Does extended thinking actually improve reasoning or just increase variance?

The mechanistic explanation for why extended thinking initially improves then degrades: it acts as a variance dial on the output distribution, not as a reasoning quality dial. As thinking tokens increase, the model's output distribution broadens. This initially helps because broader coverage increases the chance of landing on the correct answer. But beyond a point, the distribution becomes so diffuse it overshoots the reward peak — the "dilution effect" — and accuracy drops.

Formally: there's a competition between a coverage effect (broadening variance helps overlap with the reward region) and a dilution effect (too broad places mass far from the reward). This predicts the non-monotonic curve exactly.

The critical insight is that the apparent gains aren't improvements in reasoning capability — they're improvements in sampling coverage. A model that draws from a wider distribution might hit the right answer more often even if its actual reasoning hasn't improved. This is an illusion because it conflates variance with competence.

This suggests that test-time scaling through extended thinking is not an effective use of inference budget, which is why Why does parallel reasoning outperform single chain thinking? — it explicitly controls variance through independent sampling rather than letting it inflate through trace extension.

Theoretical grounding from robustness analysis: The CoT robustness bounds paper (analyzing perturbation propagation through reasoning chains) adds a theoretical dimension. Under Lipschitz continuity assumptions, longer CoT chains do dampen input perturbations — but never fully eliminate them. Even an infinite chain leaves a non-zero robustness bound. For the Linear Self-Attention model (a simplified transformer), CoT robustness depends on the norm of input embeddings and hidden state vectors: higher norm → less sensitivity to perturbations. This means variance inflation at long chains is not just an empirical finding but has a theoretical bound: you get diminishing returns on perturbation resistance, and the residual sensitivity is determined by model-level factors (embedding norms), not just chain length. The practical upshot: there is a finite chain length beyond which extending the chain provides no additional robustness benefit — which precisely defines the threshold observed empirically.