ =40.3, well aboveSwap Numbers(28.8). Increasing the window toℓ =4, 8, and 64 steadily improvesmod10 probes (κ =47.3, 51.7, Convergent Evolution8 Transformer Gated DeltaNet Mamba-2 Language Models (Muon) 96.1-4.534.963.511.0-9.19.9-5.285.4 95.1-5.337.355.210.7-6.913.6-5.277.8 93.8-2.129.350.710.7-8.412.9-4.876.7 Transformer Gated DeltaNet Mamba-2 LSTM Language Models (AdamW) 89.1-2.832.447.712.3-8.111.3-5.372.1 91.1-5.828.446.411.7-9.810.6-4.269.7 95.0-6.031.255.311.1-7.610.8-3.780.1 9.0-0.30.32.70.4-1.20.3-0.82.0 0.00.10.20.30.40.5 Fourier Frequency PPMI word2vec Classical Word Embeddings mod 2mod 3mod 4mod 5mod 6mod 7mod 8mod 9mod 10 54.4-14.519.820.8-0.1-10.67.5-7.527.1 40.7-2.318.925.07.4-2.46.0-2.029.3 1.0 1.5 2.0 3.0 4.0 Norm. Magnitude 0 20 40 60 80 100 Cohen's (%) Fourier Magnitude Spectrum Cohen's of Linear Probe (%) Figure 5: Geometric convergence depends on architecture and optimizer. (Left) Fourier magnitude spectra of number embeddings across architectures and optimizers; all exhibit spikes at periodsT =2, 5, 10, including theLSTMand classical word embeddings. All three architectures produce similar Fourier spectra under both optimizers. (Right) Mod-Tprobe accuracy separates the models into tiers. With both Muon and AdamW,Transformer,Gated DeltaNet, andMamba-2all achieve strong geometric convergence, while theLSTMremains near zero. Muon outperforms AdamW forTransformerandGated DeltaNet, butMamba-2 with AdamW slightly outperforms its Muon counterpart.PPMIandword2vecfall in between. and 72.0 respectively), showing that the model accumulates richer co-occurrence statistics from broader context. Cross-number interaction provides an additional signal as well.Isolate-kdirectly controls cross-number interaction by restricting each packed sequence to contain at most k number tokens. The extreme limit ofk =1 isolates text-number co-occurrence completely, ensuring no two number tokens can interact within the same attention window. Under this setting, it achievesκ =45.0 atT =10 and 85.9 atT =2. Notably, evenk =1 with a Transformer sur- passes PPMI (κ =27.1) and word2vec (κ =29.3), suggesting that autoregressive language modeling with text-number co-occurrence alone extracts richer modular structure than classical embedding methods. Allowing more numbers to co-occur within each attention block improves probing:κ =53.0 atk =2 and 77.2 atk =8 forT =10. The fact that Isolate(k =1) still outperformsSwap Numbersconfirms that text-number co-occurrence alone provides strong signal, but the gap toOriginalalmost closed byIsolate(k =8) shows that cross-number interaction contributes on top of it. In all cases, Fourier spikes are fully preserved while probing degrades, reinforcing the two-tiered hierarchy between spectral and geometric convergence and they are driven by different mechanisms. Probing accuracy varies sharply across moduli. Across all conditions that achieve geometric convergence, the probing accuracy depends strongly on the modulus. Mod 2, 5, and 10 are consistently the easiest (κ =96.1, 63.5, 85.4 forOriginal), mod 4 achieves nontrivial probing (κ =34.9), while moduli that share no common factor with 10 (e.g., 3, 7, 9) remain near chance. This pattern is stable across all perturbations that preserve geometric convergence. In Section 5, we show that this structure can be traced to the tokenizer. 4.2 Structural Attribution to Architecture and Optimizer We now fix the pretraining data, and vary the architecture and optimizer. We train a 300M parameter Transformer model and two linear RNNs (Gated DeltaNet and Mamba-2 (Dao & Gu, 2024)) whose architectural and training details are shown in § B.1. We vary two different optimizers for training LLMs: AdamW and Muon. Figure 5 shows the results. Transformers and linear RNNs achieve geometric convergence; LSTMs do not. Under both Muon and AdamW,Transformer,Gated DeltaNet, andMamba-2all achieve strong probing, with the Transformer performing best. All three produce nearly identical Fourier spectra regardless of the optimizer, confirming that spectral convergence is architecture- 9Fu, Zhou, Belkin, Sharan, & Jia Muon AdamW 9-Digit Addition 99.7-10.259.4100.07.0-9.97.7-9.2100.0 100.0-24.253.6100.0-6.6-13.13.4-11.5100.0 0.00.10.20.30.40.5 Fourier Frequency Muon (seed 42) Muon (seed 123) AdamW (seed 42) AdamW (seed 123) 3-Digit Addition mod 2mod 3mod 4mod 5mod 6mod 7mod 8mod 9mod 10 5.9-2.3-0.5-10.4-4.4-2.1-5.4-6.6-4.5 -4.9-12.0-8.42.8-5.3-6.6-7.2-5.90.3 3.7-10.1-10.5-14.3-9.9-12.2-11.1-10.7-4.8 1.9-0.4-2.10.40.91.1-1.9-1.4-3.3 1.0 1.5 2.0 3.0 4.0 Norm. Magnitude 0 20 40 60 80 100 Cohen's (%) Fourier Magnitude Spectrum Cohen's of Linear Probe (%) Figure 6: Tokenization determines convergence in arithmetic. (Top) In 9-digit addition, both Muon and AdamW converge to the same spectral structure with sharp Fourier peaks and near-perfectκfor mod 2, 5, and 10, showing both spectral and geometric convergence. (Bottom) In 3-digit addition, where every operand and sum fits in a single token, the Fourier spectra vary across optimizers and random seeds, andκremains near chance for all moduli. Multi-token tokenization forces modular subproblems that produce convergent representations; single-token tokenization leaves the representation unconstrained. independent and optimizer-independent in language pretraining task. The 12-layerLSTM, trained with AdamW, develops even more prominent Fourier spikes but its probing accu- racy remains near chance across all moduli. A 4-layer LSTM shows no improvement nor degradation, indicating that the failure is architectural rather than a matter of capacity (see Figure 8). We note that as shown in Figure 2, the Fourier spectrum of the LSTM embeddings closely resembles that of the number token marginal distribution, suggesting that the LSTM embeddings capture little beyond unigram frequency statistics for numbers. Classical word embeddings fall in between:PPMIandword2vec, trained on the same 10B tokens, achieve moderate probing (κ =27.1 and 29.3 atT =10) with clear Fourier spikes yet weaker probing, illustrating the dissociation between spectral and geometric convergence. The effect of optimizer is architecture-dependent. Comparing the top and middle blocks of Figure 5 isolates the effect of the optimizer. Muon produces stronger probing for the Transformer (κ =85.4 vs 72.1 atT =10) and for Gated DeltaNet (77.8 vs 69.7), but Mamba-2 trained with AdamW actually outperforms Mamba-2 trained with Muon (80.1 vs 76.7). We find that the Transformer trained with Muon has the best probing performance. The optimizer’s effect on geometric convergence thus depends on the architecture, and we do not observe a universal advantage for either optimizer in language pretraining task. Spectral and geometric convergence co-emerge gradually. Figure 9 tracksΦ T and probe accuracy throughout Transformer pretraining forT =2, 5, 10. Both increase smoothly with no phase transition, unlike the grokking observed in modular arithmetic (Nanda et al., 2023). We will discuss more on model behavior when trained directly on arithmetic in Section 5. 