Reliable Chain-of-Thought via Prefix Consistency — part 4/8
Cleaned-text code points [47671, 62901). Bibliography omitted here; source.md includes it.
###### Assumption 2 (Pooled-mass dominance) .
For every a ≠ a ⋆ a\neq a^{\star} , π ( a ⋆ ) + π → ( a ⋆ ) > π ( a ) + π → ( a ) . \pi(a^{\star})+\pi^{\rightarrow}(a^{\star})>\pi(a)+\pi^{\rightarrow}(a). (20)
##### Interpretation.
π ( a ) + π → ( a ) \pi(a)+\pi^{\rightarrow}(a) is the total population mass on candidate a a , combining initial occurrences and regeneration arrivals. Assumption 2 requires that a ⋆ a^{\star} have the largest such total. By the same decomposition, it makes the pooled-mass term strictly positive whenever w ( 1 2 ) > 0 w(\tfrac{1}{2})>0 .
###### Theorem 2 (Asymptotic convergence of PC-WMV) .
Let w w satisfy w ( 0 ) = 0 w(0)=0 , w ( 1 ) > 0 w(1)>0 , and λ w := w ( 1 ) − 2 w ( 1 2 ) ≥ 0 \lambda_{w}:=w(1)-2w(\tfrac{1}{2})\geq 0 . Then: (a) If Assumptions 1 and 2 hold, then a ^ N PC → a ⋆ \hat{a}_{N}^{\mathrm{PC}}\to a^{\star} almost surely.
(b) If w ( 1 2 ) = 0 w(\tfrac{1}{2})=0 , then Assumption 2 is unnecessary: Assumption 1 alone implies a ^ N PC → a ⋆ \hat{a}_{N}^{\mathrm{PC}}\to a^{\star} almost surely.
###### Proof.
Fix any wrong a ≠ a ⋆ a\neq a^{\star} . By Assumption 1 , the self-reproduction bracket of Eq. ( 18 ) is strictly positive, so the self-reproduction term is nonnegative and is strictly positive whenever λ w > 0 \lambda_{w}>0 .
For part (a), Assumption 2 makes the pooled-mass bracket of Eq. ( 18 ) strictly positive, so the pooled-mass term is nonnegative and is strictly positive whenever w ( 1 2 ) > 0 w(\tfrac{1}{2})>0 . Since w ( 1 ) = λ w + 2 w ( 1 2 ) > 0 w(1)=\lambda_{w}+2w(\tfrac{1}{2})>0 , at least one of λ w \lambda_{w} and w ( 1 2 ) w(\tfrac{1}{2}) is strictly positive, so Φ w ( a ⋆ ) > Φ w ( a ) \Phi_{w}(a^{\star})>\Phi_{w}(a) for every wrong a a . Hence a ⋆ a^{\star} is the unique maximizer of Φ w \Phi_{w} , and Proposition 1 yields a ^ N PC → a ⋆ \hat{a}_{N}^{\mathrm{PC}}\to a^{\star} almost surely.
For part (b), if w ( 1 2 ) = 0 w(\tfrac{1}{2})=0 the pooled-mass term vanishes identically and λ w = w ( 1 ) > 0 \lambda_{w}=w(1)>0 , so the strict positivity of the self-reproduction term suffices to give Φ w ( a ⋆ ) > Φ w ( a ) \Phi_{w}(a^{\star})>\Phi_{w}(a) for every wrong a a . Thus a ^ N PC → a ⋆ \hat{a}_{N}^{\mathrm{PC}}\to a^{\star} almost surely without Assumption 2 . ∎
### C.3 Proof of Theorem 1
Theorem 1 follows from Theorem 2 by specializing to the binary case 𝒜 = { a ⋆ , a ′ } \mathcal{A}=\{a^{\star},a^{\prime}\} , where a ⋆ a^{\star} is the correct answer and a ′ a^{\prime} is the only wrong answer (so π ( a ′ ) = 1 − π ( a ⋆ ) \pi(a^{\prime})=1-\pi(a^{\star}) ). In this case, the two assumptions of Theorem 2 reduce to the same condition and the exact margin takes a one-bracket form, from which the asymptotic boundary π ( a ⋆ ) > r W / ( r C + r W ) \pi(a^{\star})>r_{W}/(r_{C}+r_{W}) follows.
