--- abstract: | Large language models (LLMs) can perform remarkably complex tasks, yet the fine-grained details of how these capabilities emerge during pretraining remain poorly understood. Scaling laws on validation loss tell us how much a model improves with additional compute, but not what skills it acquires in which order. To remedy this, we propose the *Implicit Curriculum Hypothesis*: pretraining follows a compositional and predictable curriculum across models and data mixtures. We test this by designing a suite of simple, composable tasks spanning retrieval, morphological transformations, coreference, logical reasoning, and mathematics. Using these tasks, we track emergence points across four model families spanning sizes from 410M--13B parameters. We find that *emergence orderings* of when models reach fixed accuracy thresholds are strikingly consistent ($\rho = .81$ across 45 model pairs), and that composite tasks most often emerge after their component tasks. Furthermore, we find that this structure is encoded in model representations: tasks with similar function vector representations also tend to follow similar trajectories in training. By using the space of representations derived from our task set, we can effectively predict the training trajectories of simple held-out compositional tasks throughout the course of pretraining ($R^2 = .68$--$.84$ across models) without previously evaluating them. Together, these results suggest that pretraining is more structured than loss curves reveal: skills emerge in a compositional order that is consistent across models and readable from their internals.[^1] author: - | Emmy Liu$^{1}$, Kaiser Sun$^{2}$, Millicent Li$^{3}$, Isabelle Lee$^{4}$, Lindia Tjuatja$^{1}$,\ **Jen-tse Huang$^{2}$, Graham Neubig$^{1}$**\ $^{1}$Language Technologies Institute, Carnegie Mellon University\ $^{2}$Department of Computer Science, Data Science and AI Institute,\ Johns Hopkins University\ $^{3}$Khoury College of Computer Science, Northeastern University\ $^{4}$Department of Computer Science, University of Southern California\ `emmy@cmu.edu` bibliography: - reference.bib - model.bib title: What Do Language Models Learn and When? The Implicit Curriculum Hypothesis --- ```{=latex} \newcommand{\cA}[1]{\cellcolor{rhoA}{#1}} ``` ```{=latex} \newcommand{\cB}[1]{\cellcolor{rhoB}{#1}} ``` ```{=latex} \newcommand{\cC}[1]{\cellcolor{rhoC}{#1}} ``` ```{=latex} \newcommand{\cD}[1]{\cellcolor{rhoD}{#1}} ``` ```{=latex} \newcommand{\cE}[1]{\cellcolor{rhoE}\textcolor{white}{#1}} ``` ```{=latex} \newcommand{\cF}[1]{\cellcolor{rhoF}\textcolor{white}{#1}} ``` ```{=latex} \newcommand{\diag}{\cellcolor{diaggray}---} ``` ```{=latex} \newcommand{\igl}[1]{\textcolor{blue}{\textbf{[igl: #1]}}} ``` ```{=latex} \newcommand{\el}[1]{\textcolor{purple}{\textbf{[el: #1]}}} ``` ```{=latex} \newcommand{\ml}[1]{\textcolor{red}{\textbf{[ml: #1]}}} ``` ```{=latex} \newcommand{\lt}[1]{\textcolor{cyan}{\textbf{[lt: #1]}}} ``` ```{=latex} \newcommand{\ks}[1]{\textcolor{orange}{\textbf{[ks: #1]}}} ``` ```{=latex} \newcommand{\numModels}{9} ``` ```{=latex} \newcommand{\numFamilies}{4} ``` ```{=latex} \newcommand{\fix}{\marginpar{FIX}} ``` ```{=latex} \newcommand{\new}{\marginpar{NEW}} ``` ```{=latex} \ifcolmsubmission ``` ```{=latex} \linenumbers ``` ```{=latex} \fi ``` ```{=latex} \maketitle ``` # Introduction Large language models (LLMs) exhibit predictable improvements in performance with scale, a phenomenon characterized by well-established scaling laws [@hoffmann2022an; @gadre2025language; @muennighoff2023scaling]. These scaling laws tell us how much models are expected to improve in predicting the next token on the pretraining distribution given additional compute, but not what skills the model acquires, or when during pretraining it acquires them specifically. In practice, training runs may cost millions of dollars, yet are primarily monitored through aggregate cross-entropy loss, or through evaluating at intervals on downstream benchmarks such as MMLU [@mmlu]. However, neither approach provides actionable diagnostic information. Cross-entropy loss decreases smoothly even as qualitatively different skills are acquired at sudden transition points [@kangaslahti2025hidden]. Downstream benchmarks compose many prerequisite skills, making failures opaque: when GSM8k [@gsm8k] performance stalls, it is unclear whether the bottleneck is numerical fluency, multi-step planning, or natural language understanding. For instance, scoring well on GSM8k may require numerical fluency, multi-step planning, as well as natural language understanding, making it difficult to diagnose which prerequisite skills are missing when performance stalls [@meister-cotterell-2021-language]. A growing body of theoretical work suggests that neural networks learn functions sequentially, acquiring simpler patterns before more complex ones [@lee2025distinctcomputationsemergecompositional; @zhang2026saddletosaddledynamicsexplainssimplicity]. Recent work has built on these insights, hypothesizing that complex behaviors and scaling laws themselves emerge from the combination of more elementary sub-tasks that serve as fundamental building blocks [@khandelwal2025languagemodelscomposefunctions] or *quanta* [@michaud2023the]. However, much of this theoretical work has focused on simplified modeling settings, leaving open questions about how these insights translate to large-scale language model pretraining [@srivastava2023beyond]. Prior empirical work has shown that certain knowledge categories (e.g., syntactic vs. factual) are acquired at different rates [@liu-etal-2021-probing-across], and that grammatical phenomena are learned in a consistent order across architectures [@friedman-etal-2022-finding]. However, these studies have not examined whether the ordering reflects *compositional* dependencies between skills, nor whether it is legible in the model's internal representations. Based on this, we propose the **Implicit Curriculum Hypothesis**: **during pretraining, skills emerge in a stable compositional order that is consistent across models**. This is a stronger claim than the quanta hypothesis alone. It predicts not only that simple precedes complex, but also that the specific ordering is reproducible across models and reflects compositional dependencies between skills. To test the Implicit Curriculum Hypothesis, we design a suite of simple tasks that probe a wide range of skills. We track emergence across `\numModels{}`{=latex} models from `\numFamilies{}`{=latex} families (410M-13B parameters) and find: 1. **The emergence ordering is consistent across model families.** Spearman correlations between emergence orderings range from $\rho = .64$ to $.93$ (mean $.81$) across all 45 model pairs, including cross-family comparisons. Copying is the first skill to emerge, followed by many simple string operations, fact extraction and coreference, then logic operations, simple world knowledge, then multistep arithmetic and more complex reasoning tasks. Composite tasks emerge after their elemental prerequisites. However, this consistency holds only when emergence is defined by fixed accuracy thresholds, not relative ones. 2. **The ordering is legible in model representations.** Tasks whose internal representations are nearby in the model's residual stream, measured via function vectors [@todd2024function], follow similar learning trajectories. This proximity is sufficient to predict the full training trajectory of held-out composite tasks (mean $R^2$ of $.68$--$.84$ across models, with per-task $R^2$ exceeding $.95$) without ever evaluating them during training. `\noindent `{=latex}Together, these results suggest that pretraining is more structured than loss curves reveal: skills emerge in an order that is consistent across models, respects compositional dependencies, and is readable from model internals. ```{=latex} \begin{figure*} \includegraphics[width=0.8\linewidth]{Figures/figure1_teaser_multisize_pythia14b_olmo27b_amber_custom6_notitle_300b_jitter_v7.pdf} \caption{Emergence order across model families and sizes, smoothed with a Gaussian kernel ($\sigma=1.0$). Dots represent the point at which the model reaches a fixed 50\% accuracy threshold. While the absolute emergence time varies across models, the ordering shows regularity.