5 Convergent Evolution in Training on Arithmetic Sections 3 and 4 studied models trained on general language, where Fourier features emerge from the statistics of number tokens in natural text. We now ask whether convergent evolution also occurs when models are trained directly on arithmetic, where the training signal is purely numerical, and the prior given by human language is absent. Experimental setup. We train 300M Transformers from random initialization on integer addition using the same architecture as in §3. Each example has the forma + b = c, with loss masked on prompt tokens up to=sign. We train for 3B tokens under both Muon and AdamW. In 9-digit addition, operands have 1-9 digits with stratified digit-count sampling. Each operand may span multiple number tokens. In 3-digit addition, we enumerate all pairs (a,b)witha,b∈ [0, 999]anda + b≤999. Every operand and sum is a single token. Training for 3 billion tokens amounts to roughly 1,000 epochs; we run two seeds per optimizer. We additionally train circular probes that project embeddings onto the unit circle (see Figure 13). Convergent Evolution10 9-digit addition: convergence across optimizers. Both Muon and AdamW converge to the same spectral structure (Figure 6, top), with sharp Fourier peaks at the expected harmonics and near-perfectκfor mod 2, 5, and 10. The two optimizers produce nearly identical Fourier spectra and probing accuracy, suggesting that the multi-token setting imposes constraints strong enough for both spectral and geometric convergence. Models determine the representation regardless of optimizer. 3-digit addition: no convergence without modular pressure. The single-token setting produces a different outcome (Figure 6, bottom). With 500K unique pairs repeated over 1,000 epochs, the learned representations vary across optimizer and random seed. Muon develops Fourier peaks at frequencies that do not align with modular periods, andκremains near chance. AdamW exhibits grokking under one seed: training accuracy reaches 100% early while test accuracy remains low until a phase transition after training achieves roughly 1.6B to 2B tokens (Figure 10); under another seed, generalization never occurs. Unlike Nanda et al. (2023), who train on mod-113 addition where modular structure is explicit, single-token addition imposes no modular constraint: the sequencesa + b = care identical whether interpreted as mod-1000 or mod-1111. Without this constraint, convergent evolution does not occur and the learned representation is seed-dependent. Why tokenization determines learned representation. In 9-digit addition after tokenization [a 2 ,a 1 ,a 0 ] + [b 2 ,b 1 ,b 0 ] = [c 2 ,c 1 ,c 0 ], each output token satisfiesc i = (a i + b i + γ i ) mod1000 whereγ i ∈ 0, 1is the carry. Each output position is therefore a mod-1000 classifica- tion problem, especially the least significant position with no carry, i.e.γ 0 =0. Since all our models use tied embeddings (Table 2), the output logits depend directly on the input embedding matrix, creating pressure for the embeddings to develop highΦ 1000 to distinguish residue classes at each output position. This in turn implies non-trivialΦ T for allT |1000=2 3 ·5 3 . In single-token addition, no modular constraint is imposed, soΦ T is unconstrained and depends on the optimizer and random seed. This reveals a second route through the two-tiered hierarchy: multi-token tokenization creates modular subproblems that produce both spectral and geometric convergence through carry propagation, while single-token tokenization guarantees neither. 6 Conclusion and Discussion We have shown that periodic number representations in language models exhibit a two- tiered convergence: Fourier spikes are universal, but linearly separable mod-Tclasses emerge only when data, architecture, and optimizer align. The central lesson is that visible structure in representations does not guarantee functional organization: the LSTM and even the raw token distribution develop more prominent Fourier spikes than the Transformer yet achieve chance-level probing. More generally, any representation-level diagnostic could mistake statistical artifacts of the training distribution for learned structure. Our controlled perturbation approach offers a complementary lens to instance-level attribution methods such as influence functions (Koh & Liang, 2017): rather than attributing predictions to individual training examples, we attribute learned representations to structural properties of the data distribution. Analogous periodic representations have been found for days of the week and months of the year (Engels et al., 2025; Karkada et al., 2026); whether the spectral-geometric dissociation extends to these and other cyclic concepts is a natural next step. More broadly, the spectral- geometric hierarchy introduced here provides a concrete framework for distinguishing superficial from functional feature learning. This distinction may prove important well be- yond numerical representations as we increasingly rely on representation-level diagnostics to understand large language models. Acknowledgments The authors acknowledge the Center for Advanced Research Computing (CARC) at the University of Southern California for providing computing resources that have contributed to the research results reported within this publication. We also acknowledge the use of the USC NLP cluster provided by the USC NLP Group. DF and RJ were also supported by a 11Fu, Zhou, Belkin, Sharan, & Jia gift from the USC-Capital One Center for Responsible AI and Decision Making in Finance (CREDIF). RJ was supported in part by the National Science Foundation under Grant No. IIS-2403436. VS was supported by National Science Foundation award CCF-2239265, an Amazon Research Award, a Google Research Scholar Award and an Okawa Foundation Research Grant. The work was done in part while some of the authors were visiting the Simons Institute for the Theory of Computing. This work used the Delta system at the National Center for Supercomputing Applications through allocation CIS250737 from the Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS) program, which is supported by National Science Foundation grants #2138259, #2138286, #2138307, #2137603, and #2138296. This work was supported in part by the NVIDIA Academic Grant Program. The GPU resources provided by NVIDIA were essential for training and analyzing the 300M-parameter models examined in this study. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not reflect the views of the funding agencies. References Zeyuan Allen-Zhu. Physics of language models: Part 4.1, architecture design and the magic of canon layers. arXiv preprint arXiv:2512.17351, 2025. Maximilian Beck, Korbinian Pöppel, Markus Spanring, Andreas Auer, Oleksandra Prud- nikova, Michael Kopp, Günter Klambauer, Johannes Brandstetter, and Sepp Hochreiter. xl- stm: Extended long short-term memory, 2024. URLhttps://arxiv.org/abs/2405.04517. Piotr Bojanowski, Edouard Grave, Armand Joulin, and Tomas Mikolov. Enriching word vec- tors with subword information. Transactions of the Association for Computational Linguistics, 5:135–146, 2017. doi: 10.1162/tacl_a_00051. URLhttps://aclanthology.org/Q17-1010/. Tri Dao and Albert Gu. Transformers are SSMs: Generalized models and efficient algorithms through structured state space duality. In Forty-first International Conference on Machine Learning, 2024. URL https://openreview.net/forum?id=ztn8FCR1td. DeepSeek. Deepseek-v3 technical report, 2025. URL https://arxiv.org/abs/2412.19437. Joshua Engels, Eric J Michaud, Isaac Liao, Wes Gurnee, and Max Tegmark. Not all language model features are one-dimensionally linear. In The Thirteenth International Conference on Learning Representations, 2025. URL https://openreview.net/forum?id=d63a4AM4hb. Pierre-Étienne H Fiquet and Eero P Simoncelli. A polar prediction model for learning to represent visual transformations. In Thirty-seventh Conference on Neural Information Processing Systems, 2023. URL https://openreview.net/forum?id=hyPUZX03Ks. Ronald Aylmer Fisher. The use of multiple measurements in taxonomic problems. Annals of Eugenics, 7(2):179–188, 1936. doi: https://doi.org/10.1111/j.1469-1809.1936.tb02137.x. URLhttps://onlinelibrary.wiley.com/doi/abs/10.1111/j.1469-1809.1936.tb02137. x. Amirata Ghorbani and James Zou. Data shapley: Equitable valuation of data for machine learning. In International Conference on Machine Learning, p. 2242–2251, 2019. Andrey Gromov. Grokking modular arithmetic. arXiv preprint arXiv:2301.02679, 2023. Albert Gu and Tri Dao. Mamba: Linear-time sequence modeling with selective state spaces, 2024. URL https://arxiv.org/abs/2312.00752. Keji He, Chenyang Si, Zhihe Lu, Yan Huang, Liang Wang, and Xinchao Wang. Frequency-enhanced data augmentation for vision-and-language navigation. In A. Oh, T. Naumann, A. Globerson, K. Saenko, M. Hardt, and S. Levine (eds.), Advances in Neural Information Processing Systems, volume 36, p. 4351–4364. Curran Asso- ciates, Inc., 2023.URLhttps://proceedings.neurips.c/paper_files/paper/2023/ file/0d9e08f247ca7fbbfd5e50b7f9cf357-Paper-Conference.pdf. Convergent Evolution12 Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural Computation, 9(8):1735–1780, 11 1997. ISSN 0899-7667. doi: 10.1162/neco.1997.9.8.1735. URLhttps: //doi.org/10.1162/neco.1997.9.8.1735. Ermo Hua, Che Jiang, Xingtai Lv, Kaiyan Zhang, Ning Ding, Youbang Sun, Biqing Qi, Yuchen Fan, Xuekai Zhu, and Bowen Zhou. Fourier position embedding: Enhancing attention’s periodic extension for length generalization. arXiv preprint arXiv:2412.17739, 2024. Minyoung Huh, Brian Cheung, Tongzhou Wang, and Phillip Isola. Position: The platonic representation hypothesis. In Ruslan Salakhutdinov, Zico Kolter, Katherine Heller, Adrian Weller, Nuria Oliver, Jonathan Scarlett, and Felix Berkenkamp (eds.), Proceedings of the 41st International Conference on Machine Learning, volume 235 of Proceedings of Machine Learning Research, p. 20617–20642. PMLR, 21–27 Jul 2024. URLhttps://proceedings.mlr.press/ v235/huh24a.html. Keller Jordan, Yuchen Jin, Vlado Boza, Jiacheng You, Franz Cesista, Laker Newhouse, and Jeremy Bernstein. Muon: An optimizer for hidden layers in neural networks, 2024. URL https://kellerjordan.github.io/posts/muon/. Subhash Kantamneni and Max Tegmark. Language models use trigonometry to do addition. arXiv preprint arXiv:2502.00873, 2025. Dhruva Karkada, Daniel J Korchinski, Andres Nava, Matthieu Wyart, and Yasaman Bahri. Symmetry in language statistics shapes the geometry of model representations. arXiv preprint arXiv:2602.15029, 2026. Kimi. Kimi linear: An expressive, efficient attention architecture, 2025. URLhttps://arxiv. org/abs/2510.26692. Pang Wei Koh and Percy Liang. Understanding black-box predictions via influence functions. In Doina Precup and Yee Whye Teh (eds.), Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine Learning Research, p. 1885–1894. PMLR, 06–11 Aug 2017. URL https://proceedings.mlr.press/v70/koh17a.html. Jannik Kossen, Jiatong Han, Muhammed Razzak, Lisa Schut, Shreshth Malik, and Yarin Gal. Semantic entropy probes: Robust and cheap hallucination detection in llms. arXiv preprint arXiv:2406.15927, 2024. Amit Arnold Levy and Mor Geva. Language models encode numbers using digit represen- tations in base 10. In Proceedings of the 2025 Conference of the Nations of the Americas Chapter of the Association for Computational Linguistics: Human Language Technologies (Volume 2: Short Papers), p. 385–395, 2025. Anton Lozhkov, Loubna Ben Allal, Leandro von Werra, and Thomas Wolf. Fineweb-edu: the finest collection of educational content, 2024. URLhttps://huggingface.co/datasets/ HuggingFaceFW/fineweb-edu. George R. McGhee. Convergent Evolution: Limited Forms Most Beautiful. The MIT Press, 2011. ISBN 9780262016421. URL http://w.jstor.org/stable/j.ctt5hhhwt. Meta. The llama 3 herd of models, 2024. URL https://arxiv.org/abs/2407.21783. Meta. The llama 4 herd: The beginning of a new era of natively multimodal ai innovation, 2025. URL https://ai.meta.com/blog/llama-4-multimodal-intelligence/. Neel Nanda, Lawrence Chan, Tom Lieberum, Jess Smith, and Jacob Steinhardt. Progress measures for grokking via mechanistic interpretability. In The Eleventh International Conference on Learning Representations, 2023. URLhttps://openreview.net/forum?id= 9XFSbDPmdW. Chris Olah, Nick Cammarata, Ludwig Schubert, Gabriel Goh, Michael Petrov, and Shan Carter. An overview of early vision in inceptionv1. Distill, 5(4):e00024–002, 2020. 