##### The two assumptions coincide.
With the only wrong answer being a ′ a^{\prime} and π ( a ′ ) = 1 − π ( a ⋆ ) \pi(a^{\prime})=1-\pi(a^{\star}) , Assumption 1 becomes the binary boundary π ( a ⋆ ) T ( a ⋆ → a ⋆ ) > ( 1 − π ( a ⋆ ) ) T ( a ′ → a ′ ) . \pi(a^{\star})\,T(a^{\star}\rightarrow a^{\star})>(1-\pi(a^{\star}))\,T(a^{\prime}\rightarrow a^{\prime}). (21) For Assumption 2 , expand π → \pi^{\rightarrow} via Eq. ( 14 ), using T ( a ⋆ → a ′ ) = 1 − T ( a ⋆ → a ⋆ ) T(a^{\star}\rightarrow a^{\prime})=1-T(a^{\star}\rightarrow a^{\star}) and T ( a ′ → a ⋆ ) = 1 − T ( a ′ → a ′ ) T(a^{\prime}\rightarrow a^{\star})=1-T(a^{\prime}\rightarrow a^{\prime}) (each row of T T sums to one): π → ( a ⋆ ) \displaystyle\pi^{\rightarrow}(a^{\star}) = π ( a ⋆ ) T ( a ⋆ → a ⋆ ) + ( 1 − π ( a ⋆ ) ) ( 1 − T ( a ′ → a ′ ) ) , \displaystyle=\pi(a^{\star})\,T(a^{\star}\rightarrow a^{\star})+(1-\pi(a^{\star}))\bigl(1-T(a^{\prime}\rightarrow a^{\prime})\bigr), π → ( a ′ ) \displaystyle\pi^{\rightarrow}(a^{\prime}) = π ( a ⋆ ) ( 1 − T ( a ⋆ → a ⋆ ) ) + ( 1 − π ( a ⋆ ) ) T ( a ′ → a ′ ) . \displaystyle=\pi(a^{\star})\bigl(1-T(a^{\star}\rightarrow a^{\star})\bigr)+(1-\pi(a^{\star}))\,T(a^{\prime}\rightarrow a^{\prime}). Adding π ( a ⋆ ) \pi(a^{\star}) and π ( a ′ ) = 1 − π ( a ⋆ ) \pi(a^{\prime})=1-\pi(a^{\star}) and simplifying, π ( a ⋆ ) + π → ( a ⋆ ) \displaystyle\pi(a^{\star})+\pi^{\rightarrow}(a^{\star}) = 1 + [ π ( a ⋆ ) T ( a ⋆ → a ⋆ ) − ( 1 − π ( a ⋆ ) ) T ( a ′ → a ′ ) ] , \displaystyle=1+\bigl[\pi(a^{\star})\,T(a^{\star}\rightarrow a^{\star})-(1-\pi(a^{\star}))\,T(a^{\prime}\rightarrow a^{\prime})\bigr], π ( a ′ ) + π → ( a ′ ) \displaystyle\pi(a^{\prime})+\pi^{\rightarrow}(a^{\prime}) = 1 − [ π ( a ⋆ ) T ( a ⋆ → a ⋆ ) − ( 1 − π ( a ⋆ ) ) T ( a ′ → a ′ ) ] . \displaystyle=1-\bigl[\pi(a^{\star})\,T(a^{\star}\rightarrow a^{\star})-(1-\pi(a^{\star}))\,T(a^{\prime}\rightarrow a^{\prime})\bigr]. Subtracting, [ π ( a ⋆ ) + π → ( a ⋆ ) ] − [ π ( a ′ ) + π → ( a ′ ) ] = 2 [ π ( a ⋆ ) T ( a ⋆ → a ⋆ ) − ( 1 − π ( a ⋆ ) ) T ( a ′ → a ′ ) ] . \bigl[\pi(a^{\star})+\pi^{\rightarrow}(a^{\star})\bigr]-\bigl[\pi(a^{\prime})+\pi^{\rightarrow}(a^{\prime})\bigr]=2\,\bigl[\pi(a^{\star})\,T(a^{\star}\rightarrow a^{\star})-(1-\pi(a^{\star}))\,T(a^{\prime}\rightarrow a^{\prime})\bigr]. Hence Assumptions 1 and 2 reduce to the same condition Eq. ( 21 ), and Theorem 2 (a) gives a ^ N PC → a ⋆ \hat{a}_{N}^{\mathrm{PC}}\to a^{\star} a.s. whenever Eq. ( 21 ) holds.