} \label{fig:teaser_figure} \end{figure*} ``` # Preliminaries ## Background We provide a summary of the work that we directly build on in this paper. Further related work can be found in `\autoref{appendix:related_work_extended}`{=latex}. #### Scaling Laws Scaling laws characterize the relationship between a model's held-out validation loss $L$ and the compute budget allocated to training, typically decomposed into model size $N$ and data size $D$. These relationships are well-approximated by power laws of the form $L(N, D) \propto N^{\alpha} + D^{-\beta} + L_\infty$ [@kaplan2020scaling; @hoffmann2022an], and hold across many orders of magnitude. However, this aggregate loss curve does not directly correlate with downstream performance [@lourie2025scalinglawsunreliabledownstream; @isik2026scalinglawsdownstreamtask; @liu2026notjustscalinglawsbetterunderstanding], and it is not clear what the model is learning as loss decreases. **Quantization Hypothesis** [@michaud2023the] offers a hypothesis that these smooth scaling curves arise from the learning of discrete skills, termed *quanta*. Under this framework, a model acquires these quanta in an order optimized to reduce total loss, hiding the discrete transitions that correspond to the model learning. One practical quandary is that quanta require a post-hoc discovery method, the results of which often do not correspond to interpretable skills [@michaud2023the]. While compelling, the Quantization Hypothesis typically treats these skills as independent, additive contributions, leaving the structural dependencies and compositional nature of these skills largely unexplored. #### Simplicity Bias Secondly, works show that neural networks trained with gradient-descent-based methods tend to exhibit **simplicity bias**, a tendency to learn simpler functions before more complex ones [@saxe2014exactsolutionsnonlineardynamics; @nakkiran2019sgdneuralnetworkslearns; @shah2020pitfallssimplicitybiasneural]. In the context of language modeling, this is reflected in models learning lower-order n-grams before higher-order ones [@michaelov2025languagemodelbehavioralphases]. However, we note that the notions of simplicity are often underspecified, and moreover, these function-level notions of complexity cannot be used to quantify task complexity. #### Compositional Skill Structure A recent line of work has investigated whether skills acquired by language models follow explicit dependency structures. @chen2023skillitdatadrivenskillsframework represent skills as directed acyclic graphs (DAGs), where an edge from skill $A$ to skill $B$ indicates that training on data associated with $A$ reduces the amount of data needed to learn $B$. Such dependency graphs can be used to design curricula for target skills. While their work focuses on post-training and defines skills by data clusters, a natural question is whether we can also characterize the dependency structure of general web-data-based pretraining. Theoretically, @arora2023theoryemergencecomplexskills also provide a framework relating cross-entropy loss to competence on individual sets of skills, showing that a decrease in loss implies simultaneous improvement in both individual skills and their $k$-tuples. ## The Implicit Curriculum Hypothesis The four threads of work above establish that (1) capabilities are discrete and unlock progressively, (2) simpler functions are learned before more complex ones, and (3) skills may have a dependency structure. However, they leave open whether these threads combine in practice: does large-scale pretraining follow a structured, compositional ordering of skill acquisition that is consistent across models? ```{=latex} \begin{tcolorbox}[ enhanced, breakable, colback=gray!5, colframe=gray!50, title={The Implicit Curriculum Hypothesis} ]Let $\mathcal{T}$ be a set of tasks equipped with a \textit{known} dependency relation $\prec$, where $\tau_i \prec \tau_j$ indicates that task $\tau_j$ was constructed to compositionally depend on $\tau_i$. Let $P(\tau) = \{\tau' \in \mathcal{T} : \tau' \prec \tau\}$ denote the prerequisite set of task $\tau$ under this construction. We emphasize that $\prec$ reflects our \textit{design-level} task structure, not a claim about the model's internal primitives or the existence of any particular dependency in the model. Let $t^*_\tau(m)$ denote the emergence time of task $\tau$ for model $m$, defined as the first training step at which performance exceeds a threshold $\theta \in \mathbb{R}$. We hypothesize that this design-level structure is \textit{reflected empirically} in model training dynamics: \begin{enumerate} \item \textbf{Compositional ordering.} Tasks emerge no later than the tasks constructed to depend on them: $$\forall\, \tau_j \in \mathcal{T},\; \forall\, \tau_i \in P(\tau_j): \quad t^*_{\tau_i}(m) \leq t^*_{\tau_j}(m)$$ \item \textbf{Cross-model stability.} The emergence ordering induces a partial order $\preceq_{\mathcal{T}}$ over tasks that is consistent across models: for models $m_1, m_2$, the rank correlation between their emergence orderings is significantly higher than chance. \item \textbf{Representational alignment.} Tasks with similar internal representations exhibit similar learning trajectories. That is, for tasks $\tau_i, \tau_j$ with representation vectors $v_i, v_j$, $\exists\, \epsilon \in \mathbb{R}^{+}$: $$\mathrm{Sim}(v_i, v_j) \text{ high} \;\implies\; d\!\left(a_{\tau_i}(\cdot),\; a_{\tau_j}(\cdot)\right) < \epsilon$$ where $a_\tau(\cdot)$ is the learning trajectory of task $\tau$ and $d$ is a distance over trajectories. This further implies that trajectories of unseen tasks can be predicted from representational geometry alone, without evaluating them during training. \end{enumerate} \end{tcolorbox} ``` # Methodology ## Models and Checkpoints In order to test our hypotheses, we focus on examining open-weight models with publicly-released intermediate pre-training checkpoints. Because our hypotheses are largely about timing and emergence order, it was also important to select models with relatively dense intermediate checkpoints and larger sizes. The selected models are: - **OLMo-2** [@olmo20242]: 1B, 7B, and 13B parameter models, providing a within-family scale comparison across an order of magnitude. - **OLMo-3** [@olmo3]: 7B, offering comparison with a newer generation compared to OLMo-2. - **LLM360** [@liu2023llm360]: Crystal (7B) and Amber (7B), trained on very different data mixtures (code-oriented and natural-language-oriented, respectively), allowing us to study the effect of data composition within the same model family. - **Pythia** [@biderman2023pythia]: 410M, 1.4B, and 12B, offering a comparison with an earlier model generation trained on different data. We selected sizes spanning the full range of the suite; models below 410M were excluded due to poor performance. In order to keep checkpoint sampling consistent across families, we focused on up to the first 1T tokens of training for each model and sampled approximately 20 checkpoints for each model within this range, giving a granularity of roughly every 20B tokens. We hypothesized that this would capture the period for which most relevant simple skills emerge for the tasks we study, while the granularity would be sufficient to resolve ordering differences. ## Task Design {#sec:tasks} We design tasks with intuitive compositional relationships, diverse operation types, and unambiguous outputs, while keeping them simple enough for models as small as 1B parameters to eventually solve via in-context learning. We therefore evaluate all 91 elemental and composite tasks using exact-match accuracy; the full list is given in `\autoref{appendix:all_tasks}`{=latex}. ```{=latex} \begin{wraptable}{r}{0.60\textwidth} \tiny \setlength{\tabcolsep}{3pt} \begin{tabular}{p{2.5cm}p{4.0cm}p{0.8cm}} \toprule \textbf{Task} & \textbf{Input} & \textbf{Out} \\ \midrule \texttt{simple\_icl: present\_to\_gerund} & run & running \\ \addlinespace[0.3em] \texttt{fact\_extraction: extract\_number} & Passage: ``John gave 5 apples to Mary on Tuesday.'' How many apples? & 5 \\ \addlinespace[0.3em] \texttt{logical\_ops: conditional} & If it rains, the ground gets wet. It rains. Is the ground wet? & yes \\ \midrule \textit{Compositional tasks} \\ \midrule \texttt{comp: gerund\_upper} & run & RUNNING \\ \addlinespace[0.3em] \texttt{comp: translate\_sp\_eng\_reverse} & hola & olleh \\ \addlinespace[0.3em] \texttt{comp: extract\_verify} & Passage: Nora gave 3 apples to Ben. Ben gave 1 to Li. Claim: Ben received apples first. Does the claim follow? & True \\ \bottomrule \end{tabular} \caption{Example simple and compositional tasks. Compositional tasks require chaining multiple primitive skills.