13Fu, Zhou, Belkin, Sharan, & Jia Bruno A Olshausen and David J Field. Sparse coding with an overcomplete basis set: A strategy employed by v1? Vision research, 37(23):3311–3325, 1997. OpenAI. gpt-oss-120b & gpt-oss-20b model card, 2025. URLhttps://arxiv.org/abs/2508. 10925. Hadas Orgad, Michael Toker, Zorik Gekhman, Roi Reichart, Idan Szpektor, Hadas Kotek, and Yonatan Belinkov. LLMs know more than they show: On the intrinsic representation of LLM hallucinations. In The Thirteenth International Conference on Learning Representations, 2025. URL https://openreview.net/forum?id=KRnsX5Em3W. Kiho Park, Yo Joong Choe, and Victor Veitch. The linear representation hypothesis and the geometry of large language models. In Causal Representation Learning Workshop at NeurIPS 2023, 2023. URL https://openreview.net/forum?id=T0PoOJg8cK. Jeffrey Pennington, Richard Socher, and Christopher Manning. GloVe: Global vectors for word representation. In Alessandro Moschitti, Bo Pang, and Walter Daelemans (eds.), Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), p. 1532–1543, Doha, Qatar, October 2014. Association for Computational Linguistics. doi: 10.3115/v1/D14-1162. URL https://aclanthology.org/D14-1162/. Alec Radford, Jeff Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. Language models are unsupervised multitask learners. 2019. Adityanarayanan Radhakrishnan, Mikhail Belkin, and Dmitriy Drusvyatskiy. Linear recursive feature machines provably recover low-rank matrices, 2024. URLhttps: //arxiv.org/abs/2401.04553. Matthew Tancik, Pratul Srinivasan, Ben Mildenhall, Sara Fridovich-Keil, Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan Barron, and Ren Ng. Fourier features let networks learn high frequency functions in low dimensional domains. Advances in neural information processing systems, 33:7537–7547, 2020. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Ł ukasz Kaiser, and Illia Polosukhin. Attention is all you need. In I. Guyon, U. Von Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Gar- nett (eds.), Advances in Neural Information Processing Systems, volume 30. Curran As- sociates, Inc., 2017. URLhttps://proceedings.neurips.c/paper_files/paper/2017/ file/3f5e243547dee91fbd053c1c4a845a-Paper.pdf. Songlin Yang, Jan Kautz, and Ali Hatamizadeh. Gated delta networks: Improving mamba2 with delta rule. In The Thirteenth International Conference on Learning Representations, 2025. URL https://openreview.net/forum?id=r8H7xhYPwz. Ziqian Zhong, Ziming Liu, Max Tegmark, and Jacob Andreas. The clock and the pizza: Two stories in mechanistic explanation of neural networks. Advances in neural information processing systems, 36:27223–27250, 2023. Tianyi Zhou, Deqing Fu, Vatsal Sharan, and Robin Jia. Pre-trained large language models use fourier features to compute addition. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024. URLhttps://openreview.net/forum?id= i4MutM2TZb. Tianyi Zhou, Deqing Fu, Mahdi Soltanolkotabi, Robin Jia, and Vatsal Sharan. Fone: Precise single-token number embeddings via fourier features. arXiv preprint arXiv:2502.09741, 2025. Jingwei Zuo, Maksim Velikanov, Dhia Eddine Rhaiem, Ilyas Chahed, Younes Belkada, Guillaume Kunsch, and Hakim Hacid. Falcon mamba: The first competitive attention-free 7b language model, 2024. URL https://arxiv.org/abs/2410.05355. Convergent Evolution14 Appendix A Proof of Theorem 115 A.1 Lemmas Fourier-variance identity . . . . . . . . . . . . . . . . . . . . . . . .15 A.2 Necessary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17 A.3 Insufficiency condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17 A.4 Lower and Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19 B Experiments20 B.1 Model and Training Details . . . . . . . . . . . . . . . . . . . . . . . . . . . .20 B.2 LSTM ablation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20 B.3 Data Perturbation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20 B.4 Training Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22 B.5 Modular Probe Results with MLP and RFM Probes . . . . . . . . . . . . . .23 15Fu, Zhou, Belkin, Sharan, & Jia A Proof of Theorem 1 We first restate the theorem here. Theorem 1. Given embeddingse(n) N−1 n=0 and residue classesC r T−1 r=0 defined above, let the class meansμ r , grand meanμ, between-class scatter matrix S B , and within-class scatter matrix S W be μ r = 1 |C r | ∑ n∈C r e(n),μ = 1 N N−1 ∑ n=0 e(n), S B = 1 T T−1 ∑ r=0 (μ r −μ)(μ r −μ) ⊤ ,S W = 1 N T−1 ∑ r=0 ∑ n∈C r (e(n)−μ r )(e(n)−μ r ) ⊤ . LetΦ T = ∑ T−1 ℓ=1 ∥F ℓ/T ∥ 2 be the total power at harmonics of periodT, andH T = 0, 1/T, 2/T, . . . , (T−1)/T be the set of harmonic frequencies. (i) IfΦ T = 0, then S B = 0 and no linear probe can classify n mod T above chance. (i)For anyT ≥2,C >0, andε >0, there existNdivisible byTand embeddingse(n) satisfyingΦ T > Cyet noT-class linear classifier achieves accuracy above 1/T + ε, i.e., no more than ε above random guessing. We first start with two lemmas and then prove them in parts. A.1 Lemmas Fourier-variance identity We first establish a lemma connecting the class meansμ r to the Fourier coefficients F ν . Lemma 3. For each ℓ = 0, . . . , T− 1, define ˆμ[ℓ] = 1 √ T T−1 ∑ r=0 μ r e −2πiℓr/T . Then ˆμ[ℓ] = √ T/N· F ℓ/T . Proof.Substitutingμ r = 1 |C r | ∑ n∈C r e(n) = T N ∑ n∈C r e(n)(since|C r | = N/TwhenT|N.) and pulling the constant into the outer sum: ˆμ[ℓ] = √ T N T−1 ∑ r=0 ∑ n∈C r e(n) e −2πiℓr/T .(1) Everyn ∈ 0,. . .,N−1belongs to exactly one classC r withr = n mod T, so we can writen = mT + rfor some non-negative integerm. Thene −2πiℓr/T = e −2πiℓn/T , since the additionalmℓfull periods contribute an integer multiple of 2πto the exponent. Re-indexing the double sum as a single sum over n: ˆμ[ℓ] = √ T N N−1 ∑ n=0 e(n) e −2πiℓn/T .