Reliable Chain-of-Thought via Prefix Consistency — part 6/8

Cleaned-text code points [78790, 94236). Bibliography omitted here; source.md includes it.

Section 4.4 pools the three Math benchmarks (HMMT Feb 2026, AIME 2025, Brumo 2025) into a single Math curve per model. Since the three span different Pass@1 ranges within a fixed model, the pooled slope could in principle reflect between-benchmark Pass@1 differences rather than a within-benchmark Pass@1 effect. To check for this confound we refit the same logistic GLM on each (model, benchmark) cell separately, with the fit kernel and cluster-bootstrap protocol of Appendix D.7.1 . Figure 12 plots the four per-benchmark curves per model in a single panel, and Table 20 reports the slopes.

The two claims in Section 4.4 both survive the per-benchmark refit. (i) β ⁡ ( r C ) > 0 \beta(r_{C})>0 on all 20 20 (model, benchmark) cells with p < 0.05 p<0.05 (in fact p < 0.001 p<0.001 on 17 17 of 20 20 cells), so the rising- r C r_{C} pattern is within-benchmark and not a pooling artefact. (ii) | β ⁡ ( r W ) | < β ⁡ ( r C ) |\beta(r_{W})|<\beta(r_{C}) on all 20 20 cells, so the smaller-magnitude statement is also within-benchmark.

The per-benchmark view further clarifies the two non-zero pooled β ⁡ ( r W ) \beta(r_{W}) values from Section 4.4 . The GPT-OSS-120B Math value + 1.14 +1.14 is driven by AIME 2025 ( + 1.92 +1.92 , p = 0.002 p=0.002 ) and Brumo 2025 ( + 3.46 +3.46 , p = 0.004 p=0.004 ), with HMMT Feb 2026 separately null ( − 0.14 -0.14 , p = 0.83 p=0.83 ): the pooled value reflects a real within-benchmark Pass@1 effect on AIME and Brumo, but the effect is benchmark-specific rather than uniform across Math. The Ministral3-14B Math value − 0.75 -0.75 is uniform across all three benchmarks ( − 0.85 -0.85 , − 0.68 -0.68 , − 0.65 -0.65 ), i.e. a within-benchmark effect that holds Math-wide. In neither case is the pooled slope a between-benchmark artefact.

#### D.7.3 Robustness under Alternative Estimators

In Figure 3 , we fit a logit-linear model. To verify that the conclusions in the main paper (that is, rising r C r_{C} , a smaller and problem-dependent r W r_{W} slope within each category, and lower r W r_{W} on Science than on Math) do not depend on that form, we rerun the same plot for all five models under two alternative estimators, sharing the cluster-bootstrap protocol of Appendix D.7.1 .

##### Trial-pooled binned estimator (Figure 13 ).

This estimator reads reproduction rates directly off the data, removing the GLM’s logit-linear assumption at the cost of bin-edge sensitivity. Bin edges come from the full-data Pass@1 quantiles, with 5 5 equal-count bins per category deduplicated by np.unique . A Pass@1 mass at zero can collapse adjacent edges, e.g. Ministral3-14B Science with 53 53 problems at zero resolves to 3 3 effective bins. Each bootstrap iteration assigns problems to those fixed edges and reports the trial-pooled rate ∑ i ∈ bin k Y , i / ∑ i ∈ bin n Y , i \sum_{i\in\mathrm{bin}}k_{Y,i}\big/\sum_{i\in\mathrm{bin}}n_{Y,i} , where k Y , i k_{Y,i} and n Y , i n_{Y,i} count reproductions and trials.

##### LOWESS smoother (Figure 14 ).

Because this estimator does not assume a monotone or parametric relationship, it can reveal any non-monotonic structure in r W r_{W} that a logit-linear model would inadvertently hide. We use statsmodels.nonparametric.smoothers_lowess on trial-level ( Pass@1 j , Y j ) (\text{Pass@1}_{j},Y_{j}) pairs with span frac = 0.5 \mathrm{frac}=0.5 and robustness iterations 𝚒𝚝 = 0 \mathtt{it}=0 . The default 𝚒𝚝 = 3 \mathtt{it}=3 uses bisquare residual weights designed for continuous residuals, which collapse Bernoulli outcomes to degenerate 0 0 or 100 % 100\% curves (verified empirically on Science). The fit is linearly interpolated onto the GLM’s Pass@1 grid and masked to the observed support.

