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More Correct Mass, Worse Answers: Why Power Sampling Can Fail and How to Fix It

2026 · arxiv_cs

arXiv CS · Papers · 2026

deep-learning machine-learning neural-networks

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M ORE C ORRECT M ASS , W ORSE A NSWERS : W HY P OWER S AMPLING C AN FAIL AND H OW TO F IX I T Haohui Yang∗& Jiaxing Sun∗& Xiujun Ma† State Key Laboratory of General Artificial Intelligence, Peking University, Beijing, China {yanghaohui,sunjiaxing}@stu.pku.edu.cn maxiujun@pku.edu.cn

arXiv:2608.14420v1 [cs.LG] 14 Aug 2026

A BSTRACT Power Sampling sharpens a language model’s distribution over complete generation trajectories, offering a verifier-free way to improve reasoning at inference time. It also has the potential to serve as a general-purpose front end for a broad range of downstream sampling methods. However, we uncover a striking paradox: Power Sampling can drive more probability mass toward correct trajectories while degrading the downstream inference it is intended to enhance. Using selfconsistency as a representative case, we observe accuracy drops of up to 18.5 percentage points across models and reasoning benchmarks. We trace this paradox to two mismatches. Dose mismatch arises because a fixed exponent induces drastically different amounts of distributional change across problems. Coverage mismatch arises because global sharpening concentrates mass on a narrow set of dominant paths: high pass@k, often interpreted as evidence of preserved diversity, can therefore coexist with the loss of broad reasoning-path support required for downstream aggregation, search, and selection. Guided by this diagnosis, we replace uniform trajectory exponentiation with a deformation-controlled, supportpreserving Power target that calibrates sharpening across problems while limiting the suppression of moderate-probability paths. In a same-budget instantiation with weighted self-consistency, the repaired sampler reverses the losses caused by global Power and outperforms standard multi-sample inference across reasoning benchmarks.

I NTRODUCTION

Large language models have made substantial progress on complex tasks in mathematics, code generation, and scientific question answering (Lewkowycz et al., 2022; Chen et al., 2021). For much of this progress, the dominant recipe has been to scale the resources invested during training: model capacity, training data, and training compute (Kaplan et al., 2020). More recently, test-time scaling has emerged as a complementary axis for improving capability after training. Rather than modifying model parameters, it allocates additional inference compute to longer reasoning, search over possible solutions, or the generation of multiple candidate trajectories (Snell et al., 2025; Wu et al., 2025). In doing so, it broadens the central scaling question from how to train a stronger model to how a trained model should spend computation when solving each problem. Within this broader paradigm, repeated sampling offers a simple and broadly applicable strategy: it generates multiple reasoning trajectories and combines them through verification, selection, or aggregation (Chen et al., 2026; Khanh et al., 2026). Self-Consistency (SC), for example, selects the answer receiving the most support across independently sampled trajectories (Wang et al., 2023; Oh and Lee, 2026). Crucially, such methods rely on more than the presence of one strong trajectory. Their success also depends on whether the upstream distribution preserves the collective support on which the downstream decision is based (Tan et al., 2025; Khanh et al., 2026). ∗ †

Equal contribution. Corresponding author.

Ministral

−5 -7.0

−10

0.6

Base

1.50

Power one wrong path dominates

0.4

0.2

same α, different dose

1.5

0.00 −0.75

Prompt A Prompt B Prompt C

−15

0.0 -18.5

−20 BigCodeBench

LiveAoPS

Tail

−1.50

0.00

PHYSICS

(a) Downstream failure

(b) Coverage mismatch

Power

SoftSat

0.75

g(u)

Qwen3.5

Trajectory mass

Δ SC accuracy (pp)

Nemotron

ΔPower, j(u)

Power (α = 4) hurts SC in 7/9 settings 5

0.25

0.50

0.75

1.00

1.0

0.5

0.0 0.0

0.2

0.4

0.6

0.8

1.0

(d) Relative-Rank SoftSat

Figure 1: Power sharpening can conflict with multi-trajectory aggregation, motivating a calibrated and support-preserving alternative. Here g denotes the additive log-weight gain in q ∝ p exp(g). (a) Change in SC accuracy when Base SC8 is replaced by Power (α = 4) across three models and three benchmarks; seven of nine settings degrade. (b) Blue and red bars show Base and illustratively sharpened trajectory masses. C1 –C3 support the same correct answer, W1 supports a competing wrong answer, and Tail groups other low-mass paths; sharpening makes W1 dominant. (c) Curves plot each prompt’s centered Power gain ∆Power,j (u) against the common within-prompt rank coordinate u under a shared α. (d) Repair schematic. On the same u, dashed red and solid green curves show a linearly increasing Power reference g(u) and the saturating RelativeRank SoftSat gain, respectively.