} \label{tab:task-examples} \end{wraptable} ``` #### Simple tasks. We define a set of simple tasks spanning string manipulation (e.g., copy, uppercase, first letter), morphological transformation (e.g., singular to plural, present to gerund), knowledge retrieval (country to capital, country to currency), and translation (e.g., `en-fr`, `en-sp`). These were selected to cover distinct operation types while remaining simple enough to be plausibly atomic. Notably, several of these operations have also been investigated in the interpretability literature [@olsson2022incontextlearninginductionheads; @hendel-etal-2023-context; @todd2024function; @todd2026incontextalgebra]. We do not claim that these are the true minimal units of model computation, but they serve as a diverse set of operations from which we can construct composites with known structure. In total, we create 53 simple tasks. #### Composite tasks: synthetic chains. We construct composite tasks by chaining elemental operations in sequence. For example, `gerund_upper` applies the gerund transformation followed by uppercasing (write $\to$ WRITING). This mechanical construction guarantees that the compositional prerequisites are known exactly, yielding 38 composite tasks. The inclusion of translation-based composites (e.g., `translate_eng_fr_upper_reverse`) additionally tests whether knowledge-dependent elementals compose in the same way as rule-based ones. ## Measuring Emergence Prior work has proposed several notions of emergence, including scale-based definitions and parametric fits to learning curves [@wei2022emergentabilitieslargelanguage; @snell2024predictingemergentcapabilitiesfinetuning]. For our purposes, however, the key quantity is not the sharpness of emergence but the *relative ordering* of when tasks become feasible. Because many trajectories are noisy or irregular, we use simple threshold-based definitions. We consider two variants: #### Absolute threshold. We define emergence time $t^*_\tau(m)$ as the first checkpoint at which task $\tau$ exceeds a fixed accuracy threshold $\theta_{\text{abs}}$. #### Relative threshold. We alternatively define emergence time as the first checkpoint at which performance reaches a fraction $\alpha$ of the model's best performance on that task. ## Measuring Representational Similarity To operationalize representational alignment, we require a per-task representation that captures the computation the model performs in order to do the task. Following the methodology from @todd2024function, we extract task representations (function vectors) from the models. #### Extraction. Let a transformer have $L$ blocks and hidden dimension $d$. For block $\ell$, let $$h^{\mathrm{attn}}_{\ell} = h_{\ell-1} + \mathrm{Attn}\!\left(\mathrm{LN}(h_{\ell-1})\right)$$ denote the post-attention hidden state, and let $$h_{\ell} = h^{\mathrm{attn}}_{\ell} + \mathrm{MLP}\!\left(\mathrm{LN}(h^{\mathrm{attn}}_{\ell})\right)$$ denote the block-output hidden state. For each task $\tau$, we construct a set of ICL prompts, perform a forward pass for each prompt, and extract activations at the last non-pad token position $t_{\mathrm{last}}$ (i.e., the position from which the model begins generating its answer). We retain only prompts on which the model produces the correct answer, ensuring that the extracted representation reflects successful task execution. We consider two extraction methods, and for each model use the one that performs best (see `\autoref{appendix:fv_hyperparams}`{=latex}). *Head-based extraction.* We use causal indirect effect (CIE) analysis to identify a sparse set of attention heads $\mathcal{H} \subseteq [H] \times [L]$ with the strongest causal effects on task performance. The function vector is then the average of these heads' outputs across correctly answered prompts: $$v_\tau^{\mathcal{H}} = \frac{1}{|\mathcal{D}_\tau^+|} \sum_{x_i \in \mathcal{D}_\tau^+} \sum_{(h, j) \in \mathcal{H}} a_h^{j}(x_i),$$ where $a_h^{j}(x_i)$ is the output of attention head $h$ in block $j$, evaluated at position $t_{\mathrm{last}}$, and $\mathcal{D}_\tau^+$ denotes the set of correctly answered prompts. We additionally constrain all selected heads to come from the same block. *Hidden-state extraction.* Alternatively, we extract the block-output hidden state at block $\ell$ and position $t_{\mathrm{last}}$: $$v_\tau^{\ell} = \frac{1}{|\mathcal{D}_\tau^+|} \sum_{x_i \in \mathcal{D}_\tau^+} h_{\ell, t_{\mathrm{last}}}(x_i),$$ where $h_{\ell, t_{\mathrm{last}}}(x_i) \in \mathbb{R}^d$ is the post-MLP hidden state at block $\ell$ and position $t_{\mathrm{last}}$. #### Task similarity We measure similarity between tasks via cosine similarity between their task representations. The hypothesis predicts that tasks with higher representational similarity exhibit more similar learning trajectories. ## Evaluation Protocol We evaluate the Implicit Curriculum Hypothesis through two complementary analyses, corresponding to the behavioral claims (H1, H2) and the representational claim (H3). #### Testing compositional ordering (H1). For each composite task $c$ with a known set of prerequisite tasks $P(c)$, we check whether all prerequisites emerge no later than the composite: $$\forall\, \tau \in P(c): \quad t^*_\tau(m) \leq t^*_c(m)$$ We report the violation rate: the fraction of (composite, prerequisite, model) triples for which this ordering is violated across all models. For synthetic chain composites, $P(c)$ is known by construction. #### Testing cross-model stability (H2). For each pair of models $(m_1, m_2)$, we compute the Spearman rank correlation between their emergence orderings $\sigma_{m_1}$, $\sigma_{m_2}$ over the full task set. We report correlations separately for the absolute and relative threshold definitions. For tasks that remain unemerged by the end of training, we bin them into one bucket at the end.[^2] #### Leave-one-out prediction of composite trajectories (H3). We operationalize H3 through a leave-one-out (LOO) protocol over composite tasks. For a held-out composite task $c$, we predict its learning trajectory from the trajectories of its nearest neighbors in FV space. Before prediction, we interpolate basis task trajectories onto the held-out task's token grid, apply Gaussian smoothing ($\sigma = 1.0$), and also discard tasks with near-zero trajectory variance.[^3] We extract unit-normalized residual stream representations for all tasks at the selected layer and compute pairwise similarities using an RBF kernel: $$K(v_i, v_j) = \exp\!\Big(\!-\frac{\|v_i - v_j\|^2} {2\sigma_k^2}\Big)$$ We use kernel ridge regression to learn a predictor for the held-out task performance: Let $S$ denote the set of training tasks (excluding $c$), let $K_S \in \mathbb{R}^{|S| \times |S|}$ be the kernel matrix with entries $$(K_S)_{ij} = K(v_{\tau_i}, v_{\tau_j}),$$ and let $$k_c = \big[K(v_c, v_{\tau_j})\big]_{j \in S}$$ be the vector of similarities between the held-out task and the training tasks. For each training step $t$, we form the vector of training trajectory values $$y_t = \big[a_{\tau_j}(t)\big]_{j \in S}.$$ Kernel ridge regression solves $$\alpha_t = (K_S + \lambda I)^{-1} y_t$$ and predicts the held-out trajectory as $$\hat{a}_c(t) = k_c^\top \alpha_t.$$ We evaluate prediction quality via per-task Pearson $r^2$ and MAE against smoothed ground-truth trajectories, and report both per-task results and means across all held-out composites. To test the composition bottleneck, we compare two conditions for the function vector space: 1. **All tasks:** the basis includes both simple and composite tasks excluding the held-out target. 2. **Simple tasks only:** the basis is restricted to non-composite tasks only. If prediction quality degrades substantially under the elementals-only basis, this indicates that composite trajectories share structure with one another that is not captured by their elemental components alone, indicating a composition bottleneck. # Emergence Order Results ![Emergence order heatmap of selected tasks across models (absolute threshold = 0.8). Tasks sorted by consensus emergence order. Consistent color gradients across columns indicate stable ordering.](Figures/figure3_emergence_order_heatmap_representative_revised.png){#fig:heatmap width="70%"} We first test H1 and H2 by examining whether the emergence order of tasks is consistent across models and whether composites emerge after their constituent components. `\autoref{fig:heatmap}`{=latex} shows the emergence times of all tasks across models, sorted vertically by consensus emergence order. From inspection, it is clear that tasks that emerge early in one model tend to do so across all models. Copying and simple coreference resolution emerge early across all models, in line with previous work [@yin2025which]. These are followed by simple ICL tasks such as uppercasing and lowercasing, then morphological transformations, followed by knowledge-dependent tasks such as translation, and finally a long tail of more difficult or compositional tasks. Furthermore, we examine whether compositional tasks arise after their components. This is generally the case: among all compositional tasks, 54/76 emerged no earlier than their \`\`parent" tasks. However, we also observe a number of inversions, where the composite task emerged earlier than one (19 weak inversions) or both (3 strong inversions) parents. Notably, all three strong inversions involve the `first_letter` component task. `\autoref{tab:spearman}`{=latex} quantifies the consistency of emergence order across models. Within the OLMo-2 family, Spearman rank correlations range from .72 to .93. Cross-family correlations are also high: Amber correlates with OLMo-2 models at 0.82--0.88, while correlations with older and smaller models (e.g., OLMo-2 vs. Pythia-410M) remain substantial, ranging from 0.64 to 0.84. All correlations are highly significant and remain so after correction for multiple comparisons. Importantly, this consistency holds only under the absolute threshold definition of emergence. When using relative thresholds, cross-model correlations drop substantially (`\autoref{appendix:emergence_orders_alternate}`{=latex}). We hypothesize that this discrepancy arises because relative thresholds depend on each model's maximum performance: a weak model may reach a relative threshold early despite lacking meaningful task competence, while a stronger model may never satisfy the same