Reliable Chain-of-Thought via Prefix Consistency — part 7/8

Cleaned-text code points [94236, 109370). Bibliography omitted here; source.md includes it.

At the time of writing, no official implementation was available, so we reimplemented the method based on the description provided in the original paper. Our implementation segments each reasoning trace into sequential subthoughts at the linguistic cues described in Section 3 of their paper, regenerates a continuation from the end of each subthought, and takes an unweighted majority vote over the resulting pool of answers. To bound the per-problem cost of constructing this pool, we process initial samples in order, accumulating each sample’s initial tokens together with its subthought-regenerated continuations, and stop once the cumulative cost reaches that of the N N initial samples for that problem. Otherwise, the per-problem cost of regenerating subthought continuations from all N N initial samples would be several times that of the initial generation alone. We evaluated this baseline on only a small subset of (model, benchmark) conditions.

##### Prefix consistency (ours).

For prefix consistency, ℓ i = ∅ \ell_{i}=\emptyset : it reads only generated tokens and regenerated answers, with no log-probability access. Unlike the per-sample signal s ⁡ ( y i , ℓ i ) s(y_{i},\ell_{i}) used by the baselines above, our signal c i ( τ ) ​ ( a ) c_{i}^{(\tau)}(a) is defined per distinct candidate a ∈ A i ( τ ) a\in A_{i}^{(\tau)} (Section 3.1 ). Moreover, we use the power-family weighting w ( n ) ​ ( c ) = c n w^{(n)}(c)=c^{n} for n ∈ { 1 , 2 , 3 } n\in\{1,2,3\} (PC-linear, PC-quadratic, PC-cubic). PC-linear ( n = 1 n{=}1 ) admits a simple interpretation: substituting w ⁡ ( c ) = c w(c)=c and Eq. ( 7 ) into the PC-WMV vote gives ∑ i v i ​ ( a ) = 1 K + 1 ​ ∑ i | { a ′ ∈ A i ( τ ) : a ′ = a } | \sum_{i}v_{i}(a)=\tfrac{1}{K+1}\sum_{i}|\{a^{\prime}\in A_{i}^{(\tau)}:a^{\prime}=a\}| , which is proportional to the count of a a in the combined pool of N N initial answers and N ​ K NK regenerations. PC-linear’s argmax therefore coincides with unweighted majority voting on this ( K + 1 ) ​ N (K{+}1)N pool, treating each regeneration as one additional vote of equal weight. PC-quadratic and PC-cubic depart from this baseline by giving super-linear weight to candidates that the same group reproduces.

## Appendix F Examples of Prefix Consistency

Figure 17 shows two representative examples of how regeneration behaves after truncation. When the initial answer is correct, the regenerated continuation often recovers the same core reasoning structure and reproduces the correct answer. In contrast, when the initial reasoning is already flawed, regeneration typically does not repair it. Instead, it continues along a similar erroneous line of reasoning and may even produce a different wrong answer. These examples help illustrate why regeneration amplifies the consistency of correct traces while still failing to escape incorrect ones.

## Appendix G Evaluation Protocol

This appendix consolidates the analysis pipeline shared across the experimental sections of the main paper. Implementation choices specific to each baseline are in Appendix E . Hardware and inference settings are in Appendix I .

### G.1 Setup

##### Benchmarks.

We evaluate on one science benchmark (FrontierScience-Olympiad) and three math benchmarks (HMMT Feb 2026, AIME 2025, Brumo 2025). Table 21 lists the problem count, answer format, and scoring rule of each. We use the full released test split for every benchmark. Answers are extracted from the boxed expression, then math-normalized for the math benchmarks (fraction unification, degree-marker removal) or used with minimal normalization for FrontierScience-Olympiad. Equivalence to the gold answer is decided by exact match for AIME 2025 and by an LLM judge for the other three benchmarks (Appendix H.2 ). HuggingFace URLs and licenses are in Table 24 .