This dependence makes the upstream trajectory distribution a central design choice. Power Sampling offers a verifier-free way to reshape that distribution: given a language model distribution p over complete trajectories, it targets a sharpened distribution proportional to pα , favoring trajectories that the model already considers likely without additional training or a correctness verifier (Huang et al., 2025; Karan and Du, 2026). Recent advances, including Scalable Power Sampling and Power-SMC, substantially reduce the cost of approximating this sequence-level target, making Power Sampling practical for large-scale inference (Ji et al., 2026; Azizi et al., 2026). Because the target is independent of any particular downstream rule, it appears to be a natural front end for multi-trajectory reasoning. However, this seemingly natural composition exposes a fundamental tension: assigning more probability to individually correct trajectories does not guarantee a better aggregated answer. Indeed, Power Sampling can move more probability mass onto correct trajectories while weakening the answer-level support on which aggregation relies, producing worse final answers. The downstream consequences are substantial. Across nine model–benchmark settings, fixed-exponent Power reduces downstream scores in seven, with a largest drop of 18.5 percentage points; notably, the degradation appears in all six LiveAoPSBench and PHYSICS settings shown in Figure 1(a). Thus, trajectory-level sharpening is not reliably aligned with the multi-trajectory decision it is intended to support. Why does this misalignment arise? We identify two mismatches between global sharpening and downstream aggregation. First, coverage mismatch occurs because Power rewards trajectories individually, whereas aggregation depends on their collective answer-level support. Figure 1(b) gives a minimal example: C1 –C3 are distinct moderate-probability trajectories that map to the same correct answer, while W1 is the individually strongest trajectory and maps to a competing wrong answer. Under Base, the correct answer wins through the combined support of C1 –C3 . Because Power exponentiates each trajectory before answer-level aggregation, it magnifies W1 ’s individual advantage and suppresses the moderate correct paths; after renormalization, W1 becomes dominant and the aggregated decision can flip. The low-mass Tail emphasizes that this failure is a loss of broadly distributed support, rather than the disappearance of every alternative path. Consequently, high pass@k certifies that a correct trajectory remains reachable, but not that the distribution preserves the effective support required by aggregation, search, or selection (Karan and Du, 2026; Yue et al., 2025; Tan et al., 2025; Khanh et al., 2026). Second, dose mismatch occurs because a fixed exponent applies the same algebraic transformation, not the same amount of distributional change. Since likelihood geometry varies across prompts, the same α can produce a mild adjustment on one problem but nearcollapse on another. Figure 1(c) places prompts on the common rank coordinate u: prompt-specific likelihood spreads turn the same α into different centered gain slopes ∆Power,j (u). Together, coverage mismatch explains why sharpening can damage aggregation, while dose mismatch explains why that damage is uneven and difficult to control. 2