criterion. In contrast, our absolute thresholds are set above chance for all tasks, effectively capturing the point at which the underlying computation becomes functional. This plausibly corresponds to the formation of a task-relevant circuit. In this view, the relative consistency of the absolute order suggests that what is shared across models is *the order in which computations become feasible* under standard pretraining, even when trained on differing data distributions. ```{=latex} \scriptsize ``` ```{=latex} \setlength{\tabcolsep}{3.5pt} ``` ```{=latex} \renewcommand{\arraystretch}{1.02} ``` ::: {#tab:spearman} OLMo2 OLMo3 LLM360 Pythia ------------------------------------------------------------------------------------ ------------------ --------------------- --------------------- --------------------- --------------------- --------------------- --------------------- --------------------- --------------------- ------------------ 2-4`\cmidrule`{=latex}(lr)5-5`\cmidrule`{=latex}(lr)6-7`\cmidrule`{=latex}(lr)8-11 1B 7B 13B 7B Amber Crystal 410M 1.4B 2.8B 12B O2-1B `\diag `{=latex} `\cD`{=latex} `\cB`{=latex} `\cC`{=latex} `\cE`{=latex} `\cC`{=latex} `\cD`{=latex} `\cF`{=latex} `\cC`{=latex} `\cD`{=latex} O2-7B `\diag `{=latex} `\cF`{=latex} `\cF`{=latex} `\cE`{=latex} `\cF`{=latex} `\cB`{=latex} `\cB`{=latex} `\cC`{=latex} `\cD`{=latex} O2-13B `\diag `{=latex} `\cF`{=latex} `\cD`{=latex} `\cE`{=latex} `\cA`{=latex} `\cA`{=latex} `\cC`{=latex} `\cD`{=latex} O3-7B `\diag `{=latex} `\cD`{=latex} `\cF`{=latex} `\cB`{=latex} `\cB`{=latex} `\cC`{=latex} `\cE`{=latex} Amber `\diag `{=latex} `\cE`{=latex} `\cC`{=latex} `\cD`{=latex} `\cE`{=latex} `\cF`{=latex} Crystal `\diag `{=latex} `\cB`{=latex} `\cC`{=latex} `\cD`{=latex} `\cE`{=latex} P-410M `\diag `{=latex} `\cE`{=latex} `\cB`{=latex} `\cC`{=latex} P-1.4B `\diag `{=latex} `\cC`{=latex} `\cD`{=latex} P-2.8B `\diag `{=latex} `\cF`{=latex} P-12B `\diag `{=latex} : Spearman rank correlation ($\rho$) of emergence orderings between model pairs (absolute threshold = 80%). All 45 correlations are significant ($p<10^{-7}$). ::: ```{=latex} \footnotesize ``` ::: {style="background-color: rhoA"} ::: .64`\;`{=latex} ::: {style="background-color: rhoB"} ::: .70`\;`{=latex} ::: {style="background-color: rhoC"} ::: .75`\;`{=latex} ::: {style="background-color: rhoD"} ::: .80`\;`{=latex} ::: {style="background-color: rhoE"} ::: .85`\;`{=latex} ::: {style="background-color: rhoF"} ::: .90+ # Representational Similarity and Prediction Results Having established that skill acquisition during pretraining is both structured and consistent (H1, H2), we next ask whether this structure is reflected in the model's internal representations (H3). Namely, if two tasks have similar function vectors, do they exhibit similar learning trajectories in pretraining? Rather than testing correlations in isolation, we consider a stronger version: can the learning trajectory of a held-out composite task be predicted solely from its representational similarity to other tasks, without further evaluation during training? ```{=latex} \scriptsize ``` ::: {#tab:loo} **All tasks** **Sim. only** ------------------------------------------ --------------- --------------- ------- ------ 2-3 `\cmidrule`{=latex}(lr)4-5 **Model** $R^2$ MAE $R^2$ MAE Pythia-410M .681 .195 .717 .301 OLMo2-1B .723 .070 .602 .289 Pythia-1.4B .778 .086 .755 .193 `\addlinespace`{=latex}Amber (7B) .751 .082 .725 .205 Crystal (7B) .676 .133 .568 .315 OLMo2-7B .767 .068 .693 .208 OLMo3-7B .692 .079 .491 .215 `\addlinespace`{=latex}Pythia-12B .812 .136 .789 .194 OLMo2-13B .838 .099 .860 .242 : Leave-one-out prediction of held-out composite task trajectories (26 tasks). *All tasks* includes simple and composite tasks. *Sim. only* includes only simple tasks. Restricting to elementals degrades MAE for every model (mean $\Delta$MAE $= +.135$), indicating a composition bottleneck. ::: `\autoref{tab:loo}`{=latex} reports leave-one-out prediction results for composite task trajectories using kernel ridge regression in function vector space. When the basis includes all other tasks (elemental and composite), prediction quality is strong: $R^2$ ranges from .67 (Crystal) to .838 (OLMo2-13B), with MAE between .068 and .195 on a 0-1 accuracy scale. These results provide strong evidence that representational geometry is closely linked to learning dynamics, supporting H3. As a case study of specific predicted trajectories, `\autoref{fig:loo_examples}`{=latex} shows representative predicted trajectories compared to ground truth trajectories for OLMo2-7B from 0-1T tokens. For tasks such as `fr_eng_upper` ($R^2 = .99$, MAE $= .017$) and `plural_lower` ($R^2 = .89$, MAE $= .028$), the predicted curve closely tracks the actual trajectory, capturing both the onset of emergence and the subsequent rate of improvement. However, predictions are weaker for tasks such as `eng_fr_upper` ($R^2 = .51$, MAE $= .068$), where the held-out task's trajectory is less well approximated by its nearest neighbors in representation space. Full prediction results can be found in `\autoref{appendix:all_held_out_preds}`{=latex}. ![Example composite task predictions for OLMo2-7B between 0-1T tokens.](Figures/composite_horizontal_3_diverse_v3.png){#fig:loo_examples width="0.8\\linewidth"} # Conclusion In this paper, we unify several threads of discussion on the emergence of LM capabilities during pretraining in the Implicit Curriculum Hypothesis -- that skills acquired during pretraining emerge in a stable order driven by concrete compositions. We test this hypothesis empirically across several model families and with models spanning 410M-13B parameters. Our empirical findings support both the behavioural and representational aspects of the hypothesis: emergence orders of tasks under absolute thresholds are quite consistent across models, even across families and in models trained on different data. Furthermore, similarity in the function vector space predicts similarity of learning trajectories, such that it is possible to predict the trajectories of held-out compositional tasks from function vector similarity without evaluating them. This indicates that the developmental structure visible in behavioral evaluations may also be legible in the model's internal representations. Our results open several avenues for further investigation. One practical application is *pretraining monitoring* -- if emergence orders are stable and predictable, our task suite can serve as a basis for monitoring whether models are developing capabilities ahead of or behind schedule. Furthermore, understanding this task structure could also inform data mixture decisions. More broadly, we hope that the framework of studying pretraining as a structured developmental process will prove useful for understanding, predicting, and ultimately steering what language models learn. # Acknowledgments {#acknowledgments .unnumbered} EL was supported by the National Sciences and Engineering Research Council of Canada (NSERC), \[funding reference number 578085\], as well as the SoftBank-ARM Fellowship. ML is supported by a NSF Graduate Research Fellowship. IL is supported by a Technical AI Safety Research Grant from Coefficient Giving via Berkeley Existential Risk Initiative. This work used the Delta system at the National Center for Supercomputing Applications \[award OAC 2005572\] through allocation \[CIS250578\] from the Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS) program, which is supported by National Science Foundation grants #2138259, #2138286, #2138307, #2137603, and #2138296. ```{=latex} \bibliographystyle{colm2026_conference} ``` ```{=latex} \onecolumn ``` ```{=latex} \appendix ``` # LLM Usage Disclosure Claude Opus 4.6 and Sonnet 4.6 were used to format tables and conduct minor writing edits. All outputs were reviewed, and verified by the authors. # Extended Related Work {#appendix:related_work_extended} #### Skill Emergence and Scaling Laws. Theoretical work has sought to explain how capabilities emerge with scale. @arora2023theoryemergencecomplexskills propose that scaling laws arise from slingshot generalization, where competence at $k$-tuples of skills emerges at the same rate as elementary skills themselves. Similarly, @michaud2023the introduce the quanta hypothesis, modeling skills as discrete units whose power-law frequency distribution explains smooth scaling curves. Both theories predict that complex behaviors emerge from simpler building blocks, but leave open the question of what these building blocks are and how they compose in practice. Our work provides empirical grounding for these theories by tracking probe tasks designed to be compositionally combined, finding that compositional skills reliably emerge after their constituent components. #### Skill Evaluation and Structure. Several approaches characterize LLM capabilities through evaluation-time analysis. @burnell2023revealingstructurelanguagemodel apply factor analysis across 29 models and 27 tasks, finding three latent factors, reasoning, comprehension, and language modeling, that explain performance variation; @maimon2025iqtestllmsevaluation