Reliable Chain-of-Thought via Prefix Consistency — part 5/8

Cleaned-text code points [62901, 78790). Bibliography omitted here; source.md includes it.
We define Oracle (Standard MV) as the accuracy of an ideal selector that, at each budget B B , always picks the correct answer from the initial-sample pool whenever one exists, and Oracle (Prefix Consistency) as the same ideal selector applied to the combined pool of initial and regenerated answers. Both oracles are unattainable in practice, since the correct answer is unknown at inference time; they serve as upper bounds on what Standard MV and PC-WMV can achieve, respectively. Figure 7 replots Figure 6 restricted to Standard MV, PC-cubic, and the two oracle upper bounds, with the y-axis chosen so the oracles stay in view at high token budgets. Oracle (Standard MV) and Oracle (Prefix Consistency) coincide within the 2  σ 2\sigma confidence band on most of the 20 cells, and the residual gap between them is small relative to the PC-cubic vs. Standard MV gap in the same cells. At a fixed token budget B B , the union of correct candidates reachable from sampled groups is therefore essentially the same as that reachable from the equivalent number of initial samples drawn under the Standard MV protocol. Three factors related to pool coverage explain the small residuals. First, at τ = 0.75 \tau{=}0.75 , K = 1 K{=}1 each PC group costs ≈ 1.25 × {\approx}1.25{\times} a Standard MV sample, 2 2 2 The factor 1.25 = 1 + ( 1 − τ ) 1.25=1+(1-\tau) is the truncation-implied expectation. Continuation/generation token ratios in Table 26 give empirical per-group cost factors of 1.25 1.25 – 1.34 × 1.34{\times} on four of the five models and 1.49 1.49 – 1.60 × 1.60{\times} on Ministral3-14B. so at budget B B PC pools roughly 1.6 × 1.6{\times} as many candidate answers as Standard MV. Second, a ~ i \tilde{a}_{i} is correlated with a i a_{i} through the shared prefix y i [ : ⌈ τ | y i | ⌉ ] y_{i}[{:}\lceil\tau|y_{i}|\rceil] , so the effective number of independent answers in the group pool is well below 1.6 × 1.6{\times} , nearly canceling the first effect. Third, the asymptotic pool of correct candidates is by construction weakly larger for the group pool than for the initial pool, which puts Oracle (Prefix Consistency) weakly above Oracle (Standard MV) at high B B in a subset of cells. None of these effects involves the reweighting, so they cannot account for the PC-cubic vs. Standard MV gap in the same panels. PC-WMV’s advantage therefore comes from the weighting itself, with the prefix-consistency score c i  ( a ) c_{i}(a) acting as a per-candidate reliability signal (Section 3.1 ).

##### Per-problem marginal correctness.

Figure 8 gives a complementary per-problem view. Each point is the empirical estimate of the per-problem marginals π ( a ⋆ ) = Pr [ a i = a ⋆ ] \pi(a^{\star})=\Pr[a_{i}=a^{\star}] over the initial answers and π → ( a ⋆ ) = Pr [ a ~ i = a ⋆ ] \pi^{\rightarrow}(a^{\star})=\Pr[\tilde{a}_{i}=a^{\star}] over the regenerations (notation as defined in Appendix C.1 , Eq. ( 14 )). The plotted estimates π ^  ( a ⋆ ) \hat{\pi}(a^{\star}) and π → ^  ( a ⋆ ) \widehat{\pi^{\rightarrow}}(a^{\star}) cluster tightly on the y = x y=x diagonal in every cell, and their cell-level means agree within a few percentage points on most cells. The largest cell-mean difference is for Nemotron3-30B on AIME 2025, where the cell mean of π ^  ( a ⋆ ) \hat{\pi}(a^{\star}) is 0.90 0.90 and that of π → ^  ( a ⋆ ) \widehat{\pi^{\rightarrow}}(a^{\star}) is 0.77 0.77 , and across the 20 cells the differences run in both directions (e.g., Ministral3-14B has the cell mean of π → ^  ( a ⋆ ) \widehat{\pi^{\rightarrow}}(a^{\star}) above that of π ^  ( a ⋆ ) \hat{\pi}(a^{\star}) on AIME 2025 and Brumo 2025). A paired Wilcoxon signed-rank test on the per-problem differences π ^ q  ( a ⋆ ) − π → ^ q  ( a ⋆ ) \hat{\pi}_{q}(a^{\star})-\widehat{\pi^{\rightarrow}}_{q}(a^{\star}) rejects equality at the Bonferroni-corrected significance level α sig = 0.05 / 20 \alpha_{\mathrm{sig}}=0.05/20 on 5 of the 20 cells, with absolute cell-mean differences of at most 0.126 0.126 (the Nemotron3-30B AIME 2025 outlier, with the other four cells within 0.061 0.061 ). Among these 5 detected cells, 4 have the cell mean of π ^  ( a ⋆ ) \hat{\pi}(a^{\star}) above the cell mean of π → ^  ( a ⋆ ) \widehat{\pi^{\rightarrow}}(a^{\star}) , that is, regenerated answers are statistically less correct on average than initial answers (the only exception is Ministral3-14B FrontierScience-Olympiad). Across all 20 cells the cell-mean difference is positive on 13 cells and negative on 7, indistinguishable from a 50 / 50 50{/}50 split (two-sided binomial sign test, p = 0.26 p=0.26 ). The marginal correctness rate is therefore comparable for initial and regenerated answers, and the cell-mean differences do not systematically favor the regenerations. This is a stronger statement than mere equality of rates: PC-WMV’s advantage cannot come from regeneration raising the per-answer correctness rate, because in the cells where the rate differs the regenerations have the lower marginal correctness.

