Assign and Add: A Mechanistic Study of Compositional Arithmetic

Large language models are able to compose skills in order to perform complex tasks, many of which might not have been seen during training. The details of how exactly this composition occurs remain elusive. In this paper, we study a mechanism for compositional generalization in transformers by considering a simple controlled setting involving variable assignment and modular addition. By partitioning our training data into disjoint sets, we observe that small transformers are able to generalize to previously unseen combinations of variables and numbers. Our mechanistic analysis shows that the same ``modular addition'' MLP module is used whether the inputs are given directly or indirectly through a separate variable assignment mechanism. We also analyze the training dynamics from an empirical lens, which reveals three phases of learning: first, modular addition is learned, then the structure required for variable assignment, and finally a refinement phase where the model generalizes to some hard sequences not seen in training. Finally, we provide a theoretical framework to explain how compositionality emerges from training dynamics. These results suggest that compositional generalization can be a natural consequence of the compositionality of internal mechanisms in~transformers.

Asymmetric Scaling Laws from Sparse Features

We introduce a model for neural scaling laws under sparse activations. In the model, test loss is often dominated by rare coordinates that are never observed in the training input. This mechanism induces a novel bottleneck absent from dense models. We derive the asymptotic population loss in both the underparameterized and overparameterized regimes, and show that the loss exhibits a double-descent peak near the interpolation threshold -- where the number of parameters is just sufficient to fit the training data -- resulting in a loss curve governed by two distinct scaling exponents -- one for the overparameterized regime and one for the underparameterized regime -- with a gap determined by the degree of sparsity. Additionally, we derive a compute-optimal frontier that favors increasing dataset size over model capacity under fixed compute budgets. We also analyze gradient-descent dynamics and identify a scaling law for the probability that fixed-step gradient descent becomes unstable. We further show that the sparsity-induced effect persists under nonlinear activations.

Memory Inception: Latent-Space KV Cache Manipulation for Steering LLMs

Steering large language models (LLMs) is usually done by either instruction prompting or activation steering. Prompting often gives strong control, but caches guidance tokens at every layer and can clutter long interactions; activation steering is compact but typically weaker and does not support large structured reminders. We introduce memory inception (MI), a training-free method that steers in latent attention space by inserting text-derived key-value (KV) banks only at selected layers. Rather than materializing reminder content throughout the prompt cache, MI treats steering as selective KV allocation, injecting latent slots only where the model routes to them. On matched personality-steering tasks, MI gives the best overall control--drift trade-off, remaining competitive with prompting while consistently outperforming CAA. On updateable guidance, MI supports mid-conversation behavior shifts without rewriting the visible transcript, achieving the highest post-shift alignment on Qwen3. On structured reasoning, MI outperforms visible prompting on HARDMath and PHYSICS (10/12 subject$\times$mode cells), serving as proxies for structured reasoning in verifiable domains, while cutting content-matched KV storage by up to 118$\times$. These results position MI as a powerful steering method when guidance is persistent, structured, or expensive to keep in the visible transcript.

Spectral Lens: Activation and Gradient Spectra as Diagnostics of LLM Optimization

Training loss and throughput can hide distinct internal representation in language-model training. To examine these hidden mechanics, we use spectral measurements as practical and operational diagnostics. Using a controlled family of decoder-only models adapted from the modded NanoGPT codebase, we introduce an empirical protocol based on activation covariance and per-sample gradient SVD spectra. This dual-view reveals three empirical findings and one mechanistic explanation. First, batch size acts as a latent determinant of representation geometry: runs that reach equal loss settle into systematically distinct activation spectra. Second, the activation covariance tail measured early in training reliably forecasts downstream token efficiency. Third, movement of the activation spectrum head (leading modes), together with gradient spectra, characterizes underlying learning-dynamics changes, separating learning-side architectural improvements from primarily execution-side gains. These predictive and diagnostic signals persist across the 12-, 36-, and 48-layer model tiers. Finally, a mechanistic model proves the main observations and explains how activation covariance spectra correlate with task-aligned feature learning.

(Im)possibility of Automated Hallucination Detection in Large Language Models

Is automated hallucination detection possible? In this work, we introduce a theoretical framework to analyze the feasibility of automatically detecting hallucinations produced by large language models (LLMs). Inspired by the classical Gold-Angluin framework for language identification and its recent adaptation to language generation by Kleinberg and Mullainathan, we investigate whether an algorithm, trained on examples drawn from an unknown target language $K$ (selected from a countable collection) and given access to an LLM, can reliably determine whether the LLM's outputs are correct or constitute hallucinations. First, we establish an equivalence between hallucination detection and the classical task of language identification. We prove that any hallucination detection method can be converted into a language identification method, and conversely, algorithms solving language identification can be adapted for hallucination detection. Given the inherent difficulty of language identification, this implies that hallucination detection is fundamentally impossible for most language collections if the detector is trained using only correct examples from the target language. Second, we show that the use of expert-labeled feedback, i.e., training the detector with both positive examples (correct statements) and negative examples (explicitly labeled incorrect statements), dramatically changes this conclusion. Under this enriched training regime, automated hallucination detection becomes possible for all countable language collections. These results highlight the essential role of expert-labeled examples in training hallucination detectors and provide theoretical support for feedback-based methods, such as reinforcement learning with human feedback (RLHF), which have proven critical for reliable LLM deployment.