(2) BecauseT | N, the ratioℓ/Tis one of theNFourier frequencies, so comparing with F ℓ/T = 1 √ N ∑ n e(n) e −2πi(ℓ/T)n gives ˆμ[ℓ] = √ T N · √ N· F ℓ/T = r T N · F ℓ/T . Next, we state our second lemma on the identities for Tr(S B ) and Tr(S W ). Lemma 4. Let the between-class and within-class variancesS B andS W be defined as in Theorem 1, they satisfy Tr(S B ) = Φ T N ,Tr(S W ) = 1 N ∑ ν /∈H T ∥F ν ∥ 2 . Convergent Evolution16 Proof. Define the T× d matrix M whose rows are the class means: M = —μ ⊤ 0 — —μ ⊤ 1 — . . . —μ ⊤ T−1 — ∈ R T×d , and letUbe theT× Tmatrix with entriesU ℓr = 1 √ T e −2πiℓr/T . Define ˆ M = U M ∈ C T×d , whose rows are ˆμ[ℓ] ⊤ : ˆ M = — ˆμ[0] ⊤ — — ˆμ[1] ⊤ — . . . — ˆμ[T− 1] ⊤ — ∈ C T×d . Since U is the DFT matrix in T dimensions, U is unitary. Therefore, T−1 ∑ ℓ=0 ∥ ˆμ[ℓ]∥ 2 = ∥ ˆ M∥ 2 F = ∥U M∥ 2 F = ∥M∥ 2 F = T−1 ∑ r=0 ∥μ r ∥ 2 . This is simply the fact that a unitary change of basis preserves the sum of squared norms. Forℓ =0:ˆμ[0] = √ T/N· F 0 . SinceF 0 = 1 √ N ∑ n e(n) = √ Nμ, we haveˆμ[0] = √ Tμ, and therefore 1 T ∥ ˆμ[0]∥ 2 =∥μ∥ 2 . The between-class variance is: Tr(S B ) = 1 T ∑ r ∥μ r −μ∥ 2 = 1 T ∑ r ∥μ r ∥ 2 − 2μ ⊤ 1 T ∑ r μ r ! +∥μ∥ 2 ! (3) = 1 T ∑ r ∥μ r ∥ 2 −∥μ∥ 2 ! = 1 T T−1 ∑ ℓ=1 ∥ ˆμ[ℓ]∥ 2 ! (4) = 1 T T−1 ∑ ℓ=1 T N ∥F ℓ/T ∥ 2 ! = 1 N T−1 ∑ ℓ=1 ∥F ℓ/T ∥ 2 = Φ T N ,(5) where the second line uses 1 T ∑ r μ r =μ , which holds because the classesC r partition 0, . . . , N− 1 into equal-sized groups. ForTr(S W ), the total variance decomposes asV = Tr(S B ) + Tr(S W ). Define theN× dmatrix Ewhose rows aree(0) ⊤ ,. . .,e(N−1) ⊤ , and letU N be theN× NDFT matrix with entries (U N ) kn = 1 √ N e −2πikn/N . Then ˆ E = U N E has rows F ⊤ ν , and since U N is unitary, N−1 ∑ n=0 ∥e(n)∥ 2 =∥E∥ 2 F =∥ ˆ E∥ 2 F = ∑ ν ∥F ν ∥ 2 . Since F 0 = √ Nμ, we have∥F 0 ∥ 2 = N∥μ∥ 2 , and therefore V = 1 N N−1 ∑ n=0 ∥e(n)−μ∥ 2 = 1 N ∑ n ∥e(n)∥ 2 − N∥μ∥ 2 ! = 1 N ∑ ν̸=0 ∥F ν ∥ 2 . Therefore: Tr(S W ) = V− Tr(S B ) = 1 N ∑ ν̸=0 ∥F ν ∥ 2 − 1 N T−1 ∑ ℓ=1 ∥F ℓ/T ∥ 2 = 1 N ∑ ν /∈H T ∥F ν ∥ 2 , whereH T = 0, 1/T, 2/T,. . .,(T−1)/Tcollects the zero-frequency and harmonic fre- quencies. 17Fu, Zhou, Belkin, Sharan, & Jia A.2 Necessary condition Proof of Part (i) of Theorem 1.IfΦ T =0, thenTr(S B ) =0 by Part (i). SinceS B is positive semidefinite,Tr(S B ) =0 impliesS B =0, which requiresμ r =μfor allr =0,. . .,T−1. When all class means coincide, the class-conditional distributions ofe(n)share the same first moment, so no linear probe (or any probe relying on mean separation) can distinguish the T classes above the chance rate 1/T. A.3 Insufficiency condition Proof of Part (i).We construct embeddings whose residue classes interleave periodically on the real line, then show that the geometry of linear decision boundaries prevents any T-class linear classifier from exceeding chance-level accuracy by more thanε. FixT ≥2, C >0, andε >0. SetK = ⌈(T−1)/(Tε)⌉andN = KT. Every indexn ∈ 0,. . .,N−1 has a unique decompositionn = mT + rwith residuer ∈ 0,. . .,T−1and block index m∈0, . . . , K−1. Define e(n) = A· (n mod T) | z residue r + B· j n T k |z block index m , withA,B >0 to be chosen. Intuitively,Acontrolsn mod T: enlargingApulls numbers sharing the same residue classC r together. The parameterBcontrols⌊n/T⌋, which measures how many full copies ofTfit belown: increasingBclusters numbers of similar magnitude together, regardless of their residue. We now show howAdetermines the Fourier power while B controls the interleaving that defeats linear classifiers. Fourier power. Within classC r =r,r+T,. . .,r+(K−1)T, the termAris constant and only the block term varies, so the class mean is μ r = 1 K K−1 ∑ m=0 (Ar + Bm) = Ar + B(K−1) 2 . The grand mean isμ = A(T−1)/2+ B(K−1)/2, givingμ r − μ = A r− (T−1)/2 . Note that the block termB⌊n/T⌋contributes nothing toS B : its class meanB(K−1)/2 is identical across all classes and cancels in μ r − μ. Therefore S B = 1 T T−1 ∑ r=0 (μ r − μ) 2 = A 2 T T−1 ∑ r=0 r− T−1 2 2 = A 2 (T 2 − 1) 12 , where the last equality uses the standard identity for the second central moment ofT consecutive integers. By Lemma 4,Φ T = N· S B = A 2 KT(T 2 −1)/12. SettingA = p 12C/(KT(T 2 − 1)) givesΦ T = C. Periodic interleaving. ChooseB > (T−1)Aso that consecutive blocks separate on the real line. The largest value in blockmise((T−1) + mT) = (T−1)A + Bm, and the smallest value in blockm+1 ise(0+ (m+1)T) = B(m+1). SinceB > (T−1)A, we haveB(m+1) > (T−1)A + Bm, so blockmand blockm+1 occupy disjoint intervals on the real line. Within each block, theTpoints are sorted by residue:e(mT) = Bm < e(mT+1) = A + Bm < · < e(mT+T−1) = (T−1)A + Bm, sincee(n)is increasing inn mod Tfor fixed⌊n/T⌋. Combining both observations, sorting allNembeddings by value yields the natural ordering: the i-th smallest embedding (0-indexed) is x (i) = e(i) = A· (i mod T) + B·⌊i/T⌋,i = 0, 1, . . . , N−1, whose class label isi mod T. The sorted class label sequence is therefore(0, 1,. . .,T−1) repeated K times: 0, 1, . . . , T−1 |z block 0 , 0, 1, . . . , T−1 |z block 1 , . . . , 0, 1, . . . , T−1 |z block K−1 . Any contiguous subsequence of lengthTor longer contains at least one complete cycle and therefore includes at least one point from every class. Convergent Evolution18 02040 e(n) A=5, B=0.03, T =206250, best linear acc=100.0% N=1000 Points. Period T=10 e(n) = A (n mod T) + Bn/T 0.00.10.20.30.40.5 frequency 0 25000 50000 75000 100000 125000 F 2 T=10 0100020003000 e(n) A=5, B=31, T =206250, best linear acc=10.9% 0.00.10.20.30.40.5 frequency 0 25000 50000 75000 100000 125000 F 2 T=10 mod 10 0123456789 Figure 7: Emergence of Fourier structure in a constructed embeddinge(n) = A(n mod T) + B⌊n/T⌋withT =10,A =5,N =1000. Each dot is a numbernplaced at its scalar embedding value on the horizontal axis and colored by its classn mod T; sincee(n)is one- dimensional, the vertical coordinate carries no information and is jittered purely for visibility so that overlapping points of different classes remain distinguishable. Top (B =0.03): within-block drift is small relative to between-class spacing, so the sorted line separates into 10 clean color bands: a linear classifier one(n)recovers the mod-10 residue at near-perfect accuracy, and the DFT concentrates energy at the fundamental