### D.5 Sensitivity to τ \tau and K K

Section 3.1 fixes K = 1 K=1 to keep notation succinct. Here we extend it to the general- K K multiset of Eq. ( 4 ) and the corresponding score c i ( τ , K )  ( a ) = | { a ′ ∈ A i ( τ , K ) : a ′ = a } | / ( K + 1 ) c_{i}^{(\tau,K)}(a)=|\{a^{\prime}\in A_{i}^{(\tau,K)}:a^{\prime}=a\}|/(K+1) , sweeping K ∈ { 1 , 2 , 3 } K\in\{1,2,3\} and τ ∈ { 0.25 , 0.50 , 0.75 } \tau\in\{0.25,0.50,0.75\} .

Table 16 sweeps the truncation fraction τ ∈ { 0.25 , 0.50 , 0.75 } \tau\in\{0.25,0.50,0.75\} at K = 1 K{=}1 for GPT-OSS-20B. Both r C r_{C} and r W r_{W} decrease with deeper truncation, with r W r_{W} generally decreasing faster (e.g., on AIME 2025, r C r_{C} goes 87.8 % → 73.0 % 87.8\%\to 73.0\% and r W r_{W} goes 42.6 % → 17.1 % 42.6\%\to 17.1\% as τ \tau moves 0.75 → 0.25 0.75\to 0.25 ). The discrimination gap D ⁡ ( τ ) D(\tau) varies by benchmark: it widens monotonically on AIME 2025 ( 45.2 → 50.0 → 55.9 45.2\to 50.0\to 55.9 ), peaks at τ = 0.50 \tau{=}0.50 on HMMT and Brumo 2025, and narrows on FrontierScience-Olympiad ( 41.4 → 39.5 → 39.1 41.4\to 39.5\to 39.1 ). Deeper truncation also costs more per group, since regeneration covers a longer suffix. Cost-equivalent accuracy across ( τ , K ) (\tau,K) is reported in Table 17 .

Table 17 reports PC-linear, PC-quadratic, and PC-cubic accuracy across all ( τ , K ) (\tau,K) configurations on the four GPT-OSS-20B benchmarks at budgets B ∈ { 250  k , 1  M , 5  M } B\in\{250\text{k},1\text{M},5\text{M}\} , with Standard MV at the top for reference. The main paper fixes τ = 0.75 \tau{=}0.75 , K = 1 K{=}1 (Section 4 ) as a minimal default that preserves most of the trace and uses a single regeneration per sample. We use the sweep below to characterize how PC-WMV accuracy varies with ( τ , K ) (\tau,K) , not to select the operating point.

Two observations follow. At 250k tokens, high-cost configurations (large K K and small τ \tau ) underperform Standard MV because each regeneration eats into the budget for new groups: τ = 0.25 \tau{=}0.25 , K = 3 K{=}3 PC-cubic (cost ≈ 3.25 × {\approx}3.25{\times} ) reaches .493, .709, .874, and .882 on FrontierScience-Olympiad, HMMT, AIME, and Brumo, underperforming Standard MV on the latter three (.736, .881, .901), while the low-cost τ = 0.75 \tau{=}0.75 , K = 1 K{=}1 (cost ≈ 1.25 × {\approx}1.25{\times} ) outperforms Standard MV on all four. At 1M–5M budgets, the default τ = 0.75 \tau{=}0.75 , K = 1 K{=}1 stays close to the per-cell best on the three math benchmarks (HMMT, AIME, Brumo). FrontierScience-Olympiad is the exception, where the larger discrimination gap at deeper truncation repays the higher per-group cost (PC-cubic at τ = 0.25 \tau{=}0.25 , K = 3 K{=}3 reaches .547 at 1M and .562 at 5M on FrontierScience-Olympiad, against .537 and .545 at τ = 0.75 \tau{=}0.75 , K = 1 K{=}1 ). K = 1 K{=}1 is sufficient on the math benchmarks at 1M, while K = 3 K{=}3 yields modest improvements on HMMT at 5M (PC-cubic .822 at K = 3 K{=}3 vs. .814 at K = 1 K{=}1 , both at τ = 0.75 \tau{=}0.75 ). Overall, the default τ = 0.75 \tau{=}0.75 , K = 1 K{=}1 remains effective across the four benchmarks without being tuned to them, and the results of the main paper in Table 3 are therefore not sensitive to a particular choice of ( τ , K ) (\tau,K) .