All three estimators agree qualitatively across all five models: r C r_{C} rises across Pass@1 on every (model, category) pair, and r W r_{W} has a smaller, problem-dependent slope with the category-specific levels described in Section 4.4 .

#### D.7.4 Baseline Confidence Signals vs. Pass@1

For comparison with Figure 3 , we repeat the per-class, Pass@1-conditioned view on the two baseline confidence signals named in the caption of Figure 2 : DeepConf tail (Figure 15 ) and P(True) (Figure 16 ). For each signal, we split per-trial values by whether the trace’s answer matches gold (Correct, solid) or not (Wrong, dashed), and fit each class per category with a Gaussian linear model s = β 0 + β ⋅ Pass@1 s=\beta_{0}+\beta\cdot\text{Pass@1} , replacing the Bernoulli logistic GLM of Appendix D.7.1 since the response is now a continuous score. The cluster-bootstrap protocol ( B = 1000 B=1000 resamples over problems, 2 ​ σ 2\sigma percentile CIs) is unchanged. Scatter overlays show per-problem mean confidence (one dot per problem per class), the continuous-signal analogue of the per-problem rate k / n k/n in Figure 3 .

## Appendix E Baseline Implementation Details

We document here the exact confidence scores and voting rules used for each baseline, since different papers use slightly different weighting conventions.

##### Glossary of baseline labels.

The baselines summarized in Section 4 appear under the following labels in our tables and figures (e.g. Table 3 and the per-model tables of Appendix D.3 ). The per-baseline paragraphs below give the full implementation details. • DeepConf [ Fu et al., 2026 ] : a family of trace-level log-probability scores ( first-token , Mean (the Self-certainty signal of Kang et al. [2025] ), bottom-10% , block-min , tail ). Each is plugged into Eq. ( 3 ) with w w as the identity. Filtered variants ( top-10% , top-90% ) keep only the top- η % \eta\% of traces by the same score before voting.

• CISC [ Taubenfeld et al., 2025 ] : Confidence-Informed Self-Consistency, a per-sample weighting scheme that draws its raw confidence from one of four sources: Response probability [ Wang et al., 2023b ] (length-normalized geometric mean of per-token probabilities), Verbal binary [ Lin et al., 2022 ] (a 0/1 self-rating parsed from a follow-up call), Verbal 0–100 [ Lin et al., 2022 ] (a percentage self-rating parsed from a follow-up call), and P(True) [ Kadavath et al., 2022 ] (the renormalized softmax probability of the “ 1 1 ” token versus “ 0 0 ” at the rating-call confidence position).

• Adaptive stopping : cost–accuracy curves traced by sweeping the early-stopping hyperparameter of Adaptive Consistency ( AC sweep [ Aggarwal et al., 2023 ] ) and Early-Stopping Self-Consistency ( ESC sweep [ Li et al., 2024 ] ) over the same initial pool.

• SubthoughtReasoner [ Hammoud et al., 2025 ] : a single-trace refinement baseline that segments each trace at linguistic cues and votes over per-subthought regenerations. It appears only in the GPT-OSS-20B all-baselines tables (Tables 7 and 12 ).

• PC-linear, PC-quadratic, PC-cubic : PC-WMV (Algorithm 1 ) instantiated with the power-family weighting w ( n ) ​ ( c ) = c n w^{(n)}(c)=c^{n} at n = 1 , 2 , 3 n=1,2,3 . We use “PC-cubic” as shorthand for PC-WMV with n = 3 n{=}3 .