scale this psychometric approach to 60 models and 44 tasks, identifying eight core skills. Beyond identifying skills, @yu2024skillmix directly test compositional ability by evaluating whether models can combine $k$-tuples of language skills in novel ways. @polo2025sloth unify these perspectives through skill-based scaling laws where performance is driven by low-dimensional latent skills. These works analyze fully trained models; we complement them by studying *how skills develop during pretraining* and linking emergence order to representational structure. #### Training Dynamics and Phase Transitions. Understanding what models learn during training has gained increasing attention. @chen2023sudden identify sudden drops in loss corresponding to syntax acquisition and other phase transitions; @kangaslahti2025hidden show that such breakthroughs occur frequently but are obscured by aggregate loss metrics. @van2025polypythias release 50 additional training runs of Pythia models, finding consistent learning phases across seeds and sizes. Other work examines specific capabilities: @sun-dredze-2025-amuro investigate how downstream performance develops across pretraining checkpoints, @ge2025evolution track feature evolution using sparse dictionary learning, and @mishranext show that mathematical skills emerge in an order correlated with human curriculum despite random data ordering. Our work contributes to this literature by demonstrating that emergence orderings are stable across model families and can be predicted from representational geometry. #### Representations for Task Understanding. Mechanistic interpretability has revealed compact representations of tasks within model activations. Both @todd2024function and @hendel-etal-2023-context discover that in-context learning compresses task demonstrations into single directions, termed function vectors and task vectors respectively, which can trigger task execution even in zero-shot settings. Subsequent work explores the scope of this phenomenon: @todd2026incontextalgebra extend it to symbolic reasoning with variable-based tokens, while @khandelwal2025languagemodelscomposefunctions investigate compositional tasks and find both compositional and direct processing mechanisms. We build on this line of work by using residual-stream representations to predict learning trajectories of compositional tasks, connecting representational geometry to training dynamics. This complements recent work by @prasad2026featuresrewardsscalablesupervision that uses interpretable features for training monitoring in RL. # Full list of Elemental and Composite Tasks {#appendix:all_tasks} We provide a full list of tasks, categorized into reasoning types, in `\autoref{tab:all-elemental-tasks}`{=latex} and `\autoref{tab:all-compositional-tasks}`{=latex}. TextFRCT tasks are taken from the psychometrics literature [@ekstrom1976kit], while other tasks have been studied or were inspired by works in interpretability literature [@wang2023interpretability; @todd2024function; @chen2024parallel; @feucht2025dualroute]. # Full list of Elemental and Composite Tasks {#appendix:all_tasks} We provide a full list of tasks, categorized into reasoning types, in `\autoref{tab:all-elemental-tasks}`{=latex} and `\autoref{tab:all-compositional-tasks}`{=latex}. TextFRCT tasks are taken from the psychometrics literature [@ekstrom1976kit], while other tasks have been studied or were inspired by works in interpretability literature [@wang2023interpretability; @todd2024function; @chen2024parallel; @feucht2025dualroute]. ```{=latex} \scriptsize ``` ```{=latex} \setlength{\tabcolsep}{3pt} ``` ```{=latex} \renewcommand{\arraystretch}{0.96} ``` ::: {#tab:all-elemental-tasks} +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | **Task** | **N** | **Input** | **Output** | +:==========================================================+:======+:===================================================================================================================================================================================================================================================================================+:================+ | *Table `\ref`{=latex} continued from previous page* | | | | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | **Task** | **N** | **Input** | **Output** | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | *Continued on next page* | | | | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | ```{=latex} | | | | | \endfoot | | | | | ``` | | | | | ```{=latex} | | | | | \bottomrule | | | | | ``` | | | | | ```{=latex} | | | | | \endlastfoot | | | | | ``` | | | | | *String Operations* | | | | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `copying` | 20 | gTpigTHK | gTpigTHK | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`token_reversal` | 20 | cat | tac | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`string_analogy` | 10 | abc $\to$ abd, ijk $\to$ ? | ijl | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`simple_icl:uppercase` | 26 | b | B | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`simple_icl:lowercase` | 26 | B | b | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`simple_icl:first_letter` | 190 | the cat went up the tree | t | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`simple_icl:last_letter` | 190 | the cat went up the tree | e | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | ```{=latex} | | | | | \addlinespace | | | | | ``` | | | | | *Morphology* | | | | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `simple_icl:present_to_gerund` | 179 | run | running | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`simple_icl:singular_to_plural` | 165 | child | children | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | ```{=latex} | | | | | \addlinespace | | | | | ``` | | | | | *Translation* | | | | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `simple_icl:translate_eng_fr` | 173 | hello | bonjour | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`simple_icl:translate_fr_eng` | 175 | bonjour | hello | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`simple_icl:translate_eng_sp` | 178 | hello | hola | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`simple_icl:translate_sp_eng` | 178 | hola | hello | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | ```{=latex} | | | | | \addlinespace | | | | | ``` | | | | | *World Knowledge* | | | | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `simple_icl:country_to_capital` | 184 | Afghanistan | Kabul | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`simple_icl:country_to_currency` | 198 | United States | Dollar | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | ```{=latex} | | | | | \addlinespace | | | | | ``` | | | | | *Arithmetic* | | | | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `basic_arithmetic` | 10 | What is 5 + 3? | 8 | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`math` | 20 | 4 \* 1 | 4 | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`multistep_arithmetic:two_step` | 20 | 3 + 4, then multiply by 2 | 14 | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`multistep_arithmetic:three_step` | 20 | Start with 10, subtract 3, then multiply by 4 | 28 | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`textfrct:RG1` | 30 | In general, brass is made of two parts copper to one part zinc. How many pounds of zinc are needed to produce 45 pounds of brass? *(MCQ)* | B | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`textfrct:RG2` | 30 | Recipe A uses 1.5 cups of sugar; Recipe B uses 2. Making 8 cakes, how many fewer cups does Recipe A require? *(MCQ)* | E | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`textfrct:RG3` | 30 | There are 4 quarts in a gallon and 4 cups in a quart. How many cups are in a gallon? *(MCQ)* | C | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | ```{=latex} | | | | | \addlinespace | | | | | ``` | | | | | *Logic* | | | | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `logical_ops:negation` | 12 | Statement: All robots can move.`\newline `{=latex}Candidate: Some robots cannot move.`\newline `{=latex}Is this a correct logical negation? | True | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`logical_ops:conjunction` | 12 | Fact A is True. Fact B is True.`\newline `{=latex}Claim: A AND B. Is the claim true? | True | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`logical_ops:conditional` | 12 | Rule: If it rains, the ground gets wet.`\newline `{=latex}Fact: It rains. Does the conclusion follow? | True | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`textfrct:RL1` | 30 | All birds have purple tails. All cats are birds. Therefore all cats have purple tails. *(MCQ: correct/incorrect)* | G | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`textfrct:RL3` | 20 | More fatal accidents occur on highways after dark than during daylight hours. *(MCQ: which conclusion follows?)