##### The exact margin.
Applying Eq. ( 18 ) with a = a ′ a=a^{\prime} , the self-reproduction term contributes λ w \lambda_{w} times the quantity π ( a ⋆ ) T ( a ⋆ → a ⋆ ) − ( 1 − π ( a ⋆ ) ) T ( a ′ → a ′ ) \pi(a^{\star})\,T(a^{\star}\rightarrow a^{\star})-(1-\pi(a^{\star}))\,T(a^{\prime}\rightarrow a^{\prime}) . By the identity above, the pooled-mass term contributes 2 w ( 1 2 ) 2w(\tfrac{1}{2}) times the same bracket. Their sum is w ( 1 ) = λ w + 2 w ( 1 2 ) w(1)=\lambda_{w}+2w(\tfrac{1}{2}) times the bracket: Φ w ( a ⋆ ) − Φ w ( a ′ ) = w ( 1 ) [ π ( a ⋆ ) T ( a ⋆ → a ⋆ ) − ( 1 − π ( a ⋆ ) ) T ( a ′ → a ′ ) ] . \Phi_{w}(a^{\star})-\Phi_{w}(a^{\prime})=w(1)\,\bigl[\pi(a^{\star})\,T(a^{\star}\rightarrow a^{\star})-(1-\pi(a^{\star}))\,T(a^{\prime}\rightarrow a^{\prime})\bigr]. (22) The choice of w w thus enters only through w ( 1 ) w(1) .
###### Proof.
By Eq. ( 22 ) and w ( 1 ) > 0 w(1)>0 , Φ w ( a ⋆ ) > Φ w ( a ′ ) \Phi_{w}(a^{\star})>\Phi_{w}(a^{\prime}) if and only if π ( a ⋆ ) T ( a ⋆ → a ⋆ ) > ( 1 − π ( a ⋆ ) ) T ( a ′ → a ′ ) . \pi(a^{\star})\,T(a^{\star}\rightarrow a^{\star})>(1-\pi(a^{\star}))\,T(a^{\prime}\rightarrow a^{\prime}). By Proposition 1 , this is equivalent to a ^ N PC → a ⋆ \hat{a}_{N}^{\mathrm{PC}}\to a^{\star} almost surely. Identifying r C := T ( a ⋆ → a ⋆ ) r_{C}:=T(a^{\star}\rightarrow a^{\star}) and r W := T ( a ′ → a ′ ) r_{W}:=T(a^{\prime}\rightarrow a^{\prime}) via Eq. ( 13 ) rewrites the threshold as π ( a ⋆ ) > r W r C + r W . \pi(a^{\star})>\frac{r_{W}}{r_{C}+r_{W}}. Standard MV’s vote count 1 N ∑ i = 1 N 𝟏 { a i = a } \frac{1}{N}\sum_{i=1}^{N}\mathbf{1}\{a_{i}=a\} converges to π ( a ) \pi(a) a.s. by the strong law of large numbers, so Standard MV converges to a ⋆ a^{\star} if and only if π ( a ⋆ ) > 1 2 \pi(a^{\star})>\tfrac{1}{2} . Hence PC-WMV converges where Standard MV does not on the interval r W / ( r C + r W ) < π ( a ⋆ ) ≤ 1 2 r_{W}/(r_{C}+r_{W})<\pi(a^{\star})\leq\tfrac{1}{2} , which has positive length if and only if r C > r W r_{C}>r_{W} . ∎
### C.4 Empirical Verification of Assumptions
Table 5 verifies that Assumption 1 (A1) and Assumption 2 (A2) hold empirically on 𝒬 ′ := { q ∈ 𝒬 : 0 < π q ( a q ⋆ ) < 1 } \mathcal{Q}^{\prime}:=\{q\in\mathcal{Q}:0<\pi_{q}(a^{\star}_{q})<1\} , the subset of problems on which the verification is non-degenerate. At π q ( a q ⋆ ) = 1 \pi_{q}(a^{\star}_{q})=1 no wrong answer carries population mass, so A1 and A2 hold vacuously and would only inflate the reported probabilities. At π q ( a q ⋆ ) = 0 \pi_{q}(a^{\star}_{q})=0 the per-problem rate r C , q = T ( a q ⋆ → a q ⋆ ) r_{C,q}=T(a^{\star}_{q}\rightarrow a^{\star}_{q}) is undefined and a q ⋆ a^{\star}_{q} is not a population maximizer, so the framework does not apply. 𝒬 ′ \mathcal{Q}^{\prime} also matches the subset used in the AUROC ¯ \overline{\mathrm{AUROC}} analysis (Section 4.1 , Appendix G.2 ). The table further reports the per-problem indicator Δ w ( n ) := min a ≠ a ⋆ [ Φ w ( n ) ( a ⋆ ) − Φ w ( n ) ( a ) ] \Delta_{w^{(n)}}:=\min_{a\neq a^{\star}}\bigl[\Phi_{w^{(n)}}(a^{\star})-\Phi_{w^{(n)}}(a)\bigr] for w ( n ) ( c ) = c n w^{(n)}(c)=c^{n} , n ∈ { 1 , 2 , 3 } n\in\{1,2,3\} ( Δ w ( n ) > 0 \Delta_{w^{(n)}}>0 is equivalent to a ⋆ a^{\star} being the unique maximizer of Φ w ( n ) \Phi_{w^{(n)}} ).
