A global-to-local PCA-dimension ratio tracks reasoning-trajectory untangling

Anderson defines a Global-to-Local Dimension Ratio as PCA d95 measured globally over a chain-of-thought trajectory divided by PCA d95 measured locally, with the Levina-Bickel MLE intrinsic dimension computed separately as a check [anderson-2026-geometry-of-thought] A high ratio means the trajectory fills a large ambient subspace while staying locally low-dimensional and folded; a ratio near 1 means an unfolded, roughly linear trajectory [anderson-2026-geometry-of-thought] Law-reasoning trajectories untangle roughly 10-fold, the ratio falling from 9.82 to 0.98 between Llama-3-8B-Instruct and Llama-3.1-70B-Instruct [anderson-2026-geometry-of-thought] The dimension profile is tracked along the generation-time (CoT-step) axis and compared across model scale, rather than across network depth [anderson-2026-geometry-of-thought] No causal intervention is performed [anderson-2026-geometry-of-thought]