##### Pre-generated pool and cost vectors.

For each (model, benchmark) cell we draw N = 128 N{=}128 initial generations per problem ( N = 64 N{=}64 for Ministral3-14B, where the full N = 128 N{=}128 run had not completed at the time of writing), and for prefix-consistency methods we additionally draw K K regenerations per initial sample at the chosen truncation fraction τ \tau . All voting and analysis read from this fixed pool, so the same generated tokens back every comparison and the only randomness in the reported numbers is from the analysis-side resampling described below. Each generation carries its actual recorded token count. For each method, the cost of a vote is the cumulative generated-token length of every sample it actually reads. Standard MV, Self-certainty, DeepConf, Response probability, AC, and ESC consume only the initial samples. CISC’s verbalized sources (Verbal binary, Verbal 0–100) and P(True) additionally consume the secondary verbal-rating completions call per sample described in Appendix E . PC additionally reads the regenerated continuations from each sample’s prefix. SubthoughtReasoner reads the continuations regenerated from each subthought boundary.

##### Hyperparameter defaults.

Unless otherwise noted, every result in the main paper uses τ = 0.75 \tau{=}0.75 , K = 1 K{=}1 , and the power-family weighting w ( n ) ​ ( c ) = c n w^{(n)}(c)=c^{n} for n ∈ { 1 , 2 , 3 } n\in\{1,2,3\} . Sensitivity to alternative choices of τ \tau and K K is reported in Appendix D.5 .

##### Confidence intervals (CIs) and bootstrap conventions.

All CIs reported in this paper are 2 ​ σ 2\sigma , taken as twice the standard deviation of the trial or bootstrap distribution. The trial Monte Carlo (Sections 4.2 and 4.3 ) draws M = 500 M=500 replicates per (method, benchmark, budget) cell. The parametric ratio bootstrap (Section 4.3 ) and the cluster bootstrap over problems (Section 4.4 ) each use 1,000 1{,}000 resamples.

### G.2 AUROC ¯ \overline{\mathrm{AUROC}} for the Correctness-Predictor Evaluation (Section 4.1 )

##### Definition of AUROC ¯ \overline{\mathrm{AUROC}} .

AUROC denotes the Area Under the Receiver Operating Characteristic curve. Let 𝒬 ′ := { q ∈ 𝒬 : 0 < Pass ​ @ ​ 1 q < 1 } \mathcal{Q}^{\prime}:=\{q\in\mathcal{Q}:0<\mathrm{Pass@1}_{q}<1\} be the subset of problems on which AUROC q \mathrm{AUROC}_{q} and the per-problem rates r C , q , r W , q r_{C,q},r_{W,q} are defined, and set s i = c i ​ ( a i ) s_{i}=c_{i}(a_{i}) for prefix consistency and s i = s ⁡ ( y i , ℓ i ) s_{i}=s(y_{i},\ell_{i}) for the baselines (Appendix E ). We score each signal by a per-problem AUROC and macro-average over 𝒬 ′ \mathcal{Q}^{\prime} : AUROC q := Pr [ s i > s j | a i = a q ⋆ , a j ≠ a q ⋆ ] , AUROC ¯ := 1 | 𝒬 ′ | ∑ q ∈ 𝒬 ′ AUROC q , \mathrm{AUROC}_{q}:=\Pr\!\left[s_{i}>s_{j}\,\middle|\,a_{i}=a^{\star}_{q},\,a_{j}\neq a^{\star}_{q}\right],\qquad\overline{\mathrm{AUROC}}:=\frac{1}{|\mathcal{Q}^{\prime}|}\sum_{q\in\mathcal{Q}^{\prime}}\mathrm{AUROC}_{q}, (23) where i , j i,j are independent draws from the N N initial samples on problem q q , and the reported r C , r W , D r_{C},r_{W},D are means of r C , q , r W , q , D q r_{C,q},r_{W,q},D_{q} over 𝒬 ′ \mathcal{Q}^{\prime} .

##### AUROC computation.

For each problem q q with at least one correct and one wrong initial sample, we score the per-problem AUROC (Eq. ( 23 )) by computing the trapezoidal area under the empirical ROC of the signal s i s_{i} on the N N initial samples. Ties contribute 1 / 2 1/2 . We exclude problems for which all N N initial samples are correct or all N N initial samples are wrong (the AUROC and per-problem rates r C , q , r W , q r_{C,q},r_{W,q} are undefined there) and macro-average over the surviving problems 𝒬 ′ \mathcal{Q}^{\prime} , weighting each problem equally rather than weighting by the number of cross-class pairs. Table 2 reports this macro AUROC ¯ \overline{\mathrm{AUROC}} , and the per-problem rates r C , r W , D r_{C},r_{W},D in Table 1 are macro-averaged over the same 𝒬 ′ \mathcal{Q}^{\prime} .

