Do Geometry-Aware Positional Encodings Help Transformers in Spatial Imperfect-Information Games?

Understanding the geometric structure of internal representations is a central goal of mechanistic interpretability. Prior work has shown that transformers trained on sequences generated by hidden Markov models encode probabilistic belief states as simplex-shaped geometries in their residual stream, with vertices corresponding to latent generative states. Whether large language models trained on naturalistic text develop analogous geometric representations remains an open question. We introduce a pipeline for discovering candidate simplex-structured subspaces in transformer representations, combining sparse autoencoders (SAEs), $k$-subspace clustering of SAE features, and simplex fitting using AANet. We validate the pipeline on a transformer trained on a multipartite hidden Markov model with known belief-state geometry. Applied to Gemma-2-9B, we identify 13 priority clusters exhibiting candidate simplex geometry ($K \geq 3$). A key challenge is distinguishing genuine belief-state encoding from tiling artifacts: latents can span a simplex-shaped subspace without the mixture coordinates carrying predictive signal beyond any individual feature. We therefore adopt barycentric prediction as our primary discriminating test. Among the 13 priority clusters, 3 exhibit a highly significant advantage on near-vertex samples (Wilcoxon $p < 10^{-14}$) and 4 on simplex-interior samples. Together 5 distinct real clusters pass at least one split, while no null cluster passes either. One cluster, 768_596, additionally achieves the highest causal steering score in the dataset. This is the only case where passive prediction and active intervention converge. We present these findings as preliminary evidence that genuine belief-like geometry exists in Gemma-2-9B's representation space, and identify the structured evaluation that would be required to confirm this interpretation.

Neural language models process sequences of words, but the mathematical operations inside them are insensitive to the order in which words appear. Positional encodings are the component added to remedy this. Despite their importance, positional encodings have been designed largely by trial and error, without a mathematical theory of what they ought to do. This paper develops such a theory. Four results are established. First, any Transformer without a positional signal cannot solve any task sensitive to word order (Necessity Theorem). Second, training assigns distinct vector representations to distinct sequence positions at every global minimiser, under mild and verifiable conditions (Positional Separation Theorem). Third, the best achievable approximation to an information-optimal encoding is constructed via classical multidimensional scaling (MDS) on the Hellinger distance between positional distributions; the quality of any encoding is measured by a single number, the stress (Proposition 5, Algorithm 1). Fourth, the optimal encoding has effective rank r = rank(B) <= n-1 and can be represented with r(n+d) parameters instead of nd (minimal parametrisation result). Appendix A develops a proof of the Monotonicity Conjecture within the Neural Tangent Kernel (NTK) regime for masked language modelling (MLM) losses, sequence classification losses, and general losses satisfying a positional sufficiency condition, through five lemmas. Experiments on SST-2 and IMDB with BERT-base confirm the theoretical predictions and reveal that Attention with Linear Biases (ALiBi) achieves much lower stress than the sinusoidal encoding and Rotary Position Embedding (RoPE), consistent with a rank-1 interpretation of the MDS encoding under approximate shift-equivariance.

Neural networks gain capabilities during training, but the internal changes that precede capability acquisition are not well understood. In particular, the relationship between geometric change and behavioral change, and the effect of task difficulty and model scale on that relationship, is unclear. We track geometric measures and linear probes across six transformer sizes (405K--151M parameters), eight algorithmic tasks (144 task$\times$level$\times$model combinations), and three Pythia language models (160M--2.8B). Across all settings, representations first collapse to a low-dimensional state, then recover, and only then does behavioral performance improve. Linear probes show that the model's hidden states already contain task-relevant information before the model can act on it. The collapse floor is task-specific, the collapse propagates top-down through the network, and of the geometric measures tested, only \rankme reliably precedes capability acquisition for hard tasks. Whether this precursor is detectable depends on task difficulty relative to model capacity. For hard tasks, there is a clear gap: geometry changes first, behavior follows. For easy tasks, the model learns so quickly that both happen simultaneously and no precursor is detectable. On Pythia-2.8B, a logical deduction task that is genuinely hard for the model shows a precursor gap of ${\sim}$49K training steps, while easy benchmarks show none. This suggests that geometric patterns observed in small proxy models can persist at larger scale when the task remains difficult relative to model capacity.