Mikhail Belkin

Language models trained on natural text learn to represent numbers using periodic features with dominant periods at $T=2, 5, 10$. In this paper, we identify a two-tiered hierarchy of these features: while Transformers, Linear RNNs, LSTMs, and classical word embeddings trained in different ways all learn features that have period-$T$ spikes in the Fourier domain, only some learn *geometrically separable* features that can be used to linearly classify a number mod-$T$. To explain this incongruity, we prove that Fourier domain sparsity is necessary but not sufficient for mod-$T$ geometric separability. Empirically, we investigate when model training yields geometrically separable features, finding that the data, architecture, optimizer, and tokenizer all play key roles. In particular, we identify two different routes through which models can acquire geometrically separable features: they can learn them from complementary co-occurrence signals in general language data, including text-number co-occurrence and cross-number interaction, or from multi-token (but not single-token) addition problems. Overall, our results highlight the phenomenon of *convergent evolution* in feature learning: A diverse range of models learn similar features from different training signals.

Large language models (LLMs) can produce chains of thought (CoT) that do not accurately reflect the actual factors driving their answers. In multiple-choice settings with an injected hint favoring a particular option, models may shift their final answer toward the hinted option and produce a CoT that rationalizes the response without acknowledging the hint - an instance of motivated reasoning. We study this phenomenon across multiple LLM families and datasets demonstrating that motivated reasoning can be identified by probing internal activations even in cases when it cannot be easily determined from CoT. Using supervised probes trained on the model's residual stream, we show that (i) pre-generation probes, applied before any CoT tokens are generated, predict motivated reasoning as well as a LLM-based CoT monitor that accesses the full CoT trace, and (ii) post-generation probes, applied after CoT generation, outperform the same monitor. Together, these results show that motivated reasoning is detected more reliably from internal representations than from CoT monitoring. Moreover, pre-generation probing can flag motivated behavior early, potentially avoiding unnecessary generation.