frequencyν =1/T =0.1. Bottom (B =21): the between-block driftBdominates, interleaving classes along the line so that every mod-10 lane spans the full range; best linear accuracy collapses to≈1/T + ε withε =0.9%. The Fourier spectrum keeps the same peak atν =0.1 (sameΦ T ), showing that periodic energy at the fundamental is a property of the construction itself, independent of whether class identity is linearly decodable. Classification bound. AT-class linear classifier assigns each instancex ∈ Rto argmax c∈0,...,T−1 (w c x + b c )for parametersw c ,b c ∈ R. This is the hypothesis class of multi- class logistic regression. Note that this hypothesis class is expressive enough to perfectly clas- sifyTcontiguous groups: whenBis small, theTclasses cluster near 0,A, 2A,. . .,(T−1)A, and settingw c = cwith biases cascaded viab 0 =0,b c+1 = b c − (c A + A/2)places the T−1 decision boundaries between consecutive clusters, achieving 100% accuracy. Next we exploit the limitation that linear classifiers partitionRinto at mostTcontiguous intervals: any two of theTlinesx 7→ w c x + b c intersect in at most one point, yielding at mostT−1 breakpoints and hence at mostTintervals, each assigned to a single class. Now consider any such interval. Each complete cycle(0, 1,. . .,T−1)contained in the interval contributes exactly one point from every class, so the assigned label matches exactly a 1/Tfraction of those points. Only at interval boundaries can a partial cycle contribute additional correct predictions: at most one per boundary, for a total of at mostT−1 extra correct points across all T− 1 boundaries. It follows that accuracy ≤ N/T + T− 1 N = 1 T + T− 1 KT . The choice K =⌈(T−1)/(Tε)⌉ guarantees (T−1)/(KT)≤ ε, completing the proof. Figures 3 and 7 illustrates this construction forT =5,N =25, and forT =10,N =1000, where in both cases ε achieves its minimum at T−1 N , at 16% and 0.9% respectively. The token frequency distribution in Figure 2 provides an empirical counterexample: LSTM exhibits clear Fourier spikes atT =2, 5, 10 yet achieves chance-level probing for all moduli, showing that the construction above captures a phenomenon that occurs in practice. 19Fu, Zhou, Belkin, Sharan, & Jia A.4 Lower and Upper Bounds Lemma 5. Assumed ≥ T−1 andS W is invertible (which holds whend ≤ N− T, or after projecting to a subspace of dimension at mostN− T). In Fisher’s Linear Discriminant Analysis (Fisher, 1936), the optimal linear probe maximizes the ratio of between-class to within-class variance along a projection direction. The separability of this optimal discriminant is characterized by λ max (S −1 W S B ), the largest generalized eigenvalue of the scatter matrix pair, which satisfies 1 (T− 1)· cond(S W ) · Φ T N· λ min (S W ) ≤ λ max (S −1 W S B ) ≤ Φ T N· λ min (S W ) . The ratio between the upper and lower bounds is(T−1)· cond(S W ), wherecond(S W ) = λ max (S W )/λ min (S W ). Proof.The optimal linear discriminant forT-class classification projects the data onto the direction maximizing the generalized Rayleigh quotient: λ max (S −1 W S B ) = max v̸=0 v ⊤ S B v v ⊤ S W v . Upper bound. For anyv̸=0, the numerator satisfiesv ⊤ S B v≤ λ max (S B )∥v∥ 2 ≤ Tr(S B )∥v∥ 2 , where the second inequality holds becauseS B is positive semidefinite andλ max (S B ) ≤ Tr(S B ). The denominator satisfies v ⊤ S W v≥ λ min (S W )∥v∥ 2 . Therefore: λ max (S −1 W S B )≤ Tr(S B ) λ min (S W ) = Φ T N· λ min (S W ) . Lower bound. The matrixS B has rank at mostmin(d,T−1); sinced≥ T−1 by assumption, this simplifies toT−1. This holds because theTvectorsμ r −μ T−1 r=0 satisfy ∑ r (μ r −μ) = 0 and therefore span a subspace of dimension at mostT−1. It follows thatλ max (S B ) ≥ Tr(S B )/(T−1) =Φ T /(N(T−1)). Letv ∗ be the unit eigenvector ofS B corresponding to λ max (S B ). Then: λ max (S −1 W S B )≥ v ∗ ⊤ S B v ∗ v ∗ ⊤ S W v ∗ = λ max (S B ) v ∗ ⊤ S W v ∗ ≥ λ max (S B ) λ max (S W ) ≥ Φ T N· (T− 1)· λ max (S W ) . Gap between the bounds. The ratio of the upper to lower bound is: Φ T /(N· λ min (S W )) Φ T /(N· (T− 1)· λ max (S W )) = (T− 1)· λ max (S W ) λ min (S W ) = (T− 1)· cond(S W ). The Fourier power spectrum fully determinesΦ T via Part (i), butcond(S W )depends on the directional structure of within-class variation, which the power spectrum∥F ν ∥ 2 does not capture. Specifically,∥F ν ∥ 2 aggregates power across alldembedding dimensions at frequencyν, discarding any information about which dimensions carry the periodic signal versus which carry within-class noise. As a result, two embeddings with identical power spectra (and hence identicalΦ T ) but different within-class covariance structures can yield λ max (S −1 W S B )values differing by up to a factor of(T−1)· cond(S W ), producing vastly different probe accuracies. Convergent Evolution20 B Experiments B.1 Model and Training Details Table 2: Model architectures and training configurations. TransformerGated DeltaNetMamba-2LSTM Architecture Total parameters320M318M316M232M → Embedding131M131M131M131M → Non-embedding189M186M185M101M Tied embeddings✓ Layers12122812 Hidden dim d1024102410241024 Heads16 (8 KV)1632— Head dim646464— MLP intermediate40964096— MLP activationSwiGLUSwiGLU— Positional encodingRoPENoneNoneNone NormalizationRMSNormRMSNormRMSNorm— Sequence mechanismGQALinear Attn + Gate SSM (d state =128) Recurrence Short convolution—size 4size 4— SSM expand factor—1.5× (value)2×— Dropout—0.1 Muon optimizer 2D weight LR3× 10 −3 , momentum = 0.95 Embed/norm/bias LR3× 10 −4 (AdamW, β 2 =0.95) Weight decay0.01 (2D weights only) AdamW optimizer Learning rate3× 10 −4 (β 1 , β 2 )(0.9, 0.95) Weight decay0.01 Shared training Context length1024 Batch size512 sequences (∼524K tokens/step) LR scheduleCosine decay, 500 warmup steps, min = 10% of peak Training tokens∼9.4B (1 epoch of FineWeb-Edu 10BT) Precisionbfloat16 mixed precision B.2 LSTM ablation We ablate the number of layers of LSTMs and find reducing the number of layers to 4 instead of 12 does not change the phenomenon: both will learn fourier spikes but no probing performance. Both have huge condition number on S W yet hugeΦ T as well. B.3 Data Perturbation Details In this section, we describe in detail how we perturb data for each configurations in Table 1. Isolate-k configuration. We design an isolate configuration to test whether Fourier features and