### D.6 Robustness to Judge Choice

We show that PC’s per-cell advantage over the best baseline holds regardless of whether answer equivalence is evaluated by the model itself or by a stronger external grader.

The main paper uses each evaluated model itself to determine answer equivalence (Appendix H.2 ). One possible concern is that the evaluation may be biased by the limited capability of the model itself. As a robustness check, this section reports results using a stronger external grader. Namely, we re-scored the released pool with Claude Sonnet 4.6 [ Anthropic, 2026 ] on four models (GPT-OSS-120B, GPT-OSS-20B, Nemotron3-30B, Nemotron2-9B) across the three LLM-judged benchmarks (Brumo 2025, HMMT Feb 2026, FrontierScience-Olympiad), giving 12 cells. AIME 2025 is exact-match and is therefore judge-free.

We re-cluster the released generation pool using the external judge’s YES/NO equivalence judgments instead of the model’s own, then recompute per-sample correctness labels along with AUROC ¯ \overline{\mathrm{AUROC}} , PC-WMV accuracy, and the baseline accuracies. No new samples are drawn from the model.

##### PC’s AUROC ¯ \overline{\mathrm{AUROC}} advantage is preserved on every cell.

Figure 9 plots, per cell, the gap Δ  AUROC ¯ := AUROC ¯ PC − AUROC ¯ best  baseline \Delta\overline{\mathrm{AUROC}}:=\overline{\mathrm{AUROC}}_{\mathrm{PC}}-\overline{\mathrm{AUROC}}_{\mathrm{best\;baseline}} under self-judge ( x x ) against the same quantity under the external judge ( y y ). All 12 points fall in the same quadrant under both judges: 9 cells with Δ  AUROC ¯ > 0 \Delta\overline{\mathrm{AUROC}}>0 (PC ahead of the best baseline) and 3 cells with Δ  AUROC ¯ < 0 \Delta\overline{\mathrm{AUROC}}<0 . No cell flips. Per-cell numbers are in Table 18 .

##### WMV cost–accuracy curves under both judges nearly overlap.

Figure 10 overlays the WMV cost–accuracy curves under self-judge (faded) and the external judge (saturated) for the four primary methods (PC-cubic, Standard MV, DeepConf tail, P(True)). On Brumo and HMMT, the two layers are essentially indistinguishable. On FrontierScience-Olympiad, the external-judge curves sit above their self-judge counterparts by a roughly uniform vertical offset; PC-cubic stays the top (or tied-top) curve in every panel where it leads under self-judge, and stays below where it trails. The per-cell ordering of methods at any operating point is therefore preserved.

##### Pool-level shifts are concentrated on FrontierScience-Olympiad.

On Brumo and HMMT, the two judges disagree on fewer than 4 % 4\% of per-sample correctness labels per cell, and Pass@1 differs by at most 0.06 0.06 . On FrontierScience-Olympiad the per-sample flip rate is higher ( 9 % 9\% to 17 % 17\% ) and Pass@1 rises by up to 0.09 0.09 on three of the four cells; the fourth (GPT-OSS-20B) has compensating up- and down-flips that leave Pass@1 within 0.01 0.01 despite an 8.6 % 8.6\% flip rate. The shift reflects the external judge recognizing more equivalent surface forms than the model’s own. The added equivalences strengthen PC’s relative position rather than weaken it: in Figure 9 , three of the four FrontierScience-Olympiad points (blue) sit above the y = x y{=}x diagonal, and the leftmost column of Figure 10 shows PC’s gap over the baselines widening on those cells under the external judge.

### D.7 Reliability Signals vs. Pass@1

This subsection backs the analysis of Section 4.4 (rising r C r_{C} and a smaller, problem-dependent r W r_{W} slope within each category) with full panels and slope tables for the reproduction rates r C , r W r_{C},r_{W} under the GLM estimator (Appendix D.7.1 ), verifies that the qualitative pattern survives per-benchmark refits (Appendix D.7.2 ) and two alternative estimators (Appendix D.7.3 ), and reports the corresponding Pass@1 view for the per-sample baseline signals (Appendix D.7.4 ).