##### Deep Think with Confidence (DeepConf) [ Fu et al., 2026 ] .

We compute five trace-level confidence scores from the top- 20 20 token log-probabilities saved at generation, following the original implementation. 3 3 3 https://github.com/facebookresearch/deepconf For a trace y i y_{i} , let P t , j P_{t,j} be the j j -th largest token probability at position t ≤ | y i | t\leq|y_{i}| . The readout ℓ i = { P t , j } t ≤ | y i | , j ≤ 20 \ell_{i}=\{P_{t,j}\}_{t\leq|y_{i}|,\,j\leq 20} is the top- 20 20 log-probability record along y i y_{i} . Write C t = − 1 20 ∑ j = 1 20 log P t , j C_{t}=-\frac{1}{20}\sum_{j=1}^{20}\log P_{t,j} for the negative top- 20 20 mean at position t t . The five trace-level signals s ⁡ ( y i , ℓ i ) s(y_{i},\ell_{i}) are:

The Mean score above is a top- 20 20 implementation of the self-certainty signal originally proposed by Kang et al. [2025] on the full vocabulary. We adopt it as our Self-certainty baseline and report it under that label throughout the paper. DeepConf is the case of Eq. ( 3 ) where s ⁡ ( y i , ℓ i ) s(y_{i},\ell_{i}) is one of the five signals above and w w is the identity, giving v i ( a ) = s ( y i , ℓ i ) ⋅ 𝟏 [ a i = a ] v_{i}(a)=s(y_{i},\ell_{i})\cdot\mathbf{1}[a_{i}=a] , where a i a_{i} is the answer extracted from trace i i . Confidence filtering (Section 3.2 of their paper) optionally restricts the sum to the top- η % \eta\% of traces by s ⁡ ( y i , ℓ i ) s(y_{i},\ell_{i}) before aggregation. The unfiltered variant ( DeepConf-<score> ) sums v i ​ ( a ) v_{i}(a) over all traces, for each of the five scores above. In addition, we evaluate four filtered variants that combine the two stronger scores with the two retention levels proposed in the paper:

The DeepConf online early-termination mechanism is not used: our evaluation is offline over a pre-generated pool under a shared token budget.

##### Confidence-Informed Self-Consistency (CISC) [ Taubenfeld et al., 2025 ] .

CISC (Definition 3.1 of their paper) extracts a per-sample raw confidence s i = s ⁡ ( y i , ℓ i ) s_{i}=s(y_{i},\ell_{i}) from one of four sources drawn from prior work, then applies a softmax weighting w ⁡ ( s i ) = exp ⁡ ( s i / T ) / ∑ j exp ⁡ ( s j / T ) w(s_{i})=\exp(s_{i}/T)/\sum_{j}\exp(s_{j}/T) (over samples) with tunable temperature T T :

The original CISC release 4 4 4 https://github.com/google-research/google-research/tree/master/cisc targets non-reasoning models and forces a single-token response ( temperature=0 , max_tokens=1 ). This setting is incompatible with our reasoning models, whose chat template (e.g., Harmony for GPT-OSS) leaves the final channel open at the end of the saved trace: the appended rating suffix sits inside that open channel, and the model may emit channel-control tokens or brief reasoning before producing the digit, and occasionally reopens an analysis or commentary channel after the digit. For our reasoning models we therefore make a separate completions call per initial sample, continuing from the saved trace y i y_{i} appended with a short rating suffix and using each model’s recommended sampling parameters. Verbal binary and P(True) share the binary suffix “ Now I will rate my confidence in the proposed answer as either 0 or 1. Proposed confidence: ( ”, and the Verbal 0–100 suffix is “ Now I will rate my confidence in the proposed answer on a scale of 0–100. Proposed confidence: ( ” (with leading newline). The follow-up call is allowed to reason. We then detect the CoT-to-final delimiter and read the confidence from the final portion (first token whose top- 20 20 candidates contain “ 0 0 ” or “ 1 1 ” for Verbal binary and P(True), first integer up to “)” for Verbal 0–100). A single binary call serves both Verbal binary (text parse of the digit) and P(True) (top- 20 20 logprob softmax of “ 1 1 ” versus “ 0 0 ” at the same token). The suffix wording mirrors the CISC public release. Parse failures are imputed with the per-problem median of the non-missing scores, which keeps the answer in the vote at a neutral weight (Appendix H.1 discusses why this is preferred over dropping the sample from the pool). To avoid a baseline-specific T T search that could over-fit our benchmarks, we report the untuned linear weighting w ⁡ ( s ) = s w(s)=s for each source (verbal 0 0 – 100 100 rescaled to [ 0 , 1 ] [0,1] ), so v i ( a ) = s ( y i , ℓ i ) ⋅ 𝟏 [ a i = a ] v_{i}(a)=s(y_{i},\ell_{i})\cdot\mathbf{1}[a_{i}=a] . Softmax with T = 1 T{=}1 is also computed by our pipeline but omitted from the tables because it consistently underperforms the linear variant. A full T T sweep is left to future work.

