Chain-of-thought trajectories collapse to under ten intrinsic dimensions

Ma et al. estimate intrinsic dimension with the TLE estimator (k=20) for the static vocabulary-embedding cloud (D_world) and for autoregressive last-token chain-of-thought trajectories (D_stim) across Qwen2.5, Qwen3, Gemma 3, and DeepSeek-R1-Distill-Qwen families [ma-etal-2026-constrained-inference-manifolds] Deep-layer reasoning trajectories collapse onto a manifold of fewer than ten intrinsic degrees of freedom despite thousands of ambient dimensions, while the vocabulary cloud stays near ambient dimension [ma-etal-2026-constrained-inference-manifolds] A composite reasoning-health diagnostic H = log(D_world)*V/exp(0.1*D_stim), combining an information-volume term V, correlates with benchmark scores at Spearman rho>0.9 [ma-etal-2026-constrained-inference-manifolds] Token-shuffle, non-cognitive-prompt, and truncation controls rule out trivial confounds, and no causal intervention is performed [ma-etal-2026-constrained-inference-manifolds]