For prefix consistency, the binary score c i ​ ( a i ) ∈ { 1 / 2 , 1 } c_{i}(a_{i})\in\{1/2,1\} has only one non-trivial operating point on the false-positive/true-positive rate plane, ( FPR , TPR ) = ( r W , q , r C , q ) (\mathrm{FPR},\mathrm{TPR})=(r_{W,q},r_{C,q}) , so the trapezoidal area gives AUROC q = ( 1 + D q ) / 2 \mathrm{AUROC}_{q}=(1+D_{q})/2 . Tables 1 and 2 therefore encode the same information for prefix consistency, and the AUROC ¯ \overline{\mathrm{AUROC}} ordering across benchmarks matches the D D ordering.

### G.3 Cost-Accuracy Evaluation (Sections 4.2 and 4.3 )

##### Sample-with-replacement trials.

At each token budget B B we draw samples from the pool with replacement until the cumulative cost reaches B B , then run each method’s voting rule on the drawn set and report the mean accuracy over the M M trials. The trial-mean variance used to derive the CI is σ 2 = 1 M ​ | 𝒬 | 2 ​ ∑ q p ^ q ​ ( 1 − p ^ q ) \sigma^{2}=\frac{1}{M|\mathcal{Q}|^{2}}\sum_{q}\hat{p}_{q}(1-\hat{p}_{q}) , where p ^ q \hat{p}_{q} is the trial-mean correctness on problem q q and | 𝒬 | |\mathcal{Q}| is the number of problems. This Monte Carlo CI is what is shown as “ ± \pm ” in Table 3 .

##### Dense token-budget grid.

Token-efficiency ratios in Table 4 require evaluating the cost–accuracy curve at arbitrary target budgets, so we evaluate every method on the log-uniform grid ℬ = { 10 3 + k / 100 : k = 0 , 1 , … , 400 } . \mathcal{B}=\bigl\{10^{3+k/100}\;:\;k=0,1,\ldots,400\bigr\}. The dense grid is used for all methods that admit a continuous budget (Standard MV, PC variants, DeepConf, CISC, P(True), Response probability). AC and ESC instead contribute their natural-stopping points (one per swept hyperparameter) as described in Appendix E .

##### Standard MV plateau and Pass@1.

The Standard MV plateau is Standard MV’s bootstrap-saturated accuracy on the N N -sample initial pool, taken as Standard MV’s stored accuracy at max ⁡ ℬ = 10 7 \max\mathcal{B}=10^{7} tokens. We use this finite-budget anchor rather than the unbounded i.i.d. asymptote (which is pool-determined but, on slow-converging problems, can sit above what any finite budget reaches) so that the target stays reachable by Standard MV at every α ∈ [ 0 , 1 ] \alpha\in[0,1] . Pass@1 is the closed-form expected accuracy of a single uniformly drawn pool sample, computed directly from the pool.

##### Monotone envelope and log-budget interpolation.

Each method’s grid of (budget, accuracy) points is reduced to its running-max envelope along sorted budget (so accuracy is non-decreasing in budget). This gives the “min budget at which the method has ever reached this accuracy” semantics that the token-efficiency ratio is meant to compare. To read off the budget at a target accuracy acc tgt \mathrm{acc}_{\mathrm{tgt}} , we find the first envelope segment ( B ( j − 1 ) , acc ( j − 1 ) ) → ( B ( j ) , acc ( j ) ) (B^{(j-1)},\mathrm{acc}^{(j-1)})\to(B^{(j)},\mathrm{acc}^{(j)}) that brackets acc tgt \mathrm{acc}_{\mathrm{tgt}} and linearly interpolate in (accuracy, log-budget): log ⁡ B tgt = log ⁡ B ( j − 1 ) + acc tgt − acc ( j − 1 ) acc ( j ) − acc ( j − 1 ) ⋅ ( log ⁡ B ( j ) − log ⁡ B ( j − 1 ) ) . \log B_{\mathrm{tgt}}=\log B^{(j-1)}+\frac{\mathrm{acc}_{\mathrm{tgt}}-\mathrm{acc}^{(j-1)}}{\mathrm{acc}^{(j)}-\mathrm{acc}^{(j-1)}}\cdot(\log B^{(j)}-\log B^{(j-1)}). The same rule applies to AC and ESC, whose curves are the natural-stopping point lists.