Under the budget accounting defined in Appendix G.1 , we charge Verbal binary, Verbal 0–100, and P(True) a per-sample token cost of | y i | |y_{i}| plus the length of the secondary-call output up to and including the parsed digit (or its enclosing “)”). Here, we charge this minimum-parse length rather than the full secondary-call length because the model occasionally continues with extra reasoning after the digit that the parser ignores. Charging that overflow would overstate the unavoidable cost of these baselines. A tighter max_tokens cap on the secondary call would bound the overflow at the source, but we leave the secondary call’s maximum output length matched to the initial generation budget (Appendix I ) to avoid baseline-specific tuning, and rely on the parser-anchored cost above to keep the budget accounting honest. Response probability has no secondary call: the per-token log-probabilities of y i y_{i} are saved at initial generation and read for free.

##### Adaptive Consistency (AC) [ Aggarwal et al., 2023 ] .

AC consumes the initial pool one sample at a time and stops at the first k k for which a Beta-binomial posterior favors the running top answer over the runner-up by at least C thresh C_{\mathrm{thresh}} . Let n 1 n_{1} and n 2 n_{2} be the top- 1 1 and top- 2 2 answer counts after k k samples. The closed-form stopping criterion is 1 − I 1 / 2 ​ ( n 1 + 1 , n 2 + 1 ) ≥ C thresh , 1-I_{1/2}(n_{1}+1,\,n_{2}+1)\;\geq\;C_{\mathrm{thresh}}, where I x ​ ( α , β ) I_{x}(\alpha,\beta) is the regularized incomplete beta function. This is the analytic equivalent of the Monte Carlo integral implemented in the official release. 5 5 5 https://github.com/Pranjal2041/AdaptiveConsistency At the stop time, the prediction is the running mode, so AC adds no per-sample weighting: it is Standard MV truncated at a natural stopping point. The original paper reports C thresh = 0.95 C_{\mathrm{thresh}}=0.95 as the default and sweeps [ 0.5 , 1 ) [0.5,1) in Figure 2 to trace a cost–accuracy frontier. We sweep C thresh ∈ { 0.5 , 0.6 , 0.7 , 0.8 , 0.9 , 0.95 , 0.97 , 0.99 , 0.995 , 0.999 } C_{\mathrm{thresh}}\in\{0.5,0.6,0.7,0.8,0.9,0.95,0.97,0.99,0.995,0.999\} . Each value yields a single (cost, accuracy) point at its natural stopping cost, and we label the resulting curve “AC sweep”.

##### Early-Stopping Self-Consistency (ESC) [ Li et al., 2024 ] .

ESC consumes the initial pool in fixed-size windows of W W samples. When a window is unanimous, ESC locks that answer as the final prediction and stops. Otherwise, the window’s votes are added to a running counter and a new window is drawn. Before any lock, the intermediate prediction is the mode of the running counter. The paper recommends W = 5 W=5 for most tasks and W = 8 W=8 for MATH. Our streaming reformulation reproduces the lock semantics of the official batch release. 6 6 6 https://github.com/Yiwei98/ESC We sweep W ∈ { 2 , 3 , … , 10 } W\in\{2,3,\ldots,10\} and label the resulting natural-stopping curve “ESC sweep”. Like AC, ESC is Standard MV with a window-based stopping rule and adds no per-sample weight.

##### Cost accounting for AC and ESC.

Both methods consume only initial samples. Their per-trial budget is the cumulative generated-token length of the consumed initial samples, the same cost accounting used by Standard MV, Self-certainty, DeepConf, and Response probability. Because each hyperparameter value yields one (cost, accuracy) point at its natural stopping cost rather than a curve over a shared budget grid, we treat the points across the swept hyperparameter as the method’s cost–accuracy curve and interpolate it for any required budget. The pool size N N bounds how many tokens AC and ESC can spend, so their plateaus are bounded by N N -sample Standard MV.

##### SubthoughtReasoner [ Hammoud et al., 2025 ] .