Reliable Chain-of-Thought via Prefix Consistency — part 2/8

Cleaned-text code points [15693, 31685). Bibliography omitted here; source.md includes it.

We use the power family w ( n ) ​ ( c ) = c n w^{(n)}(c)=c^{n} for n ∈ { 1 , 2 , 3 } n\in\{1,2,3\} , denoted PC-linear, PC-quadratic, and PC-cubic, where the “PC” prefix abbreviates prefix consistency. Under K = 1 K=1 , c i ​ ( a ) ∈ { 0 , 1 / 2 , 1 } c_{i}(a)\in\{0,1/2,1\} . Thus, a candidate that is reproduced under regeneration ( c = 1 c=1 ) receives weight 1 1 , while a candidate that appears in only one of the two traces ( c = 1 / 2 c=1/2 ) receives weight 1 / 2 n 1/2^{n} . The weight ratio between a reproduced candidate and a single-trace one is therefore 2 n : 1 2^{n}:1 , that is, 2 : 1 2:1 for linear, 4 : 1 4:1 for quadratic, and 8 : 1 8:1 for cubic. The larger n n is, the more pronounced the relative weight given to reproduced answers.

##### Cost-equivalent evaluation.

We measure inference cost by the total number of generated tokens, treating each generated token as equally expensive, and log-probability access as free. For each model and benchmark, we first generate N = 128 N{=}128 initial samples per problem ( N = 64 N{=}64 for Ministral3-14B), from which all methods sample with replacement under a common token budget B B . See Appendix G for the pool construction, trial design, and confidence-interval definition.

Table 3 reports accuracy under fixed token budgets (250k, 1M, and 5M tokens) across the three models and four benchmarks. At 1M tokens, prefix consistency matches or exceeds all baselines on the more difficult benchmarks (FrontierScience-Olympiad, HMMT Feb 2026), while the advantage is smaller on AIME 2025 where Standard MV already achieves high accuracy. The improvements are consistent across weighting functions (PC-linear, PC-quadratic, and PC-cubic). PC-cubic provides the greatest improvement for the most difficult problems. Per-model tables with the full set of baselines (DeepConf variants, Response probability, verbalized confidence, P(True)) for all five models, including the two not shown above (GPT-OSS-20B, Nemotron2-9B), are reported in Appendix D.3 .

### 4.3 Token Efficiency

Section 4.2 reported accuracy under fixed budget constraints. We next compare how many tokens each method needs to reach the same target accuracy.

Table 4 shows the token-efficiency ratio B method / B MV B_{\mathrm{method}}/B_{\mathrm{MV}} , where B X B_{X} is the budget method X X needs to reach the target accuracy Pass@1 + α × +\,\alpha\times (Standard MV plateau − - Pass@1) for α ∈ { 75 % , 90 % , 99 % } \alpha\in\{75\%,90\%,99\%\} . The Standard MV plateau is Standard MV’s bootstrap-saturated accuracy on the N N -sample pool (Appendix G.3 ), so α \alpha interpolates between Pass@1 ( α = 0 \alpha{=}0 ) and this plateau ( α = 1 \alpha{=}1 ). A ratio < 1 <1 means X X is more cost-efficient than Standard MV at the target; e.g., 0.05 × 0.05\times corresponds to the 21 × 21\times saving in Figure 1 . The headline numbers in this paper (median 4.6 × 4.6\times , up to 21 × 21\times vs. Standard MV, up to 10 × 10\times vs. AC sweep) are computed at α = 99 % \alpha{=}99\% across all 20 20 (model, benchmark) cells: Table 4 together with Table 12 (GPT-OSS-20B) and Table 14 (Nemotron2-9B) in Appendix D.3 .

PC-cubic is more cost-efficient than Standard MV on 11 11 out of 12 12 model and benchmark settings at α = 99 % \alpha{=}99\% . The strongest savings are 0.05 × 0.05\times on both FrontierScience-Olympiad with GPT-OSS-120B and AIME 2025 with Ministral3-14B, and 0.08 × 0.08\times on FrontierScience-Olympiad with Ministral3-14B. All three correspond to cells with large D D (Table 1 ).

