Explain why self-consistency requires temperature > 0 and quantify the gain it produces over single pass CoT on math benchmarks.

The plan as written cannot improve over single-pass CoT. At temperature zero the language model is deterministic: given the same prompt, every run produces exactly the same tokens. Ten identical samples voted on by majority is one sample's outcome. The N=10 cost is paid; the voting benefit is zero. The fix is to raise the temperature into the range where reasoning paths actually diverge. The standard pick is temperature 0.7. At that level the model still produces high-quality reasoning, but each sample explores a slightly different path. The vote then concentrates on the answer that the most reasoning paths agree on. The gain you should expect on math benchmarks is significant. The original Wang et al. paper showed 17.9 points on GSM8K with PaLM at N=40. Reproductions on modern frontier models show 10 to 20 points on GSM8K-class reasoning benchmarks. In production, N=5 to N=10 at temperature 0.7 captures most of the lift at much lower cost than N=40. Past N=10 the marginal gain flattens fast. The reproducibility tension is real. Temperature zero is great for deterministic eval. It is incompatible with self-consistency. You have to pick one for any given route.

**Self-consistency** requires temperature greater than zero because the mechanism is voting over diverse reasoning paths, and temperature zero produces no diversity. At T=0 the decoder always picks the argmax token; N runs produce N identical sequences; voting reduces to a single sample's outcome. The majority-vote formula is straightforward: $$\text{answer} = \arg\max_y \sum_{i=1}^N \mathbb{1}[y_i = y]$$ With N independent CoT samples, each yields its own final answer `y_i`. The vote concentrates on the answer that appears most often. The asymmetry it exploits: correct answers tend to be reached via multiple reasoning paths while incorrect answers scatter. The vote picks up that signal. **Temperature recommendation.** Use 0.7 as the production default. The acceptable band is 0.5 to 0.9. Below 0.5 the diversity is too low for the vote to extract signal. Above 0.9 the reasoning quality starts to degrade. **Empirical gains.** Wang et al. 2022 reported +17.9 points on GSM8K with PaLM-540B at N=40 (56.5 to 74.4 percent). SVAMP gained 11 points; AQuA-RAT gained 12. Modern reproductions on GPT-4 class and Claude class frontier models show 10 to 20 absolute points on multi-step reasoning benchmarks at N=5 to N=10. Cost: N times the single-sample inference. The diminishing-returns curve flattens past N=10. The right production target is N=5 to N=10 unless the task is high-value enough to justify more. Now the reproducibility tension. Temperature zero is valuable for deterministic eval, where you want bit-identical output across runs. It is incompatible with self-consistency, which requires the very stochasticity reproducibility forbids. The route either accepts non-determinism for the self-consistency lift or runs single-pass CoT at T=0 for deterministic eval. You cannot have both on the same call.