mod-Tprobes can emerge when reducing the interactive between number tokens, even indirectly through intermediate text tokens across multiple layers. The key idea is to enforce a block-diagonal causal attention mask that partitions each sequence into segments, where each segment contains at mostktoken. Concretely, given a tokenized sequence, we locate all positions containing number tokens and place segment boundaries at the midpoint of the text span between each consecutive pair ofknumber tokens. This way, every number token still sees some surrounding context on both sides, but can never interact with any other number token, if they are not in the same segment. Within each segment, standard causal attention applies: positioniattends to positionjonly ifj ≤ iand both positions belong to the same segment. We also reset RoPE position IDs to zero at segment boundaries to avoid leaking positional information across segments, and mask the loss at boundaries so 21Fu, Zhou, Belkin, Sharan, & Jia 0.00.10.20.30.40.5 0 4 8 T=2 T=5 T=10 LSTM Fourier Magnitude Spectrum 2345678910 0 50 100 9 0 0 3 0 -1 0 -1 2 Cohen's of Linear Probe (%) 123456789 0 15 30 max = 0.290 Fourier 10 = 337.4 cond(S W ) = 8569.0 Eigenvalues of S 1 W S B (T=10) 0.00.10.20.30.40.5 Fourier Frequency 0 4 8 T=2 T=5 T=10 LSTM (4-layer) 2345678910 Modulus T 0 50 100 5 1 2 4 4 00 0 2 123456789 Eigenvalue Rank i 0 15 30 max = 0.698 Fourier 10 = 736.3 cond(S W ) = 20297.1 Figure 8: Ablation on the depth of LSTM models. We find that reducing the number of layers to 4 (green) instead of 12 (blue) does not change the phenomenon: both will learn fourier spikes but no probing performance. the model is not trained to predict across segment breaks. Importantly, we do not modify the training data at all. The tokenized sequences are identical to those used in the standard configuration; only the attention mask differs. Context lengthℓ. To test the role of broad context in Fourier feature formation and geometric emergence, we train models whose effective context is limited to a fixed window ofℓ consecutive tokens. Each 1024-token sequence is reshaped into⌊1024/ℓ⌋independent subsequences of lengthℓ, each processed with its own standard causal attention mask. Tokens in one window cannot attend to tokens in any other window. This is equivalent to training on a corpus of short documents of lengthℓ. We experiment withℓ∈2, 4, 8, 64: a window ofℓ =2 reduces the model to learning bigram statistics (each token sees only the immediately preceding token), whileℓ =4 and 8 permit short-range dependencies but still prevents any long-range co-occurrence patterns.ℓ =64 permits longer range dependencies but is still much shorter thanOriginal’s context length of 1024. As with the isolate configuration, the underlying token sequence is unchanged; only the effective context window differs. Swap numbers. This configuration dissociates number tokens from their original textual context while preserving natural numbern-gram statistics. For each training sequence, we keep every text token in its exact position and replace the entire subsequence of number tokens with a contiguous, order-preserving slice of number tokens drawn from other documents in the corpus. Concretely, we pre-extract all number tokens from the full training set into a single in-memory stream, and for each sequence we substitute a randomly chosen contiguous segment from this pool. The replaced number tokens therefore retain realistic sequential patterns but lose their association with the surrounding text. This tests whether the text-number co-occurrence structure, rather than the number token statistics alone, drives Fourier feature and modular probe emergence. Unigram replace. In the unigram configuration, every number token in the training data is independently replaced by a random draw from the corpus-wide marginal (unigram) distribution over number tokens. This destroys all sequential structure among numbers, such as their relationship to surrounding text and their co-occurrence with other numbers. But this type of perturbation exactly preserves the marginal frequency of each number token. If a number token n appears with probability p n in the original corpus, it appears with the same probability in the perturbed data. By comparing to the original and swap-numbers configurations, the unigram ablation isolates whether co-occurrence statistics beyond simple token frequency are necessary for Fourier features and modular arithmetic to emerge. Convergent Evolution22 B.4 Training Dynamics Figure 9 shows that during language pretraining, both Fourier powerΦ T and linear probe accuracy increase smoothly from the start of training forT =2, 5, 10, with no sudden phase transition. This contrasts with grokking in modular arithmetic (Nanda et al., 2023), where structured representations appear abruptly after prolonged memorization. In language pretraining, the model continuously encounters diverse co-occurrence statistics rather than memorizing a fixed set of examples, so both tiers of convergence emerge gradually. In contrast, Figure 10 shows training dynamics for addition. 0246810 Training Tokens (B) 0.0 1.5 3.0 4.5 6.0 Fourier Power 2 T = 2 Fourier Power 2 Linear Probe Acc 0246810 Training Tokens (B) 0 4 8 12 16 Fourier Power 5 T = 5 Fourier Power 5 Linear Probe Acc 0246810 Training Tokens (B) 0 10 20 30 40 Fourier Power 10 T = 10 Fourier Power 10 Linear Probe Acc 40 55 70 85 100 Linear Probe Accuracy (%) 15 30 45 60 75 Linear Probe Accuracy (%) 0 25 50 75 100 Linear Probe Accuracy (%) Figure 9: Spectral and geometric convergence co-emerge gradually during pretraining. Fourier powerΦ T (left axis) and linear probe accuracy (right axis) for a 300M Transformer trained with Muon, shown forT =2, 5, 10. Both metrics increase smoothly throughout training with no phase transition, in contrast to the sudden emergence observed in grokking on modular arithmetic tasks in Nanda et al. (2023). 