This diagnosis suggests that a successful repair must control both the shape and strength of sharpening. We therefore introduce Relative-Rank SoftSat. To address coverage mismatch, SoftSat promotes moderately ranked trajectories while saturating gains at the top, preventing a few dominant paths from absorbing disproportionate mass. To address dose mismatch, relative rank replaces raw likelihood with a common within-prompt coordinate in [0, 1], making the strength of reweighting more comparable across problems. Together, the two components favor strong trajectories without discarding the broader support required by downstream aggregation. Figure 1(d) makes this design response explicit through g(u): on the same relative-rank coordinate, the dashed linear Power reference keeps growing, whereas SoftSat raises moderate ranks and saturates toward the top. Finally, we exploit the candidate-level parallelism enabled by modern Power samplers (Ji et al., 2026; Azizi et al., 2026). Rather than invoking Power-SMC independently N times for SCN , we recast target sampling as importance-weighted aggregation over a shared Base candidate pool. The resulting estimator reuses the same N trajectories as Base+SCN , while our analysis shows that it asymptotically preserves the decision of direct target sampling under a standard positive-margin condition. This same-budget realization therefore retains the computational appeal of parallel sampling while allowing the repaired target to guide downstream aggregation. Our contributions are threefold: 1. We show that Power Sampling can increase correct-trajectory mass yet degrade multitrajectory decisions, and identify coverage mismatch and dose mismatch as the causes. 2. We introduce Relative-Rank SoftSat, which replaces absolute likelihood with withinprompt rank to control dose and saturates gains on dominant trajectories to preserve coverage. 3. We connect ideal target sampling to importance-weighted aggregation over a shared Base pool, enabling a low-cost realization that asymptotically preserves the target decision.

R ELATED W ORK

2.1

S EQUENCE -L EVEL D ISTRIBUTION S HARPENING

Distribution sharpening reallocates probability toward outputs already favored by a model and has been studied as a mechanism underlying language-model self-improvement (Huang et al., 2025). Within this view, Power Sampling operates at inference time by exponentiating complete-trajectory probabilities and approximating the resulting target with autoregressive Markov chain Monte Carlo (Karan and Du, 2026). Because iterative MCMC introduces substantial sequential latency, subsequent work has focused on making this target practical at inference time. Specifically, Scalable Power Sampling replaces iterative MCMC with an autoregressive approximation based on sampled continuations (Ji et al., 2026), whereas Power-SMC uses sequential importance weighting and resampling to advance a particle population in a GPU-friendly batched decode (Azizi et al., 2026). By reducing the sequential bottleneck, these methods make fixed-exponent Power distributions substantially easier to sample from. Their primary focus, however, remains how to obtain Power samples efficiently; our work instead asks how the resulting target and parallel candidate computation should interact with a downstream multi-sample decision rule. Despite this progress, existing evaluations largely characterize the resulting sampler through singleoutput accuracy and pass@k (Karan and Du, 2026; Ji et al., 2026). Yet pass@k establishes only whether a correct trajectory remains reachable; it does not describe how probability mass is distributed across reasoning paths or answer modes. This distinction matters for multi-sample inference, where limited output diversity can constrain Best-of-N reasoning (Khanh et al., 2026) and repeated generations may form a stable but incorrect consensus (Tan et al., 2025). Accordingly, our work studies the downstream compatibility of sharpening and modifies the target deformation to preserve the support required by aggregation. 2.2

M ULTI -S AMPLE C ONSISTENCY AND AGGREGATION

Multi-sample aggregation methods can be organized around two choices: how they define agreement and how much support each candidate contributes. The first choice determines which trajectories are 3

treated as evidence for the same outcome. Self-Consistency groups reasoning trajectories by their normalized final answer and selects the largest group (Wang et al., 2023), whereas CodeT extends consistency to programs by jointly evaluating generated code and tests through execution (Chen et al., 2023a). For outputs without a canonical short answer, Universal Self-Consistency asks an LLM to identify the most consistent response (Chen et al., 2023b); Latent Self-Consistency learns representations for majority-set selection across short- and long-form answers (Oh and Lee, 2026); and ModeX identifies a dominant response through a similarity graph over complete generations (Choi and Li, 2026). Although these methods use different equivalence relations, all of their readouts depend on collective support across sampled candidates. The second choice determines how much support each candidate contributes. Whereas standard SC assigns one vote per trajectory, CISC weights answers using model-reported confidence (Taubenfeld et al., 2025), RASC evaluates both rationale quality and answer consistency (Wan et al., 2025), and RPC combines answer frequency with internal sequence probability (Zhou et al., 2025). Dynamic distributional alignment instead adjusts token-level sampling temperature in response to the evolving answer distribution (Li et al., 2025). Other approaches improve sampling efficiency by adapting the sample count or switching among complementary models (Wang et al., 2026; Chen et al., 2026). In contrast to confidence or correctness scores, our weights represent the probability mass assigned by an explicit target trajectory distribution. Consequently, that mass can be accumulated under answer equality, execution-based relations, or semantic similarity without changing the task-specific definition of consistency.