The savings track the discrimination gap D D (Table 1 ): pairs with the largest D D yield the largest reductions. PC-cubic offers little advantage over Standard MV on two cells (it underperforms on Nemotron3-30B AIME 2025 and only marginally beats Standard MV on Nemotron3-30B HMMT Feb 2026), both of which have a small Pass@1-to-plateau gap of at most .11 .11 ( .902 → .967 .902\to.967 and .708 → .810 .708\to.810 ), leaving little room above Pass@1 for any reweighting irrespective of D D . The practical penalty in this regime, where Pass@1 is close to Standard MV plateau, is correspondingly small.

##### Comparison with adaptive-stopping baselines.

PC-cubic is competitive with or better than AC sweep on most cells and outperforms ESC sweep at α = 99 % \alpha{=}99\% on every (model, benchmark) cell except Nemotron3-30B on AIME 2025, despite being a non-adaptive reweighting of the same initial pool that AC and ESC consume sequentially. The advantage is most pronounced on large- D D cells (FrontierScience-Olympiad on every model, and most Ministral3-14B benchmarks), e.g. PC-cubic at 0.05 × 0.05\times vs. AC sweep at 0.29 × 0.29\times at α = 99 % \alpha{=}99\% on GPT-OSS-120B FrontierScience-Olympiad.

AC substantially outperforms PC only on the two cells noted above where Pass@1 is close to Standard MV plateau (Nemotron3-30B on AIME 2025 and HMMT Feb 2026), since the aggregated vote on wrong answers is small and AC’s early stop alone bounds the cost. However, on 2 2 out of 12 12 (model, benchmark) cells at α = 99 % \alpha{=}99\% (marked “N/A” in Table 4 ) AC’s accuracy never reaches Standard MV plateau, because its early-stop rule terminates generation before the running accuracy reaches the α = 99 % \alpha{=}99\% target. PC-cubic reaches the target on all 12 12 .

ESC stops at the first fixed-size window of samples that all share the same answer, which is too strict on benchmarks where wrong answers are diverse, so ESC either stops well after AC or fails to stop within the budget. PC and adaptive stopping act on orthogonal axes: PC reweights votes while AC and ESC decide when to stop sampling. A hybrid that votes by PC weights and stops by the AC rule would combine AC’s cost bound on easy cells with PC’s accuracy on difficult ones (left to future work).

##### Remark on the cost accounting.

Treating log-probability access as free is implementation-dependent but holds for our vLLM setup. This favors baselines that read log-probabilities of the initial trace without generating extra tokens (Self-certainty, DeepConf, Response probability), while prefix consistency spends budget on the regeneration tokens. Even so, the PC-WMV has significant advantage. Imposing any cost for log-probability retrieval only widens the margin.

Additional results on token-efficiency ratios are reported per model in Appendix D.3 .

### 4.4 How Problem Difficulty Affects the Discrimination Gap D D

The discrimination gap D D determines where prefix consistency improves WMV (Theorem 1 ), so we now study how D D varies with problem difficulty, indexed by Pass@1.

Figure 3 plots per-problem r C r_{C} and r W r_{W} as a function of Pass@1 for three of the five models (the remaining two are in Appendix D.7.1 ), stratified by category. Across all five models, r C r_{C} rises with problem easiness, r W r_{W} depends on the model and category but only weakly on Pass@1, and D = r C − r W D=r_{C}-r_{W} inherits both. We discuss r C r_{C} and r W r_{W} in turn below.

First, r C r_{C} (solid lines) increases with Pass@1 for both categories. Pass@1 here indexes problem easiness within a fixed model, and on easier problems, the correct answer is more reliably reproduced under regeneration. Logistic generalized linear model (GLM) slopes β ⁡ ( r C ) \beta(r_{C}) on logit ​ ( r ) = β 0 + β ⋅ Pass@1 \text{logit}(r)=\beta_{0}+\beta\cdot\text{Pass@1} range from + 2.7 +2.7 to + 5.1 +5.1 across the six (model, category) pairs shown in Figure 3 , all significantly positive (cluster-bootstrap p < 0.005 p<0.005 , see Appendix D.7.1 ).