0.00.51.01.52.02.53.0 Training Tokens (B) 0 20 40 60 80 100 Accuracy (%) 9-Digit Addition: Muon vs AdamW (seed 42) Muon AdamW Test Train 0.00.51.01.52.02.53.0 Training Tokens (B) 0 20 40 60 80 100 Accuracy (%) 3-Digit Addition: Muon vs AdamW (seed 42) Muon AdamW Test Train 0.00.51.01.52.02.53.0 Training Tokens (B) 0 20 40 60 80 100 Accuracy (%) 3-Digit Addition: Muon vs AdamW (seed 123) Muon AdamW Test Train Figure 10: Training dynamics for Transformers trained on addition (two seeds). (Left) In 9-digit addition, both Muon and AdamW converge smoothly to near-perfect train and test accuracy, with no grokking phase. (Right) In 3-digit addition, under seed 42, training accuracy reaches 100% for both optimizers, but generalization is optimizer- and seed- dependent. AdamW exhibits a quick grokking under seed 42 (test accuracy jumps around 1.5–2B tokens) but not under seed 123 (where training accuracy can’t reach 100% and test accuracy remains random). This confirms that single-token addition imposes no consistent pressure toward structured representations. 23Fu, Zhou, Belkin, Sharan, & Jia B.5 Modular Probe Results with MLP and RFM Probes In this section, we present the modular probes similar to Figures 4 to 6 but with RFM probes and 2-layer MLP probes with hidden layer of size 64. Original Transformer LM (Muon) 97.2-7.041.265.411.0-7.713.3-3.990.0 Context Length 64 Context Length 8 Context Length 4 Context Length 2 Reducing Context Length 94.4-8.134.447.511.9-8.014.4-4.879.1 88.8-7.130.834.711.8-7.59.9-2.855.9 86.9-9.129.833.18.8-6.412.2-3.053.7 70.8-7.228.229.97.7-4.811.5-0.339.5 Isolate (k=8) Isolate (k=2) Isolate (k=1) Reducing Cross-Number Interaction 94.7-9.035.657.610.3-8.713.6-6.082.4 91.7-6.032.937.211.2-6.314.8-3.659.4 86.91.332.035.315.8-4.812.3-0.848.5 0.00.10.20.30.40.5 Fourier Frequency Swap Numbers Unigram Replace Number Replacement mod 2mod 3mod 4mod 5mod 6mod 7mod 8mod 9mod 10 73.0-12.422.524.65.8-5.79.9-4.533.4 -1.5-0.8-4.50.8-2.01.50.20.30.6 1.0 1.5 2.0 3.0 4.0 Norm. Magnitude 0 20 40 60 80 100 Cohen's (%) Fourier Magnitude Spectrum Cohen's of RFM (%) Transformer Gated DeltaNet Mamba-2 Language Models (Muon) 97.2-7.041.265.411.0-7.713.3-3.990.0 96.1-7.840.155.310.4-7.317.3-3.685.0 95.1-5.034.151.110.5-8.315.9-3.282.4 Transformer Gated DeltaNet Mamba-2 LSTM Language Models (AdamW) 91.1-4.535.848.214.8-8.214.3-4.280.9 92.4-6.732.247.813.1-9.112.5-4.377.9 95.3-7.135.855.510.6-8.013.3-2.885.8 2.0-2.01.12.0-1.3-0.50.8-0.12.7 0.00.10.20.30.40.5 Fourier Frequency PPMI word2vec Classical Word Embeddings mod 2mod 3mod 4mod 5mod 6mod 7mod 8mod 9mod 10 50.5-9.318.119.04.7-5.87.2-4.029.3 48.0-4.522.525.88.0-1.68.5-1.432.4 1.0 1.5 2.0 3.0 4.0 Norm. Magnitude 0 20 40 60 80 100 Cohen's (%) Fourier Magnitude Spectrum Cohen's of RFM (%) Muon AdamW 9-Digit Addition 99.6-12.560.1100.07.7-9.910.1-8.1100.0 100.0-23.260.0100.0-5.1-13.89.1-11.2100.0 0.00.10.20.30.40.5 Fourier Frequency Muon (seed 42) Muon (seed 123) AdamW (seed 42) AdamW (seed 123) 3-Digit Addition mod 2mod 3mod 4mod 5mod 6mod 7mod 8mod 9mod 10 22.1-3.7-0.8-10.8-4.4-0.8-4.9-6.1-3.4 -10.0-14.0-8.93.6-4.0-6.0-6.6-5.71.1 3.2-15.7-11.6-14.1-10.4-11.5-10.1-10.7-3.5 -1.2-0.8-2.71.00.40.1-2.2-1.9-2.7 1.0 1.5 2.0 3.0 4.0 Norm. Magnitude 0 20 40 60 80 100 Cohen's (%) Fourier Magnitude Spectrum Cohen's of RFM (%) Figure 11: Structural Attribution Results with RFM probes. Convergent Evolution24 Original Transformer LM (Muon) 95.0-10.331.962.57.6-9.67.2-5.183.8 Context Length 64 Context Length 8 Context Length 4 Context Length 2 Reducing Context Length 90.9-11.228.144.28.0-7.78.6-5.669.7 86.1-4.824.534.19.0-7.57.3-4.349.7 86.5-9.626.131.37.5-6.710.6-4.544.9 67.7-7.324.527.77.0-5.511.0-1.836.9 Isolate (k=8) Isolate (k=2) Isolate (k=1) Reducing Cross-Number Interaction 92.3-10.927.957.29.0-9.78.8-5.475.6 89.5-7.628.335.09.3-7.010.0-5.151.0 84.7-0.826.932.912.9-4.810.5-2.643.6 0.00.10.20.30.40.5 Fourier Frequency Swap Numbers Unigram Replace Number Replacement mod 2mod 3mod 4mod 5mod 6mod 7mod 8mod 9mod 10 65.1-13.318.722.33.8-8.55.9-5.529.8 -4.50.7-3.23.0-2.41.1-1.6-0.20.1 1.0 1.5 2.0 3.0 4.0 Norm. Magnitude 0 20 40 60 80 100 Cohen's (%) Fourier Magnitude Spectrum Cohen's of MLP (%) Transformer Gated DeltaNet Mamba-2 Language Models (Muon) 94.8-10.932.062.67.6-9.87.8-5.583.9 93.6-9.933.453.27.5-8.611.0-5.876.1 92.0-8.026.851.05.6-9.810.0-3.874.5 Transformer Gated DeltaNet Mamba-2 LSTM Language Models (AdamW) 86.8-7.928.446.611.4-8.79.2-5.468.5 88.4-9.625.247.75.9-9.07.8-4.767.5 92.3-11.129.054.16.2-8.57.4-4.078.4 4.3-3.31.62.2-0.4-2.01.2-0.22.5 0.00.10.20.30.40.5 Fourier Frequency PPMI word2vec Classical Word Embeddings mod 2mod 3mod 4mod 5mod 6mod 7mod 8mod 9mod 10 45.9-11.311.815.20.8-8.12.8-6.622.2 44.7-6.119.225.34.5-3.84.7-3.732.3 1.0 1.5 2.0 3.0 4.0 Norm. Magnitude 0 20 40 60 80 100 Cohen's (%) Fourier Magnitude Spectrum Cohen's of MLP (%) Muon AdamW 9-Digit Addition 99.7-17.055.0100.04.8-10.14.2-9.3100.0 100.0-28.369.1100.0-2.1-13.711.3-11.5100.0 0.00.10.20.30.40.5 Fourier Frequency Muon (seed 42) Muon (seed 123) AdamW (seed 42) AdamW (seed 123) 3-Digit Addition mod 2mod 3mod 4mod 5mod 6mod 7mod 8mod 9mod 10 14.1-5.5-0.4-11.3-4.50.6-4.5-6.0-4.2 -10.4-11.8-9.918.0-6.4-7.2-6.6-6.32.9 9.2-15.2-13.0-14.1-6.4-12.3-11.0-10.2-4.9 -2.90.9-3.0-0.30.40.8-2.0-2.3-3.1 1.0 1.5 2.0 3.0 4.0 Norm. Magnitude 0 20 40 60 80 100 Cohen's (%) Fourier Magnitude Spectrum Cohen's of MLP (%) Figure 12: Structural Attribution Results with 2-layer MLP probes. 25Fu, Zhou, Belkin, Sharan, & Jia 0 1 23 4 5 6 78 9 9-digit addition (84% acc) 0 1 23 4 5 6 78 9 3-digit addition (11% acc) 0 1 2 3 4 5 6 7 8 9 Figure 13: Circular probe projections of token embeddings onto the unit circle for mod-10 classification. Each point is a number tokenn∈ [0, 999]projected to 2-D by the probe and normalized; color indicatesn mod10. (Left) embeddings from a model trained on 9-digit addition cluster sharply by residue class (84% test accuracy), indicating that the token embedding layer has learned a geometrically organised, clock-like representation. (Right) embeddings from a model trained on 3-digit addition show no angular structure (11% test accuracy, near chance), consistent with the absence of Fourier peaks in Figure 6. Cicular Probes for Transformers Trained on Addition. To probe how number tokens are represented in the embedding layer, we train circular probes on the token embeddings for numbers. BeyondT-class modular probes, which test linear separability in a(T−1)- dimensional subspace, a circular probe tests angular separability in 2-D: it learns a linear mapW ∈ R d×2 projecting each embedding onto the unit circle, and classifies by cosine similarity to m anchor directions at θ k = 2πk/m.