P RELIMINARIES

3.1

T RAJECTORY D ISTRIBUTIONS AND P OWER S AMPLING

Given a prompt, let τ = (τ1 , . . . , τT ) denote a complete generation trajectory; we omit prompt conditioning for clarity. The Base model assigns trajectory probability and log-likelihood p(τ ) =

T Y

p(τt | τ<t ),

t=1

ℓ(τ ) = log p(τ ) =

T X

(1) log p(τt | τ<t ).

t=1

Token-level temperature sampling modifies each next-token distribution separately. At temperature γ > 0, it induces T Y p(τ | τ )1/γ P t <t 1/γ . pγ (τ ) = (2) v p(v | τ<t ) t=1 Lowering γ sharpens every prefix-conditioned decision. Power Sampling instead sharpens the distribution over complete trajectories. For α > 1, its ideal target is X p(τ )α , Zα = qα (τ ) = p(τ ′ )α . (3) Zα ′ τ

The token-temperature distribution in equation 2 and the Power target in equation 3 are generally different because token-level normalization depends on the generated prefix. Power-SMC approximates the sequence-level target with M particles and returns one approximate Power trajectory per run (Azizi et al., 2026). Our analysis uses the ideal qα as its reference and separates this target from finite-M approximation error. The number of trajectories used by a downstream aggregation rule is denoted by N ; it is distinct from the internal particle count M . 3.2

F ROM T RAJECTORIES TO C ONSENSUS

Let A(τ ) be the normalized answer extracted from τ , and let R(τ ) ∈ {0, 1} indicate whether the trajectory is correct. Any trajectory distribution µ induces answer mass and total correct-trajectory mass Qµ (a) = Pr [A(τ ) = a], Cµ = Eτ ∼µ [R(τ )]. (4) τ ∼µ

Base (central 50%) Power (central 50%)

0.0

0.2

0.4

0.6

0.8

Correct-answer mass C

1.0

(b) Coverage within correct set

2 1 0 −1 −2 −3 −4 −5

Base Power

Conditional mass density

Log density-ratio gain relative to the median

(a) Answer-level coverage

Correct-trajectory Base logp (median/IQR standardized)

Gold-answer margin G

1.00 0.75 0.50 0.25 0.00 −0.25 −0.50 −0.75 −1.00

= 4)

10--90% of problems Median Highest-dose 5%

5000 4000 3000 2000 1000 0

0.5

0.6

0.7

0.8

0.9

Within-problem Base-likelihood percentile

Figure 2: Three views of why Power sharpening need not improve downstream aggregation in a representative setting. (a) Power increases correct-answer mass C while reducing the gold-answer margin G; filled markers denote across-problem centers and translucent ellipses summarize the central 50% of problems. (b) Conditional on correctness, Power concentrates mass in a narrower, highBase-likelihood region; the two densities are mirrored only for visual comparison. (c) A fixed α produces widely different within-problem log-weight gains; the shaded region spans the 10th–90th percentiles across problems, and blue-gray curves mark the 5% of problems with the largest gain at the 90th likelihood percentile. When correctness is determined by a unique gold answer a⋆ , Cµ = Qµ (a⋆ ); for code, correctness may instead be determined by execution. Given N independently sampled trajectories, SelfConsistency selects N X b aSC = arg max 1{A(τi ) = a}. (5) a

i=1

For executable or open-ended outputs without a canonical answer string, ModeX instead derives consensus from a similarity graph over complete responses (Choi and Li, 2026). These readouts use different notions of agreement, but both convert support across multiple trajectories into a final decision.

M ISMATCH A NALYSIS

A higher chance of sampling a correct trajectory need not lead to a better decision from multiple samples. To understand this gap, we ask two questions that pass@k leaves unanswered: where does Power move probability mass within a problem, and does the same exponent move different problems by comparable amounts? These questions expose two complementary failure modes, which we call coverage mismatch and dose mismatch, as summarized in Figure 2. The figure uses one representative setting; the appendix reports the same diagnostics across all three evaluated models. 4.1

C OVERAGE M ISMATCH

Pass@k captures whether at least one correct trajectory appears among k samples. For independent draws from a trajectory distribution µ, k

pass@ k(µ) = 1 − (1 − Cµ ) .

