An invariant reasoning subspace stabilizes latent chain-of-thought refinement

Malarkkan et al. introduce TILR: SVD of per-step contrastive differences between a later- and earlier-checkpoint of the same GPT-2-base Coconut latent-reasoning backbone, keeping the rank r_0.90 explaining >=90% of contrastive variance [malarkkan-etal-2026-invariant-reasoning-directions-latent-trajectories] Across 6 benchmarks r_0.90 ranges from 1 (near-rank-one, coincident with input-PCA) to 27-34 (genuinely refinement-specific structure surviving input-PCA regression) [malarkkan-etal-2026-invariant-reasoning-directions-latent-trajectories] At inference only the projection onto this fixed subspace is applied, scaled by a norm-based reliability gate giving a formal no-harm guarantee [malarkkan-etal-2026-invariant-reasoning-directions-latent-trajectories] Against random, input-PCA, trajectory-PCA and orthogonal-knockout controls it improves accuracy on all 6 benchmarks (mean +2.1%), cuts paraphrase-induced trajectory variance ~39%, and reduces cross-checkpoint variance 53-74%, replicated on GPT-2-Medium and Qwen2.5-Math-1.5B [malarkkan-etal-2026-invariant-reasoning-directions-latent-trajectories]