Pr [ A1 ] \Pr[\mathrm{A1}] is non-negligible across a wide range of benchmarks and models, so A1 \mathrm{A1} is not a rare or pathological event in practice. Conditional on A1 \mathrm{A1} , Pr [ A2 ∣ A1 ] \Pr[\mathrm{A2}\mid\mathrm{A1}] is at least 87.5 % 87.5\% on every cell and reaches 100 % 100\% on 40 % 40\% of them, so the joint occurrence of A1 \mathrm{A1} and A2 \mathrm{A2} is common. The last three columns ( Pr [ Δ w ( n ) > 0 ∣ A1 ] \Pr[\Delta_{w^{(n)}}>0\mid\mathrm{A1}] ) further show that the positive gap predicted by Theorem 2 occurs with overwhelmingly high probability for w ( n ) ( c ) = c n w^{(n)}(c)=c^{n} , n ∈ { 1 , 2 , 3 } n\in\{1,2,3\} , even under the weaker condition that does not require A2 \mathrm{A2} . These findings indicate that the assumptions are not artificially imposed but rather reflect patterns that arise naturally and frequently in real experimental data.
## Appendix D Additional Experiments
### D.1 ROC Curves for Correctness Predictors
Tables 1 and 2 report AUROC ¯ \overline{\mathrm{AUROC}} as a single number per (model, benchmark, signal). Figure 4 visualizes the underlying ROC curves on the same 5 model × \times 4 benchmark grid, overlaying prefix consistency and the WMV baselines per panel. Curves are macro-averaged: each problem’s ROC is interpolated onto a common false-positive-rate grid and then averaged over the same problem subset 𝒬 ′ \mathcal{Q}^{\prime} used in AUROC ¯ \overline{\mathrm{AUROC}} (problems with at least one correct and one wrong initial sample, Appendix G.2 ).
The visual pattern matches the AUROC ¯ \overline{\mathrm{AUROC}} numbers: prefix consistency’s curve sits at or above the baselines on FrontierScience-Olympiad and HMMT Feb 2026 in 8 of the 10 (model, benchmark) settings across the five models, while on AIME 2025 and Brumo 2025 the gap shrinks as the baselines’ curves move closer to PC. The two cells where PC is not visibly above are Nemotron2-9B FrontierScience-Olympiad (where P(True) leads, consistent with PC’s smallest discrimination gap D D in Table 1 ) and GPT-OSS-120B HMMT Feb 2026 (where DeepConf tail edges PC by .03 .03 in AUROC ¯ \overline{\mathrm{AUROC}} ).
### D.2 Cost–Accuracy Curves
Figure 5 extends the cost–accuracy analysis of Figure 1 to all 20 (model, benchmark) cells, complementing the fixed-budget tables in Appendix D.3 . Across nearly all 20 cells PC-cubic (green) sits above the other baselines once the budget exceeds the per-problem regeneration cost, and reaches Standard MV plateau at a noticeably smaller budget. The discrimination gap is largest on FrontierScience-Olympiad and shrinks as Standard MV plateau approaches its peak (AIME 2025, Brumo 2025).