(6)

Because this quantity depends only on Cµ , it says nothing about how correct mass is divided among trajectories or how incorrect mass is organized across competing answers. A high pass@k can therefore coexist with a smaller answer margin Gµ . Figure 2(a) contrasts total correct mass with the correct answer’s margin over its strongest competitor. Under Power, the problem distribution shifts toward larger correct mass but a smaller margin: the strongest incorrect answer gains still more mass. The answer-level view nevertheless leaves open where the additional correct mass goes. Figure 2(b) resolves this mass along within-problem standardized Base log-likelihood, with problems weighted equally. Power concentrates it in a narrower, higher-likelihood region of the correct set. 5

The two panels expose different aspects of coverage: panel (a) concerns competition between answers, whereas panel (b) concerns concentration within the correct set. The latter motivates examining response-level readouts such as ModeX, but does not by itself establish a change in response modes or in the winner of answer-based SC. Together, the two views show why total correct mass alone is insufficient to characterize the support available to aggregation. 4.2

D OSE M ISMATCH

Coverage mismatch concerns where mass moves. A separate failure arises because a fixed α does not impose a comparable deformation across problems. Power acts on raw sequence-level likelihood rather than a problem-relative scale. Its relative reweighting of two trajectories satisfies pj (τi )α−1 log = (α − 1) ℓj (τi ) − ℓj (τk ) . (7) pj (τk )α−1 Although α is shared across problems, the likelihood gap is not. Narrow gaps yield mild reweighting, whereas wide gaps can make the same exponent sharply concentrate the distribution. To place these deformations on a common trajectory coordinate, let Lj (u) be the u-quantile of Base log-likelihood for problem j. The Power log-weight gain at that position relative to the withinproblem median is ∆Power,j (u) = (α − 1) Lj (u) − Lj (1/2) . (8) This aligns trajectories by their within-problem position while retaining the actual log-weight gain applied by Power. This heterogeneity is not confined to selected trajectory pairs. If Eτ ∼pj [|ℓj (τ )|3 ] < ∞, then as α ↓ 1, the full target obeys (α − 1)2 Varτ ∼pj [ℓj (τ )] + O((α − 1)3 ). (9) 2 Thus, problem-specific likelihood dispersion controls the leading-order movement of the entire Power target. We defer the exact identity and derivation to the supplementary material. Figure 2(c) makes the consequence visible: the curves fan outward under a fixed α, with a small set of problems receiving markedly larger gains. DKL (qα,j ∥ pj ) =

Coverage mismatch concerns where mass moves; dose mismatch concerns how far the distribution moves. Power couples both effects through a single global exponent, motivating a repair that controls them separately.

M ETHOD

The preceding analysis points to two changes: likelihood must first be placed on a comparable within-problem scale, and the gain assigned on that scale must preserve useful coverage. Before defining these changes, we establish how a sequence-level target can be combined with a multisample readout without repeatedly running its sampler. 5.1

F ROM P OWER S AMPLING TO W EIGHTED C ONSENSUS

A single Power-SMC run uses M particles to return one approximate Power trajectory. A literal Power-SMC+SCN composition therefore requires N independent runs. We instead estimate the answer mass of the ideal Power target from one pool of N Base trajectories. By equation 3, its unnormalized importance ratio is rα (τ ) = p(τ )α−1 = exp((α − 1)ℓ(τ )) . For independent X1:N ∼ p and Y1:N ∼ qα , consider PN rα (Xi )1{A(Xi ) = a} weight b QN (a) = i=1 PN , i=1 rα (Xi ) N X b sample (a) = 1 Q 1{A(Yi ) = a}. N N i=1

(10)

(11)

Lemma 1 (Asymptotic expectation equivalence). For every fixed answer a, h i h i b weight (a) = lim E Q b sample (a) lim E Q N N N →∞

N →∞

(12)

= Qqα (a). Lemma 2 (Approximation rate). If 0 < qα (τ )/p(τ ) ≤ C almost surely for some finite C, then 2 b weight (a) − Q b sample (a) E Q = O(N −1 ), N N

