The rise of Artificial Intelligence (AI) recently empowered researchers to investigate hard mathematical problems which eluded traditional approaches for decades. Y et, the use of AI in Universal Algebra (UA)--one of the fields laying the foundations of modern mathematics--is still completely unexplored.

Modern generative models demonstrate impressive capabilities, likely stemming from an ability to identify and manipulate abstract concepts underlying their training data. However, fundamental questions remain: what determines the concepts a model learns, the order in which it learns them, and its ability to manipulate those concepts? To address these questions, we propose analyzing a model's learning dynamics via a framework we call the concept space, where each axis represents an independent concept underlying the data generating process. By characterizing learning dynamics in this space, we identify how the speed at which a concept is learned, and hence the order of concept learning, is controlled by properties of the data we term concept signal. Further, we observe moments of sudden turns in the direction of a model's learning dynamics in concept space. Surprisingly, these points precisely correspond to the emergence of hidden capabilities, i.e., where latent interventions show the model possesses the capability to manipulate a concept, but these capabilities cannot yet be elicited via naive input prompting. While our results focus on synthetically defined toy datasets, we hypothesize a general claim on emergence of hidden capabilities may hold: generative models possess latent capabilities that emerge suddenly and consistently during training, though a model might not exhibit these capabilities under naive input prompting.

Many standard TTT methods train on carefully selected data from the pre-training dataset (i.e., do not add any new privileged information; Hardt & Sun, 2024; Hübotter et al., 2025), and several works studied how to optimally select data for imitation, e.g., the early seminal work of MacKay (1992) and recent extensions (Hübotter et al., 2024; Bagatella et al., 2025b). TTT has also been extended from supervised learning to reinforcement learning (Zuo et al., 2025; Bagatella et al., 2025a; Diaz-Bone et al., 2025). So far it has not been well understood why and when TTT is effective. While many different methods have been proposed for TTT, we focus here on analyzing "semi-parametric" TTT (e.g., Hardt & Sun, 2024; Hübotter et al., 2025), where a pre-trained model is fine-tuned with a supervised loss on a small neighborhood of the test point in the training data. This is different from some other methods for test-time "adaptation", which are commonly applied with distribution shifts (e.g., Wang et al., 2021; Zhang et al., 2022; Durasov et al., 2025). Basu et al. (2023) consider a similar setting to ours, but analyze it through the lens of non-parametric estimation, relying on the smoothness of the target function in the feature space Ψ.

To address these questions, we propose analyzing a model's learning

The challenge in object-based visual reasoning lies in generating concept representations that are both descriptive and distinct.