##### Parametric bootstrap for ratio CIs.

Confidence intervals on B method / B MV B_{\mathrm{method}}/B_{\mathrm{MV}} come from a parametric bootstrap with the same trial Monte Carlo noise model used by Table 3 . Each draw perturbs every fixed-budget accuracy entry by independent 𝒩 ⁡ ( 0 , σ acc 2 ) \mathcal{N}(0,\sigma_{\mathrm{acc}}^{2}) noise with σ acc = CI acc / 2 \sigma_{\mathrm{acc}}=\mathrm{CI}_{\mathrm{acc}}/2 (one σ \sigma from the stored 2 ​ σ 2\sigma CI), and additionally perturbs each natural-stopping operating point’s cost by 𝒩 ⁡ ( 0 , σ cost 2 ) \mathcal{N}(0,\sigma_{\mathrm{cost}}^{2}) noise with σ cost = CI cost / 2 \sigma_{\mathrm{cost}}=\mathrm{CI}_{\mathrm{cost}}/2 (the trial-MC standard error of the natural-stopping cost, which varies across operating points unlike the fixed-budget cost). The plateau is perturbed by an analogous draw using its own CI. Each replicate then recomputes the envelope, the target acc tgt = Pass ​ @ ​ 1 + α ⋅ ( plateau ′ − Pass ​ @ ​ 1 ) \mathrm{acc}_{\mathrm{tgt}}=\mathrm{Pass@1}+\alpha\cdot(\mathrm{plateau}^{\prime}-\mathrm{Pass@1}) with the perturbed plateau plateau ′ \mathrm{plateau}^{\prime} , and the ratio. Pass@1 is closed-form over the pool (deterministic) and is held fixed across draws. Anchoring the plateau to Standard MV’s stored accuracy at max ⁡ ℬ \max\mathcal{B} rather than to the running-max envelope’s max breaks the upward extreme-value bias that an iid + running-max combination would otherwise inject near the plateau. Cells where the method’s monotone envelope does not reach the target on the point estimate are reported as “N/A”. The CI subscript is additionally suppressed (point estimate shown alone) when fewer than 50 % 50\% of bootstrap draws reach the target.

### G.4 Reproduction-Rate GLM (Section 4.4 )

##### Logistic GLM and cluster bootstrap.

The slopes β ⁡ ( r C ) \beta(r_{C}) and β ⁡ ( r W ) \beta(r_{W}) in Section 4.4 and Table 19 are the Pass@1 coefficients of a per-(model, category) logistic GLM fit on the expanded per-trial Bernoulli outcomes: logit ​ r C \mathrm{logit}\,r_{C} (and separately logit ​ r W \mathrm{logit}\,r_{W} ) is fit as an affine function of Pass@1, where each (correct initial sample, regeneration) pair contributes one trial to the r C r_{C} fit, each (wrong initial sample, regeneration) pair contributes one trial to the r W r_{W} fit, and every trial carries the source problem’s Pass@1 as its covariate. The fit uses statsmodels’ GLM with a binomial family. CIs and p p -values come from a joint cluster bootstrap over whole problems: each resample draws problems with replacement (preserving each problem’s full set of trials) and refits the GLM. We report band-widths from the bootstrap distribution and two-sided p p -values from the empirical sign distribution.

## Appendix H Answer Extraction and Equivalence Judging

Final answers are extracted from \boxed{} via regex and normalized.

### H.1 Answer Extraction

##### Boxed-answer parses.

Failed boxed responses receive no vote in any aggregator (Standard MV, PC, DeepConf, CISC), so they only ever lower the effective sample count, never bias the vote distribution toward a wrong answer. Seventeen of the twenty cells stay below 1 % 1\% on both Generations and Continuations (Table 22 ). The three cells that exceed 1 % 1\% on continuations all involve Nemotron3-30B: FrontierScience-Olympiad ( 0.17 % → 1.62 % 0.17\%\to 1.62\% ), AIME 2025 ( 0.00 % → 1.17 % 0.00\%\to 1.17\% ), and Brumo 2025 ( 4.14 % → 1.25 % 4.14\%\to 1.25\% , the only cell with generations also above 1 % 1\% ). We treat boxed parse failures as missing votes rather than as a separate signal because the rate is small enough that it does not move the relative comparisons reported in the paper.

##### Verbal score handling.