#### D.7.1 GLM Panels and Slopes

Figure 11 adds GLM panels for the two models not shown in Figure 3 (GPT-OSS-20B, Nemotron2-9B), and Table 19 reports the slopes for all five. The qualitative pattern is consistent: r C r_{C} rises with Pass@1 on both categories, while r W r_{W} has a smaller, problem-dependent slope and sits at a lower level on Science than on Math. The GLM fit kernel and cluster-bootstrap inference protocol below are shared with the alternative estimators of Appendix D.7.3 .

##### GLM fit kernel.

For each outcome Y ∈ { C , W } Y\in\{C,W\} (regenerations seeded from a correct ( C C ) or wrong ( W W ) trace, matching r C r_{C} and r W r_{W} in Section 3.1 ), fit a logistic regression Pr ⁡ ( Y j = 1 ∣ Pass@1 j ) = σ ⁡ ( β 0 , Y + β Y ⋅ Pass@1 j ) \Pr(Y_{j}=1\mid\text{Pass@1}_{j})=\sigma(\beta_{0,Y}+\beta_{Y}\cdot\text{Pass@1}_{j}) on trial-level data, where each trial j j inherits the Pass@1 of its problem, by maximum likelihood ( statsmodels.GLM , Binomial family, default iteratively reweighted least squares (IRLS) solver). Reported quantities are the slope β Y \beta_{Y} and the predicted curve on a fixed Pass@1 grid, masked to the cell’s observed support.

##### Inference protocol.

A (model, category) cell consists of N N per-problem records, each carrying its Pass@1 and two Bernoulli trial streams: r C , i , j ∈ { 0 , 1 } r_{C,i,j}\in\{0,1\} for the n C , i n_{C,i} regenerations seeded from a correct trace and r W , i , j r_{W,i,j} for the n W , i n_{W,i} from a wrong trace ( r = 1 r=1 if the regenerated answer matches the seed, 0 0 otherwise). Each problem i i contributes n C , i n_{C,i} trials ( Pass@1 i , r C , i , j ) \bigl(\text{Pass@1}_{i},\,r_{C,i,j}\bigr) to the r C r_{C} regression (and n W , i n_{W,i} trials ( Pass@1 i , r W , i , j ) \bigl(\text{Pass@1}_{i},\,r_{W,i,j}\bigr) to r W r_{W} ), where the same Pass@1 is shared across all trials from the same problem. We refer to this trial-level dataset as the expanded trials. Point estimates apply each estimator’s fit kernel f ^ Y \hat{f}_{Y} ( Y ∈ { C , W } Y\in\{C,W\} ) to the expanded trials.

Confidence intervals are derived using B = 1000 B=1000 cluster-bootstrap iterations. At iteration b b , we resample N N problem indices with replacement, then refit f ^ C ( b ) \hat{f}_{C}^{(b)} and f ^ W ( b ) \hat{f}_{W}^{(b)} on the corresponding trial set. Within-problem correlation among regenerations that share a prefix is corrected for by resampling problems rather than trials; trial-level Bernoulli standard errors would otherwise be underestimated by a factor of 2 2 to 4 4 on these data. Because f ^ C ( b ) \hat{f}_{C}^{(b)} and f ^ W ( b ) \hat{f}_{W}^{(b)} share the same resample within an iteration, the difference D ( b ) = f ^ C ( b ) − f ^ W ( b ) D^{(b)}=\hat{f}_{C}^{(b)}-\hat{f}_{W}^{(b)} inherits the correct joint distribution, and CIs for D D are read off { D ( b ) } \{D^{(b)}\} directly rather than summed from per-fit half-widths.

Pointwise 2  σ 2\sigma CIs are percentile intervals at level 2  ( 1 − Φ  ( 2 ) ) ≈ 0.0455 2(1-\Phi(2))\approx 0.0455 , taken per Pass@1 grid point for continuous estimators and per bin for the binned estimator. The two-sided bootstrap p p -value for H 0 : β = 0 H_{0}:\beta=0 in Table 19 is p = min { 1 , 2 min ( Pr b [ β ( b ) ≤ 0 ] , Pr b [ β ( b ) ≥ 0 ] ) } p=\min\{1,\,2\min(\Pr_{b}[\beta^{(b)}\leq 0],\,\Pr_{b}[\beta^{(b)}\geq 0])\} , with “ < < .001” indicating no replicate crossed zero. The seed is fixed at 0 0 . A small convergence check (3 seeds, several B B , GPT-OSS-120B Math r W r_{W} ) confirmed stability from B = 1000 B=1000 .

#### D.7.2 Robustness under Per-Benchmark Pooling