* | 3 | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`textfrct:RL4` | 24 | *ICL ex.: black sheep = dag kip; white dog = tin bud; black cow = dag stam*`\newline `{=latex}Query: white sheep = ? *(MCQ)* | 2 | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | ```{=latex} | | | | | \addlinespace | | | | | ``` | | | | | *Reading Comprehension* | | | | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `fact_extraction:extract_entity` | 20 | Passage: \`\`Alice gave five apples to Bob at the park." Who received the apples? | Bob | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`fact_extraction:extract_number` | 20 | Passage: \`\`John gave 5 apples to Mary on Tuesday." How many apples did John give? | 5 | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`fact_extraction:extract_location` | 20 | Passage: \`\`The cat sat on the red mat in the kitchen." Where is the mat? | the kitchen | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`coreference:pronoun_simple` | 20 | \`\`Alice told Bob that she would be late." Who does \`\`she" refer to? | Alice | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`coreference:pronoun_hard` | 20 | \`\`The trophy didn't fit in the suitcase because it was too big." What was too big? | the trophy | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`ignoring_context` | 5 | Some text here. X = 5. More text.`\newline `{=latex}Question: What is X? | 5 | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`ioi_task` | 1000 | *Instr.: Identify who should be referenced.*`\newline `{=latex}Then, Henry and Phil had a lot of fun at the harbor. Henry gave a basket to | `[Phil, Henry]` | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`part_of_speech` | 15 | The cat is in the house. The part of speech for \`\`cat" is \_ | noun | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | ```{=latex} | | | | | \addlinespace | | | | | ``` | | | | | *Verbal Closure (FRCT)* | | | | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `textfrct:CV1` | 50 | erte | tree, rete | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`textfrct:CV2` | 40 | EZIRTMODSLOWTSEXQILNECKBWOCJAKX | SLOW, NECK | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`textfrct:CV3` | 36 | \_tam\_ | stamp | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | ```{=latex} | | | | | \addlinespace | | | | | ``` | | | | | *Induction (FRCT)* | | | | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `textfrct:I1` | 30 | *Instr.: One of the five letter sets does NOT follow the same pattern as the others. Find it.*`\newline `{=latex}. QPPQ `\quad `{=latex}. HGHH `\quad `{=latex}. TTTU `\quad `{=latex}. DDDE `\quad `{=latex}. MLMM | 1 | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`textfrct:I2` | 28 | *Instr.: Each row marks one position with \`x'. Identify the pattern and find the correct position in row 5.*`\newline`{=latex} `——- x——- —- –``\newline`{=latex} `—- -x— – — ——``\newline`{=latex} `————— –x—–``\newline`{=latex} `——– —x———–``\newline`{=latex} `—-1 2—3– 4—5—–` | 3 | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | ```{=latex} | | | | | \addlinespace | | | | | ``` | | | | | *Associative Memory (FRCT)* | | | | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `textfrct:MA2` | 30 | *Instr.: Memorize 30 word--number pairs, then answer retrieval queries.*`\newline `{=latex}Query: What number corresponds to \`coat'? | 49 | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`textfrct:MA3` | 30 | *Instr.: Memorize 30 first--last name pairs, then answer retrieval queries.*`\newline `{=latex}Query: Last name: Nichols | Edward | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | ```{=latex} | | | | | \addlinespace | | | | | ``` | | | | | *Verbal Comprehension (FRCT)* | | | | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `textfrct:V1` | 36 | *Instr.: Choose the best definition (MCQ).*`\newline `{=latex}\`airtight': (1) firm `\;`{=latex} (2) light `\;`{=latex} (3) hermetically sealed `\;`{=latex} (4) plane sick | 3 | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`textfrct:V2` | 36 | *Instr.: Choose the best definition (MCQ).*`\newline `{=latex}\`handicraft': (1) cunning `\;`{=latex} (2) fast boat `\;`{=latex} (3) utility `\;`{=latex} (4) manual skill `\;`{=latex} (5) guild | 4 | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`textfrct:V3` | 48 | *Instr.: Choose the best definition (MCQ).*`\newline `{=latex}\`cottontail': (1) squirrel `\;`{=latex} (2) poplar `\;`{=latex} (3) boa `\;`{=latex} (4) marshy plant `\;`{=latex} (5) rabbit | 5 | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`textfrct:V4` | 36 | *Instr.: Choose the best definition (MCQ).*`\newline `{=latex}\`mumble': (1) speak indistinctly `\;`{=latex} (2) complain `\;`{=latex} (3) handle awkwardly `\;`{=latex} (4) fall `\;`{=latex} (5) tear apart | 1 | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ | `\addlinespace`{=latex}`textfrct:V5` | 36 | *Instr.: Choose the best definition (MCQ).*`\newline `{=latex}\`rancor': (1) forbearance `\;`{=latex} (2) ridicule `\;`{=latex} (3) malice `\;`{=latex} (4) bravery | 3 | +-----------------------------------------------------------+-------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------+ : All elemental tasks in the evaluation suite with representative examples. ::: ```{=latex} \scriptsize ``` ```{=latex} \setlength{\tabcolsep}{3pt} ``` ```{=latex} \renewcommand{\arraystretch}{0.96} ``` ::: {#tab:all-compositional-tasks} +-----------------------------------------------------------------------+-------+-------------+-------------+ | **Task** | **N** | **Input** | **Output** | +:======================================================================+:======+:============+:============+ | *Table `\ref`{=latex} continued from previous page* | | | | +-----------------------------------------------------------------------+-------+-------------+-------------+ | **Task** | **N** | **Input** | **Output** | +-----------------------------------------------------------------------+-------+-------------+-------------+ | *Continued on next page* | | | | +-----------------------------------------------------------------------+-------+-------------+-------------+ | ```{=latex} | | | | | \endfoot | | | | | ``` | | | | | ```{=latex} | | | | | \bottomrule | | | | | ``` | | | | | ```{=latex} | | | | | \endlastfoot | | | | | ``` | | | | | *Morphology $\times$ String Operation* | | | | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `compositional:gerund_lower` | 178 | RUN | running | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:gerund_upper` | 178 | run | RUNNING | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:gerund_reverse` | 178 | run | gninnur | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:gerund_upper_reverse` | 178 | run | GNINNUR | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:plural_lower` | 165 | CHILD | children | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:plural_upper` | 165 | child | CHILDREN | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:plural_reverse` | 165 | child | nerdlihc | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:plural_upper_reverse` | 165 | child | NERDLIHC | +-----------------------------------------------------------------------+-------+-------------+-------------+ | ```{=latex} | | | | | \addlinespace | | | | | ``` | | | | | *Translation $\times$ String Operation* | | | | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `compositional:translate_eng_fr_first` | 173 | hello | b | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:translate_eng_fr_last` | 173 | hello | r | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:translate_eng_fr_lower` | 173 | HELLO | bonjour | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:translate_eng_fr_reverse` | 173 | hello | ruojnob | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:translate_eng_fr_upper` | 173 | hello | BONJOUR | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:translate_eng_fr_upper_reverse` | 173 | hello | RUOJNOB | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:translate_eng_sp_first` | 178 | hello | h | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:translate_eng_sp_last` | 178 | hello | a | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:translate_eng_sp_lower` | 178 | HELLO | hola | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:translate_eng_sp_reverse` | 178 | hello | aloh | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:translate_eng_sp_upper` | 178 | hello | HOLA | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:translate_eng_sp_upper_reverse` | 178 | hello | ALOH | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:translate_fr_eng_first` | 171 | bonjour | h | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:translate_fr_eng_last` | 171 | bonjour | o | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:translate_fr_eng_lower` | 171 | BONJOUR | hello | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:translate_fr_eng_reverse` | 171 | bonjour | olleh | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:translate_fr_eng_upper` | 171 | bonjour | HELLO | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:translate_sp_eng_first` | 178 | hola | h | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:translate_sp_eng_last` | 178 | hola | o | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:translate_sp_eng_lower` | 