Second, r W r_{W} (dashed lines) depends more weakly on Pass@1 than r C r_{C} : | β ⁡ ( r W ) | ≤ 1.14 |\beta(r_{W})|\leq 1.14 across the six pairs, and among four out of six curves, we cannot statistically reject β = 0 \beta=0 at a 2 ​ σ 2\sigma confidence level. For the remaining two non-zero slopes, β \beta is + 1.14 +1.14 (GPT-OSS-120B Math) and − 0.75 -0.75 (Ministral3-14B Math), with opposite signs, both smaller in magnitude than the smallest r C r_{C} slope. The level of r W r_{W} varies by category, sitting at ∼ {\sim} 15–40% on Science and ∼ {\sim} 25–60% on Math. Math errors may reflect internally consistent miscalculations and science errors may reflect more diffuse knowledge gaps, but this is a hypothesis: our results establish r C > r W r_{C}>r_{W} as a behavioral regularity, and a mechanistic verification (e.g., calculation-heavy vs. concept-heavy subsets) is left to future work.

The finding that D D widens on easier cells may appear to conflict with the “savings track D D ” claim of Section 4.3 , since easier cells also have a smaller gap between Pass@1 and Standard MV plateau. The two reconcile by noting that PC-WMV’s advantage over Standard MV depends on both D D and the gap above Pass@1: a large D D pays off only when there is room to reweight votes, which is why the two cells where PC-cubic offers little advantage over Standard MV are precisely the cells with Pass@1 concentrated within ∼ .10 {\sim}.10 of Standard MV plateau (Nemotron3-30B on AIME 2025 and HMMT Feb 2026).

## 5 Conclusion

We introduced prefix consistency, a reliability signal for weighted majority voting that truncates each CoT and checks whether answers reproduce under regeneration. Across benchmarks, prefix consistency is a stronger correctness predictor than existing baselines, and PC-WMV improves upon existing weighted majority voting methods under cost-equivalent comparison, especially on more difficult benchmarks. Our analysis highlights the discrimination gap D = r C − r W D=r_{C}-r_{W} as the key quantity governing when the method helps: PC-WMV is most effective when D > 0 D>0 and Pass@1 leaves a meaningful gap below Standard MV plateau. Regeneration stability is thus a practically useful test-time signal for aggregating votes, not merely a descriptive property of Chain-of-Thought.

## Appendix A Related Work

### A.1 Test-Time Scaling

##### Sequential and parallel scaling.

Test-time compute scaling comes in two paradigms. Sequential scaling extends a single reasoning chain, for example by appending continuation tokens to lengthen CoT [ Muennighoff et al., 2025 ] . Parallel scaling generates multiple independent solutions and aggregates them: self-consistency is the canonical instance, while tree search [ Yao et al., 2023 ] branches over candidate steps and process-reward verifiers rerank parallel samples [ Lightman et al., 2024 ] . Parashar et al. [2025] benchmarked a range of inference-time techniques across reasoning and planning tasks and reported that no single method is consistently best. Our method sits between the two: it uses parallel samples but introduces a structured perturbation (truncation + regeneration) that probes the robustness of each sample’s reasoning. Beyond aggregation, the same perturbation could serve as a quality estimate for partial reasoning states inside richer search procedures such as Graph-of-Thoughts [ Besta et al., 2024 ] , which extends Tree-of-Thoughts to a directed graph with refinement and aggregation operations.