(13)

b weight (a) − Q b sample (a) = Op (N −1/2 ). Q N N The proofs, including a finite-sample bound, are given in the appendix. If the answer space is finite and Qqα has a unique mode separated by a positive margin, both estimators consequently select the same mode with probability tending to one. This connection is deliberately made at the ideal-target level: it does not couple the result to the finite-M approximation error of Power-SMC, nor does it turn a deterministically weighted Base pool into fresh target samples. 5.2

R ELATIVE -R ANK S OFT S AT

Power reads the absolute gap in trajectory log-likelihood. To control dose across problems, we replace it with the within-problem rank up (τ ) = Pr [ℓ(X) ≤ ℓ(τ )] ∈ [0, 1]. X∼p

(14)

This coordinate depends only on a trajectory’s likelihood ordering within the same problem. We then shape its gain with n u o2 SoftSatm (u) = 1 − 1 − min ,1 , 0 < m ≤ 1, (15) m and define the corresponding reweighting multiplier rβ,m (τ ) = exp[β SoftSatm (up (τ ))] .

(16)

The resulting target is qβ,m (τ ) ∝ p(τ )rβ,m (τ ). The importance ratios in equation 10 and equation 16 expose the repair directly in the weighting view. Power reads the raw coordinate ℓ(τ ), so the same coefficient α − 1 inherits problem-specific likelihood gaps; replacing it with up (τ ) addresses dose mismatch. Power also applies a linear log-weight gain across that coordinate, whereas SoftSat raises the gain for moderate ranks and saturates it for u ≥ m; this change of shape addresses coverage mismatch. Because the SoftSat potential lies in [0, 1], the target-to-Base density ratio is bounded, so the preceding approximation applies to this target as well. 5.3

F INITE -P OOL W EIGHTS AND C ONSENSUS R EADOUT

Given a Base pool τ1:N , we estimate the relative rank in equation 14 by its empirical counterpart, N

u bi =

1 X 1{ℓ(τk ) ≤ ℓ(τi )}. N

(17)

k=1

We obtain the finite-pool multiplier by clipping the shaped gain, rei = clip[1−η, 1+κη] (exp[β SoftSatm (b ui ) + c]) .

(18)

The offset c makes these multipliers mean one; their normalized values are used as candidate weights W1:N . The weighted candidates are passed to a task-appropriate consensus readout D. For answer-based tasks, this gives weighted Self-Consistency, DSC (τ1:N , W1:N ) = arg max a

N X i=1

Wi 1{A(τi ) = a}.

(19)

For free-form responses, we apply the same candidate weights throughout the ModeX readout. Uniform weights recover the original readout in either case. Algorithm 1 Relative-Rank SoftSat with a Weighted Consensus Readout Require: Prompt x, Base model pθ , budget N , readout D, parameters (m, β, η, κ) Ensure: Consensus output yb 1: Sample τ1:N ∼ pθ (· | x) 2: Rank ℓ(τ1:N ) within the pool to obtain u b1:N 3: Apply SoftSat and bounded mean-one rescaling to obtain re1:N P 4: Normalize Wi ← rei / j rej 5: return yb ← D(τ1:N , W1:N ) Our method reuses the same N Base trajectories as uniform consensus, and therefore incurs the same model-generation cost with no additional model calls.

E XPERIMENTS

We evaluate whether Power’s failure persists across response spaces and whether SoftSat mitigates it without changing the Base candidate pool or generation budget. 6.1

E XPERIMENTAL S ETUP

Benchmarks and models. Our evaluation spans code, mathematical reasoning, and physics. We use the Full-Instruct split of BigCodeBench (1,140 problems) (Zhuo et al., 2025), the April 2024 slice of LiveAoPSBench (498 problems) (Mahdavi et al., 2025), and the text-only Regular subset of the PHYSICS official test split (503 problems) (Feng et al., 2025). We evaluate three models in the 4–9B range: NVIDIA-Nemotron-3-Nano-4B-BF16, Qwen3.5-9B, and Ministral-3-8B-Reasoning2512. Generation and readout. For every problem, we draw eight Base trajectories at temperature 1 with top-k = 50, using the benchmark prompt, the model’s chat template, and a maximum generation length of 16,384 tokens. All multi-sample methods reuse this pool. Uniform assigns equal mass to all trajectories, while Power uses α = 4. SoftSat uses one configuration across all model– benchmark pairs: m = 0.75, β = 1, η = 0.25, and κ = 64, giving clip bounds [0.75, 17]. Power and SoftSat therefore denote deterministic weighted readouts of the same Base pool, rather than fresh samples from either target distribution. We use a readout matched to each response space. BigCodeBench uses ModeX (Choi and Li, 2026) followed by the official execution evaluator. LiveAoPSBench and PHYSICS group equivalent extracted answers and select the group with greatest uniform or weighted mass; scoring follows their official equivalence procedures. We report each benchmark’s official score and do not average scores across benchmarks. 6.2