178 | HOLA | hello | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:translate_sp_eng_reverse` | 178 | hola | olleh | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:translate_sp_eng_upper` | 178 | hola | HELLO | +-----------------------------------------------------------------------+-------+-------------+-------------+ | ```{=latex} | | | | | \addlinespace | | | | | ``` | | | | | *Case/Reversal Chains* | | | | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `compositional:lower_first` | 971 | AFGHANISTAN | a | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:lower_last` | 971 | AFGHANISTAN | n | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:lower_reverse` | 971 | AFGHANISTAN | natsinahgfa | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:upper_first` | 971 | afghanistan | A | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:upper_last` | 971 | afghanistan | N | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:upper_reverse` | 971 | afghanistan | NATSINAHGFA | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:reverse_first` | 971 | Afghanistan | n | +-----------------------------------------------------------------------+-------+-------------+-------------+ | `\addlinespace`{=latex}`compositional:reverse_last` | 971 | Afghanistan | A | +-----------------------------------------------------------------------+-------+-------------+-------------+ : All compositional tasks in the evaluation suite with representative examples. ::: ```{=latex} \newpage ``` ```{=latex} \newpage ``` # Full learning trajectories by category {#appendix:all_learning_traj} Figures `\ref{fig:all_traj_pythia_410m}`{=latex} -- `\ref{fig:all_traj_olmo2_13b}`{=latex} show full learning trajectories of tasks for each model. ![Complete trajectories for Pythia-410M over 300B tokens.](Figures/trajectory_page_pythia_410m.png){#fig:all_traj_pythia_410m width="0.95\\linewidth"} ![Complete trajectories for OLMo2-1B over 1T tokens.](Figures/trajectory_page_olmo2_1b.png){#fig:all_traj_olmo2_1b width="0.95\\linewidth"} ![Complete trajectories for Pythia-1.4B over 300B tokens.](Figures/trajectory_page_pythia_1_4b.png){#fig:all_traj_pythia_1_4b width="0.95\\linewidth"} ![Complete trajectories for OLMo-2 7B over 1T tokens.](Figures/trajectory_page_olmo2_7b.png){#fig:all_traj_olmo2_7b width="0.95\\linewidth"} ![Complete trajectories for OLMo-3 7B over 1T tokens.](Figures/trajectory_page_olmo3_7b.png){#fig:all_traj_olmo3_7b width="0.95\\linewidth"} ![Complete trajectories for Amber (7B) over 1T tokens.](Figures/trajectory_page_amber.png){#fig:all_traj_amber_7b width="0.95\\linewidth"} ![Complete trajectories for CrystalCoder (7B) over 1T tokens.](Figures/trajectory_page_crystalcoder.png){#fig:all_traj_crystal_7b width="0.95\\linewidth"} ![Complete trajectories for Pythia-12B over 300B tokens. Note that this model exhibits some instabilities compared to others.](Figures/trajectory_page_pythia_12b.png){#fig:all_traj_pythia_12b width="0.95\\linewidth"} ![Complete trajectories for OLMo2-13b over 1T tokens. Note that this model exhibits some instabilities compared to others.](Figures/trajectory_page_olmo2_13b.png){#fig:all_traj_olmo2_13b width="0.95\\linewidth"} # Emergence Order Agreement Under Alternate Definitions {#appendix:emergence_orders_alternate} ```{=latex} \small ``` ::: {#tab:emergence_defs_summary} **Definition** **$n$ pairs** **Mean $\rho$** **Min $\rho$** **Max $\rho$** --------------------------------------------------------------------------- --------------- ----------------- ---------------- ---------------- Absolute threshold ($\theta = 0.5$) 36 0.860 0.597 0.955 Absolute threshold ($\theta = 0.8$, stable for 3 consecutive checkpoints) 36 0.790 0.599 0.961 Relative threshold ($\alpha = 0.5$, fraction of max performance) 36 0.528 0.085 0.866 Relative threshold ($\alpha = 0.8$, fraction of max performance) 36 0.500 0.077 0.773 : Summary of emergence ordering consistency under different definitions. Absolute thresholds yield substantially higher cross-model correlations than relative thresholds. ::: ```{=latex} \small ``` ::: {#tab:spearman_abs05} **Model A** **Model B** **$n$ tasks** **$\rho$** **$p$** -------------------------------------------------- ------------- --------------- ------------ ----------------------- *Within-family (OLMo-2)* OLMo2-1B OLMo2-7B 106 0.889 $4.5 \times 10^{-37}$ OLMo2-1B OLMo2-13B 105 0.832 $4.5 \times 10^{-28}$ OLMo2-7B OLMo2-13B 104 0.865 $2.9 \times 10^{-32}$ *Within-family (Pythia)* Pythia-1.4B Pythia-410M 96 0.910 $1.3 \times 10^{-37}$ Pythia-1.4B Pythia-12B 98 0.909 $3.7 \times 10^{-38}$ Pythia-410M Pythia-12B 100 0.815 $6.5 \times 10^{-25}$ *Within-family (LLM360)* Amber Crystal 102 0.905 $6.8 \times 10^{-39}$ *Cross-family (OLMo-2 $\leftrightarrow$ OLMo-3)* OLMo2-1B OLMo3-7B 106 0.925 $1.4 \times 10^{-45}$ OLMo2-7B OLMo3-7B 105 0.918 $4.7 \times 10^{-43}$ OLMo2-13B OLMo3-7B 104 0.897 $6.5 \times 10^{-38}$ *Cross-family (OLMo-2 $\leftrightarrow$ LLM360)* Amber OLMo2-1B 107 0.932 $5.5 \times 10^{-48}$ Amber OLMo2-7B 106 0.907 $7.2 \times 10^{-41}$ Amber OLMo2-13B 105 0.839 $5.6 \times 10^{-29}$ Crystal OLMo2-1B 102 0.913 $1.3 \times 10^{-40}$ Crystal OLMo2-7B 101 0.910 $1.4 \times 10^{-39}$ Crystal OLMo2-13B 100 0.889 $6.3 \times 10^{-35}$ *Cross-family (OLMo-3 $\leftrightarrow$ LLM360)* Amber OLMo3-7B 106 0.918 $1.6 \times 10^{-43}$ Crystal OLMo3-7B 102 0.955 $8.1 \times 10^{-55}$ *Cross-family (OLMo-2 $\leftrightarrow$ Pythia)* Pythia-1.4B OLMo2-1B 98 0.907 $9.6 \times 10^{-38}$ Pythia-1.4B OLMo2-7B 97 0.830 $8.7 \times 10^{-26}$ Pythia-1.4B OLMo2-13B 97 0.716 $1.6 \times 10^{-16}$ Pythia-410M OLMo2-1B 100 0.834 $4.8 \times 10^{-27}$ Pythia-410M OLMo2-7B 99 0.793 $1.2 \times 10^{-22}$ Pythia-410M OLMo2-13B 98 0.597 $8.4 \times 10^{-11}$ Pythia-12B OLMo2-1B 102 0.856 $2.1 \times 10^{-30}$ Pythia-12B OLMo2-7B 101 0.832 $5.1 \times 10^{-27}$ Pythia-12B OLMo2-13B 100 0.786 $3.4 \times 10^{-22}$ *Cross-family (OLMo-3 $\leftrightarrow$ Pythia)* Pythia-1.4B OLMo3-7B 97 0.864 $4.6 \times 10^{-30}$ Pythia-410M OLMo3-7B 99 0.799 $3.9 \times 10^{-23}$ Pythia-12B OLMo3-7B 101 0.867 $1.2 \times 10^{-31}$ *Cross-family (LLM360 $\leftrightarrow$ Pythia)* Amber Pythia-1.4B 98 0.935 $7.4 \times 10^{-45}$ Amber Pythia-410M 100 0.853 $1.9 \times 10^{-29}$ Amber Pythia-12B 102 0.930 $3.6 \times 10^{-45}$ Crystal Pythia-1.4B 93 0.824 $3.8 \times 10^{-24}$ Crystal Pythia-410M 96 0.751 $1.2 \times 10^{-18}$ Crystal Pythia-12B 97 0.850 $3.3 \times 10^{-28}$ : Pairwise Spearman rank correlations: Absolute threshold ($\theta = 0.5$). Mean $\rho = 0.860$. ::: ```{=latex} \small ``` ::: {#tab:spearman_abs08stable} **Model A** **Model B** **$n$ tasks** **$\rho$** **$p$** -------------------------------------------------- ------------- --------------- ------------ ----------------------- *Within-family (OLMo-2)* OLMo2-1B OLMo2-7B 106 0.718 $4.5 \times 10^{-18}$ OLMo2-1B OLMo2-13B 105 0.721 $4.0 \times 10^{-18}$ OLMo2-7B OLMo2-13B 104 0.934 $2.0 \times 10^{-47}$ *Within-family (Pythia)* Pythia-1.4B Pythia-410M 96 0.824 $6.1 \times 10^{-25}$ Pythia-1.4B Pythia-12B 98 0.792 $2.9 \times 10^{-22}$ Pythia-410M Pythia-12B 100 0.689 $2.3 \times 10^{-15}$ *Within-family (LLM360)* Amber Crystal 102 0.823 $2.7 \times 10^{-26}$ *Cross-family (OLMo-2 $\leftrightarrow$ OLMo-3)* OLMo2-1B OLMo3-7B 106 0.743 $7.9 \times 10^{-20}$ OLMo2-7B OLMo3-7B 105 0.961 $3.5 \times 10^{-59}$ OLMo2-13B OLMo3-7B 104 0.953 $8.3 \times 10^{-55}$ *Cross-family (OLMo-2 $\leftrightarrow$ LLM360)* Amber OLMo2-1B 107 0.785 $1.6 \times 10^{-23}$ Amber OLMo2-7B 106 0.877 $6.2 \times 10^{-35}$ Amber OLMo2-13B 105 0.875 $3.0 \times 10^{-34}$ Crystal OLMo2-1B 102 0.695 $5.0 \times 10^{-16}$ Crystal OLMo2-7B 101 0.838 $8.8 \times 10^{-28}$ Crystal OLMo2-13B 100 0.868 $1.5 \times 10^{-31}$ *Cross-family (OLMo-3 $\leftrightarrow$ LLM360)* Amber OLMo3-7B 106 0.877 $6.5 \times 10^{-35}$ Crystal OLMo3-7B 102 0.853 $6.1 \times 10^{-30}$ *Cross-family (OLMo-2 $\leftrightarrow$ Pythia)* Pythia-1.4B OLMo2-1B 98 0.883 $2.3 \times 10^{-33}$ Pythia-1.4B OLMo2-7B 97 0.693 $3.6 \times 10^{-15}$ Pythia-1.4B OLMo2-13B 97 0.669 $7.3 \times 10^{-14}$ Pythia-410M OLMo2-1B 100 0.779 $1.4 \times 10^{-21}$ Pythia-410M OLMo2-7B 99 0.686 $4.4 \times 10^{-15}$ Pythia-410M OLMo2-13B 98 0.629 $4.2 \times 10^{-12}$ Pythia-12B OLMo2-1B 102 0.777 $7.4 \times 10^{-22}$ Pythia-12B OLMo2-7B 101 0.786 $2.1 \times 10^{-22}$ Pythia-12B OLMo2-13B 100 0.843 $4.4 \times 10^{-28}$ *Cross-family (OLMo-3 $\leftrightarrow$ Pythia)* Pythia-1.4B OLMo3-7B 97 0.752 $7.3 \times 10^{-19}$ Pythia-410M OLMo3-7B 99 0.689 $3.1 \times 10^{-15}$ Pythia-12B OLMo3-7B 101 0.839 $5.7 \times 10^{-28}$ *Cross-family (LLM360 $\leftrightarrow$ Pythia)* Amber Pythia-1.4B 98 0.805 $1.7 \times 10^{-23}$ Amber Pythia-410M 100 0.758 $7.4 \times 10^{-20}$ Amber Pythia-12B 102 0.889 $1.3 \times 10^{-35}$ Crystal Pythia-1.4B 93 0.703 $4.2 \times 10^{-15}$ Crystal Pythia-410M 96 0.599 $1.1 \times 10^{-10}$ Crystal Pythia-12B 97 0.830 $7.1 \times 10^{-26}$ : Pairwise Spearman rank correlations: Absolute threshold ($\theta = 0.8$, stable for 3 consecutive checkpoints). Mean $\rho = 0.790$. ::: ```{=latex} \small ``` ::: {#tab:spearman_relmax05} **Model A** **Model B** **$n$ tasks** **$\rho$** **$p$** -------------------------------------------------- ------------- --------------- ------------ ----------------------- *Within-family (OLMo-2)* OLMo2-1B OLMo2-7B 106 0.579 $8.2 \times 10^{-11}$ OLMo2-1B OLMo2-13B 105 0.433 $3.9 \times 10^{-6}$ OLMo2-7B OLMo2-13B 104 0.563 $4.8 \times 10^{-10}$ *Within-family (Pythia)* Pythia-1.4B Pythia-410M 96 0.866 $5.3 \times 10^{-30}$ Pythia-1.4B Pythia-12B 98 0.828 $8.1 \times 10^{-26}$ Pythia-410M Pythia-12B 100 0.748 $3.5 \times 10^{-19}$ *Within-family (LLM360)* Amber Crystal 102 0.489 $1.8 \times 10^{-7}$ *Cross-family (OLMo-2 $\leftrightarrow$ OLMo-3)* OLMo2-1B OLMo3-7B 106 0.588 $3.5 \times 10^{-11}$ OLMo2-7B OLMo3-7B 105 0.702 $7.1 \times 10^{-17}$ OLMo2-13B OLMo3-7B 104 0.659 $2.7 \times 10^{-14}$ *Cross-family (OLMo-2 $\leftrightarrow$ LLM360)* Amber OLMo2-1B 107 0.587 $3.0 \times 10^{-11}$ Amber OLMo2-7B 106 0.516 $1.5 \times 10^{-8}$ Amber OLMo2-13B 105 0.279 $3.9 \times 10^{-3}$ Crystal OLMo2-1B 102 0.743 $3.9 \times 10^{-19}$ Crystal OLMo2-7B 101 0.668 $2.3 \times 10^{-14}$ Crystal OLMo2-13B 100 0.571 $5.5 \times 10^{-10}$ *Cross-family (OLMo-3 $\leftrightarrow$ LLM360)* Amber OLMo3-7B 106 0.513 $1.9 \times 10^{-8}$ Crystal OLMo3-7B 102 0.764 $8.9 \times 10^{-21}$ *Cross-family (OLMo-2 $\leftrightarrow$ Pythia)* Pythia-1.4B OLMo2-1B 98 0.492 $2.7 \times 10^{-7}$ Pythia-1.4B OLMo2-7B 97 0.476 $8.6 \times 10^{-7}$ Pythia-1.4B OLMo2-13B 97 0.085 $0.41$ Pythia-410M OLMo2-1B 100 0.445 $3.5 \times 10^{-6}$ Pythia-410M OLMo2-7B 99 0.452 $2.6 \times 10^{-6}$ Pythia-410M OLMo2-13B 98 0.129 $0.21$ Pythia-12B OLMo2-1B 102 0.511 $4.1 \times 10^{-8}$ Pythia-12B OLMo2-7B 101 0.531 $1.1 \times 10^{-8}$ Pythia-12B OLMo2-13B 100 0.304 $2.1 \times 10^{-3}$ *Cross-family (OLMo-3 $\leftrightarrow$ Pythia)* Pythia-1.4B OLMo3-7B 97 0.314 $1.8 \times 10^{-3}$ Pythia-410M OLMo3-7B 99 0.355 $3.1 \times 10^{-4}$ Pythia-12B OLMo3-7B 101 0.460 $1.3 \times 10^{-6}$ *Cross-family (LLM360 $\leftrightarrow$ Pythia)* Amber Pythia-1.4B 98 0.763 $7.3 \times 10^{-20}$ Amber Pythia-410M 100 0.702 $4.0 \times 10^{-16}$ Amber Pythia-12B 102 0.753 $6.9 \times 10^{-20}$ Crystal Pythia-1.4B 93 0.381 $1.6 \times 10^{-4}$ Crystal Pythia-410M 96 0.341 $6.6 \times 10^{-4}$ Crystal Pythia-12B 97 0.409 $3.1 \times 10^{-5}$ : Pairwise Spearman rank correlations: Relative threshold ($\alpha = 0.5$, fraction of max performance). Mean $\rho = 0.528$. ::: ```{=latex} \small ``` ::: {#tab:spearman_relmax08} **Model A** **Model B** **$n$ tasks** **$\rho$** **$p$** -------------------------------------------------- ------------- --------------- ------------ ----------------------- *Within-family (OLMo-2)* OLMo2-1B OLMo2-7B 106 0.491 $8.9 \times 10^{-8}$ OLMo2-1B OLMo2-13B 105 0.359 $1.7 \times 10^{-4}$ OLMo2-7B OLMo2-13B 104 0.707 $4.6 \times 10^{-17}$ *Within-family (Pythia)* Pythia-1.4B Pythia-410M 96 0.716 $2.3 \times 10^{-16}$ Pythia-1.4B Pythia-12B 98 0.547 $5.5 \times 10^{-9}$ Pythia-410M Pythia-12B 100 0.521 $2.7 \times 10^{-8}$ *Within-family (LLM360)* Amber Crystal 102 0.632 $1.0 \times 10^{-12}$ *Cross-family (OLMo-2 $\leftrightarrow$ OLMo-3)* OLMo2-1B OLMo3-7B 106 0.498 $5.7 \times 10^{-8}$ OLMo2-7B OLMo3-7B 105 0.773 $4.8 \times 10^{-22}$ OLMo2-13B OLMo3-7B 104 0.698 $1.8 \times 10^{-16}$ *Cross-family (OLMo-2 $\leftrightarrow$ LLM360)* Amber OLMo2-1B 107 0.556 $5.3 \times 10^{-10}$ Amber OLMo2-7B 106 0.612 $3.0 \times 10^{-12}$ Amber OLMo2-13B 105 0.544 $2.1 \times 10^{-9}$ Crystal OLMo2-1B 102 0.634 $8.8 \times 10^{-13}$ Crystal OLMo2-7B 101 0.714 $5.0 \times 10^{-17}$ Crystal OLMo2-13B 100 0.603 $3.1 \times 10^{-11}$ *Cross-family (OLMo-3 $\leftrightarrow$ LLM360)* Amber OLMo3-7B 106 0.590 $2.8 \times 10^{-11}$ Crystal OLMo3-7B 102 0.674 $8.6 \times 10^{-15}$ *Cross-family (OLMo-2 $\leftrightarrow$ Pythia)* Pythia-1.4B OLMo2-1B 98 0.580 $4.0 \times 10^{-10}$ Pythia-1.4B OLMo2-7B 97 0.286 $4.5 \times 10^{-3}$ Pythia-1.4B OLMo2-13B 97 0.077 $0.45$ Pythia-410M OLMo2-1B 100 0.502 $1.0 \times 10^{-7}$ Pythia-410M OLMo2-7B 99 0.309 $1.9 \times 10^{-3}$ Pythia-410M OLMo2-13B 98 0.159 $0.12$ Pythia-12B OLMo2-1B 102 0.523 $1.7 \times 10^{-8}$ Pythia-12B OLMo2-7B 101 0.527 $1.5 \times 10^{-8}$ Pythia-12B OLMo2-13B 100 0.383 $8.3 \times 10^{-5}$ *Cross-family (OLMo-3 $\leftrightarrow$ Pythia)* Pythia-1.4B OLMo3-7B 97 0.177 $0.08$ Pythia-410M OLMo3-7B 99 0.298 $2.8 \times 10^{-3}$ Pythia-12B OLMo3-7B 101 0.436 $5.1 \times 10^{-6}$ *Cross-family (LLM360 $\leftrightarrow$ Pythia)* Amber Pythia-1.4B 98 0.517 $4.9 \times 10^{-8}$ Amber Pythia-410M 100 0.437 $5.5 \times 10^{-6}$ Amber Pythia-12B 102 0.622 $3.0 \times 10^{-12}$ Crystal Pythia-1.4B 93 0.399 $7.3 \times 10^{-5}$ Crystal Pythia-410M 96 0.376 $1.6 \times 10^{-4}$ Crystal Pythia-12B 97 0.512 $8.5 \times 10^{-8}$ : Pairwise Spearman rank correlations: Relative threshold ($\alpha = 0.8$, fraction of max performance). Mean $\rho = 0.500$. ::: ```{=latex} \newpage ``` # Function vector hyperparameters {#appendix:fv_hyperparams} `\autoref{tab:fv-hyperparams}`{=latex} shows the hyperparameters selected for each model's task representation. Hyperparameters (representation type -- between a fixed set of heads and a full residual stream, layers, and number of heads) were chosen via a three-criterion search over candidate configurations. A fixed calibration set of elemental and composite tasks was used, and three criteria were considered: (1) *within-task consistency*, measured as split-half cosine similarity between FVs extracted from random partitions of correct examples; (2) *inter-task discriminability*, the ratio of within-task to between-task cosine similarity; and (3) *compositional structure*, measured as the cosine similarity between each compositional task's FV and its least-squares reconstruction from component FVs. Final selection used a rank-sum policy over these three criteria, with ties broken by raw metric values. Note that in all cases, only correct examples were used when constructing the final function vectors. ```{=latex} \small ``` ::: {#tab:fv-hyperparams} **Model** **Representation** **Layer** ***k* heads** **$\sigma$** **$\lambda$** -------------------------------------- -------------------- ----------- --------------- -------------- --------------- *LLM360* `amber` hidden states 21 --- 6.02568 0.0001 `\addlinespace`{=latex}`crystal` hidden states 8 --- 6.25822 0.0001 `\addlinespace`{=latex}*Pythia* `pythia_410m` hidden states 3 --- 0.33991 0.001 `\addlinespace`{=latex}`pythia_1.4b` hidden states 12 --- 5.93639 0.001 `\addlinespace`{=latex}`pythia_12b` hidden states 9 --- 4.02777 0.0001 `\addlinespace`{=latex}*OLMo-2* `olmo2_1b` hidden states 8 --- 3.46810 0.0001 `\addlinespace`{=latex}`olmo2_7b` hidden states 16 --- 1.05641 0.005 `\addlinespace`{=latex}`olmo2_13b` cie_heads 10 15 0.96582 0.005 `\addlinespace`{=latex}*OLMo-3* `olmo3_7b` hidden states 16 --- 4.37314 0.001 : Function vector hyperparameters selected per model. The full residual stream was chosen as the representation for all models besides OLMo2-13B, in which the top 10 heads by causal indirect effect in layer 10 were chosen. $\sigma$ and $\lambda$ are the parameters used in ridge regression. ::: # All Held-out trajectory predictions {#appendix:all_held_out_preds} Figures `\ref{fig:held_out_pred_pythia_410m}`{=latex} -- `\ref{fig:held_out_pred_olmo2_13b}`{=latex} show the results of the leave-one-out prediction setup for each compositional task. ![Predictions of held-out compositional tasks for Pythia-410M.](Figures/all_26_tasks_grid_pythia_410m.png){#fig:held_out_pred_pythia_410m width="0.95\\linewidth"} ![Predictions of held-out compositional tasks for OLMo2-1B.](Figures/all_26_tasks_grid_olmo2_1b.png){#fig:held_out_pred_olmo2_1b width="0.95\\linewidth"} ![Predictions of held-out compositional tasks for Pythia-1.4B.](Figures/all_26_tasks_grid_pythia_1_4b.png){#fig:held_out_pred_pythia_1_4b width="0.95\\linewidth"} ![Predictions of held-out compositional tasks for OLMo2-7B.](Figures/all_26_tasks_grid_olmo2_7b.png){#fig:held_out_pred_olmo2_7b width="0.95\\linewidth"} ![Predictions of held-out compositional tasks for OLMo3-7B.](Figures/all_26_tasks_grid_olmo3_7b.png){#fig:held_out_pred_olmo3_7b width="0.95\\linewidth"} ![Predictions of held-out compositional tasks for Amber.](Figures/all_26_tasks_grid_amber.png){#fig:held_out_pred_amber width="0.95\\linewidth"} ![Predictions of held-out compositional tasks for CrystalCoder.](Figures/all_26_tasks_grid_crystal.png){#fig:held_out_pred_crystal width="0.95\\linewidth"} ![Predictions of held-out compositional tasks for Pythia-12B.](Figures/all_26_tasks_grid_pythia_12b.png){#fig:held_out_pred_pythia_12b width="0.95\\linewidth"} ![Predictions of held-out compositional tasks for OLMo2-13B.](Figures/all_26_tasks_grid_olmo2_13b.png){#fig:held_out_pred_olmo2_13b width="0.95\\linewidth"} [^1]: Data and code available at [^2]: i.e. their emergence time is considered to be 1001B tokens for a 1T training run [^3]: in practice, this ended up being compositions of the `reverse` task as they were usually 0 throughout training.