##### Majority voting.

Wang et al. [2023b] introduced self-consistency for CoT reasoning: sample multiple reasoning traces and take the majority vote over extracted answers. A line of work targets the cost of self-consistency by stopping early. Aggarwal et al. [2023] introduced Adaptive Consistency, which evaluates a Beta-binomial posterior over the running top-1 and top-2 answer counts and stops once the posterior margin of the top answer crosses a threshold. Li et al. [2024] introduced Early-Stopping Self-Consistency, which inspects fixed-size windows of samples for unanimity and stops at the first unanimous window. Sharma and Chopra [2025] showed that sequential, entropy-aware voting can outperform parallel self-consistency at matched compute, highlighting the importance of cost-equivalent comparisons. Komiyama et al. [2026] analyzed the asymptotic behavior of majority voting (best-of- ∞ \infty ) and proposed a Bayesian nonparametric stopping rule. Our work is complementary: we assess the quality of each vote through regeneration weighting, which is compatible with the adaptive stopping rules above.

##### Confidence-based weighting.

Kang et al. [2025] proposed self-certainty , a trace-level quality score defined as the mean KL divergence of the per-token output distribution from uniform, and used it to rerank best-of- N N candidates. Fu et al. [2026] then extended self-certainty. In addition to the global trace mean (equivalent to self-certainty), they introduced a sliding-window group confidence and proposed the use of their bottom 10% (bottom-10% group confidence) and the mean over the final tokens of a trace (tail confidence) as alternative trace-level scores for confidence-weighted majority voting. Taubenfeld et al. [2025] demonstrated that combining self-assessed confidence (most effectively via P(True), the model’s own probability that its answer is correct) with self-consistency improves accuracy, though this still relies on the model’s introspective calibration. Xiong et al. [2024] studied whether LLMs can express uncertainty and found that verbalized confidence is often poorly calibrated. Scalena et al. [2025] used per-token entropy to allocate compute adaptively during generation, a token-level compute-allocation complement to our sample-level reliability estimation. All of these signals rely on the model’s own log-probabilities or verbalized self-assessment. We show that such confidence signals are unreliable on difficult problems and propose an alternative based on prefix self-reproduction that requires neither.

##### Prefix continuation as a per-sample signal.

Several recent methods regenerated from a truncated reasoning prefix or extracted a structural signal from a CoT and used it to select or weight samples. Hammoud et al. [2025] proposed SubthoughtReasoner, which segments a single greedy trace into sequential subthoughts at linguistic cues and takes the mode over per-subthought regenerations as the final answer (single-trace refinement, +13pp on AIME 2024 vs. the same trace’s last answer). Faria and Smith [2025] also regenerated from intermediate states, but for test-time alignment via Metropolis-Hastings (building on quality-aware machine-translation sampling [ Faria et al., 2024 ] ) rather than answer aggregation. Zhu et al. [2024] iteratively promoted a high-confidence prefix among N N initial partial answers via Beta-gated agreement on the extracted-answer distribution and resampled subsequent branches conditioned on it (up to 40.5% latency reduction at matched accuracy). Jindal et al. [2026] clustered the first 256 tokens of N N initial samples and expanded only the dominant cluster (up to 60% token reduction at matched accuracy). Feng et al. [2025] scored each of N = 64 N{=}64 traces by its Failed-Step Fraction (the proportion of abandoned reasoning branches) and selected the lowest-FSF trace, yielding 5% to 13% improvements over random selection on AIME 2025. The latter three collapsed N N samples onto a subset (a single best prefix, the dominant cluster, or one selected trace) and therefore inherited the failure mode of Standard MV when the subset misses the correct answer: Jindal et al. [2026] reported a 10pp drop on AIME 2025 (76.7% to 66.7%) where the largest prefix cluster need not contain the correct answer. PC-WMV moves in the converse direction: it keeps all N N samples and reweights the vote by each sample’s prefix self-reproduction, so a minority sample whose prefix continuation reproduces its own answer is upweighted rather than discarded. The two directions are complementary, since prefix-cluster pruning could feed PC-WMV reweighting on the surviving traces. Under cost-equivalent comparison, SubthoughtReasoner does not consistently improve over Standard MV in our experiments, filling the comparison left open by Hammoud et al. [2025] .