### D.3 Full Baseline Comparison
Figure 6 extends Figure 5 to the full baseline set plus the oracle upper bounds. The three PC variants form a tight green band near the plateau across nearly every cell, while the DeepConf variants and especially their filtered variants spread out widely.
Tables 6 , 7 , 8 , 9 , and 10 report fixed-budget accuracy at B ∈ { 250 k , 1 M , 5 M } B\in\{250\text{k},1\text{M},5\text{M}\} tokens for every evaluated baseline (Standard MV, all five DeepConf aggregation strategies [ Fu et al., 2026 ] and their filtered variants, Response probability, verbalized confidence and P(True) [ Taubenfeld et al., 2025 , Kadavath et al., 2022 ] , and PC-linear/quadratic/cubic), one table per model. See Appendix E for baseline definitions. Tables 11 , 12 , 13 , 14 , and 15 report the corresponding token-efficiency ratios at α ∈ { 75 % , 90 % , 99 % } \alpha\in\{75\%,90\%,99\%\} .
Two patterns emerge. First, on the more difficult science benchmark (FrontierScience-Olympiad) PC-cubic is the best non-oracle method at B = 250 k B{=}250\text{k} and B = 1 M B{=}1\text{M} for the four models with non-trivial discrimination gap (GPT-OSS-120B, GPT-OSS-20B, Nemotron3-30B, Ministral3-14B): at B = 1 M B{=}1\text{M} it reaches .508, .537, .486, and .174 respectively, against the strongest non-PC baseline at .503, .525, .472, and .143. At B = 5 M B{=}5\text{M} PC-cubic remains best on three of these four, with PC-quadratic narrowly ahead on Nemotron3-30B (.504 vs. .503, within 2 σ 2\sigma ). The PC-linear ≤ \leq PC-quadratic ≤ \leq PC-cubic ordering holds in 11 of the 12 (model, budget) FrontierScience-Olympiad cells over these four models, with the Nemotron3-30B 5M cell as the only inversion, consistent with the convex-weight analysis: in Eq. ( 17 ), the coefficient λ w := w ( 1 ) − 2 w ( 1 2 ) \lambda_{w}:=w(1)-2w(\tfrac{1}{2}) on the self-reproduction term π ( a ) T ( a → a ) \pi(a)\,T(a\rightarrow a) (the joint probability that a a is both the initial and the regenerated answer) takes values 0 0 , 1 2 \tfrac{1}{2} , 3 4 \tfrac{3}{4} for PC-linear, PC-quadratic, PC-cubic respectively, so larger n n upweights self-reproducing candidates more strongly. The exception is Nemotron2-9B, whose FrontierScience-Olympiad discrimination gap is the smallest in our suite ( D = 7.4 % D=7.4\% , Table 1 ). PC-WMV there roughly matches Standard MV (PC-cubic .199 vs. Standard MV .203 at B = 1 M B{=}1\text{M} ), consistent with Theorem 1 predicting no advantage when D D is small.
Second, on the easier benchmarks (AIME 2025, Brumo 2025) where Standard MV plateau already sits close to its peak, verbalized confidence (Verbal binary, P(True)) and DeepConf tail (top-90%) become competitive or take the lead in a few cells (e.g. Verbal binary on GPT-OSS-20B Brumo reaches .951 at B = 1 M B{=}1\text{M} vs. .931 for PC-cubic). This matches the cost–quality trade-off discussed in Section 4 , where PC’s larger per-group cost is no longer amortized once Standard MV is near its plateau.
SubthoughtReasoner [ Hammoud et al., 2025 ] appears only in the GPT-OSS-20B tables, the only model for which it has been re-evaluated under the unified incremental path. SubthoughtReasoner does not consistently improve over Standard MV (vs. Standard MV at B = 1 M B{=}1\text{M} : .778 vs. .775 on HMMT, .518 vs. .523 on FrontierScience-Olympiad, .886 vs. .896 on AIME 2025), and the GPT-OSS-20B Brumo cell and other-model cells are not evaluated.
### D.4 Pool Coverage vs. Reweighting
We show that PC-WMV’s advantage over the non-PC baselines comes from reweighting, not from an enlarged pool of correct candidates.
##### Oracle upper bounds.