M AIN R ESULTS

Table 1 shows that Power is not a reliable alternative to uniform consensus. It underperforms Uniform in all six LiveAoPSBench and PHYSICS settings, by 1.808–18.474 points on LiveAoPSBench and 1.934–6.999 points on PHYSICS. On BigCodeBench, in contrast, Power improves Qwen3.5 and Ministral but degrades Nemotron. Its effect therefore varies substantially across response spaces and models. SoftSat is substantially more stable. Relative to Uniform, it improves five of the nine settings, ties one, and has a largest regression of 1.004 points. On the two largest Power failures, both with Nemotron, the gaps to Uniform fall from 18.474 to 1.004 points on LiveAoPSBench and from 6.999 to 0.255 points on PHYSICS. SoftSat does not uniformly outperform Uniform; the main observation is instead that it removes Power’s large regressions while retaining gains in five settings. 8

Table 1: Official benchmark scores (%) with eight Base trajectories per problem. Single uses the first sampled trajectory; the remaining columns apply task-specific consensus to the same pool. ∆ is SoftSat minus Uniform. Bold marks the best multi-sample result in each row. Benchmark

Model

Single

Uniform

Power (α=4)

SoftSat (β=1)

∆

BigCodeBench

Nemotron-4B Qwen3.5-9B Ministral-8B

17.456 34.386 29.123

24.386 36.579 33.421

22.544 38.333 34.825

24.561 37.719 34.912

+0.175 +1.140 +1.491

LiveAoPSBench

Nemotron-4B Qwen3.5-9B Ministral-8B

21.486 59.237 44.177

34.137 64.659 50.201

15.663 62.851 46.386

33.133 64.659 51.606

−1.004 0.000 +1.405

PHYSICS

Nemotron-4B Qwen3.5-9B Ministral-8B

18.869 69.422 35.957

23.891 72.913 43.164

16.892 70.979 39.332

23.636 72.822 43.686

−0.255 −0.092 +0.522

(a) Budget-dependent strength

(b) Accuracy across budgets

2.0 1.5 1

1.0 0.5

0.25

37.37

34.91

37.98

37.28

37.37

34 33.33 33.42

32 30

0.0

SC + SoftSat

Accuracy (%)

Best strength, β *

Uniform SC

31.32

Number of trajectories, N

Figure 3: Budget-dependent SoftSat behavior for Ministral on BigCodeBench. (a) The best sharpening strength β ∗ for each trajectory budget N . (b) Accuracy of uniform SCN and SoftSat-weighted SCN at the corresponding β ∗ . Results for all three benchmarks appear in the appendix. 6.3

B UDGET-D EPENDENT S TRENGTH

The appropriate sharpening strength changes with the number of available trajectories. Figure 3 reports a budget sweep for Ministral on BigCodeBench. The best β rises from 0.25 at N = 4 to 2 at N = 32, while SoftSat improves over uniform SC at each evaluated budget. Thus, a larger candidate pool can support a stronger reweighting rule, but the resulting accuracy must still be read against the corresponding uniform-consensus baseline. 6.4

R EPAIRING THE D IAGNOSED M ISMATCHES

The accuracy gains in Table 1 do not by themselves show that SoftSat addresses the failure identified in Section 4. Figure 4 therefore returns to the same representative setting as Figure 2 and overlays Base, Power, and SoftSat. At the answer level, Power moves the across-problem center from (C, G) = (0.433, 0.132) to (0.464, −0.072), whereas SoftSat gives (0.439, 0.133) and retains a coverage profile close to Base. Within the correct set, Power shifts conditional mass toward the highest-Base-likelihood trajectories, while SoftSat remains much closer to the Base profile. We do not repeat a dose panel for SoftSat. Its gain is defined on a bounded relative-rank coordinate, so fixed hyperparameters impose the same gain curve across problems, up to likelihood ties. The empirical question left for Figure 4 is therefore whether this controlled dose preserves useful coverage. 6.5

C HECKING THE W EIGHTED R EALIZATION

On all 1,118 human-verified MoreHopQA Case-5 examples (Schnitzler et al., 2024), repeated Power-SMC sampling and its weighted realization select the same normalized answer in 99.55% 9

(a) Answer-level coverage

1.00

(b) Coverage within correct set

Correct-trajectory Base logp (median/IQR standardized)

Gold-answer margin G

0.50 0.25 0.00

−0.25 −0.50

Base (central 50%) Power (central 50%) SoftSat (central 50%)

−0.75 −1.00

Base Power SoftSat

0.75

0.0

0.2

0.4

0.6

Correct-answer mass C

0.8

1 0

−1 −2 −3 −4 −5

1.0

Conditional mass density

Figure 4: SoftSat repairs the coverage distortions in the representative setting from Figure 2. Each panel overlays Base, Power, and SoftSat on the same candidate pool. (a) SoftSat restores the goldanswer margin without discarding correct-answer mass. (b) SoftSat avoids Power’s concentration on a narrow high-likelihood portion of the correct set. Full three-model diagnostics appear in the appendix.

of cases. Their accuracies differ by 0.089 percentage points, with a paired-bootstrap 95% confidence interval of [−0.268, 0.447], while mean inference time falls by 7.42×. Appendix C gives the full protocol and distributional diagnostics. This check supports the weighted realization in this finite setting; it is not a claim of universal equivalence or a test of the asymptotic convergence rate.

C ONCLUSION

We studied why sequence-level Power Sampling can improve the probability of correct trajectories yet degrade decisions formed from multiple samples. Our analysis identified two sources of this failure: coverage mismatch concentrates probability mass on a narrow set of dominant trajectories and answers, while dose mismatch makes a fixed exponent induce sharply different deformations across problems. This diagnosis led to Relative-Rank SoftSat, which replaces absolute likelihood with a problem-relative coordinate and saturates the gain assigned to high-probability trajectories. We also connected sampling from an ideal Power target to weighted consensus over a Base candidate pool, providing a practical way to combine sequence-level targets with multi-sample inference. Across models and reasoning benchmarks, standard Power produced large and inconsistent regressions, whereas SoftSat removed these failures while retaining gains in several settings; a finite-sample check further showed close agreement between repeated Power-SMC sampling and its weighted realization. More broadly, our results show that a sharpened distribution should be judged not only by the quality of individual trajectories or pass@k, but also by whether it preserves the support structure required by the downstream decision rule.

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F ULL M ISMATCH D IAGNOSTICS

Figure 5 extends the three diagnostics in Figure 2 to every evaluated model on LiveAoPSBench. Figure 6 similarly extends the coverage comparison in Figure 4.

B UDGET S WEEP ACROSS B ENCHMARKS

Figure 7 extends Figure 3 to the three evaluated benchmarks.

F INITE -S AMPLE C HECK OF THE W EIGHTED R EALIZATION

The preceding lemmas motivate a weighted realization of target answer mass, but they do not imply identical decisions for a finite particle system. We therefore perform a narrow check on all 1,118 human-verified MoreHopQA Case-5 examples (Schnitzler et al., 2024). With Qwen3.5-9B and α = 4, the sampling reference runs an M = 8 particle Power-SMC system independently N = 8 12

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Figure 5: Full Power mismatch diagnostics on LiveAoPSBench. Rows correspond to Nemotron-4B, Qwen3.5-9B, and Ministral-8B; columns follow Figure 2. times and applies uniform SC to the eight returned answers. The weighted realization instead runs Power-SMC once and aggregates the answer mass of its M = 8 terminal particles using their normalized weights. Figure 8 shows that 1,113 of 1,118 final decisions are unchanged. Weighting corrects three errors and introduces two, yielding 91.14% accuracy versus 91.06% for repeated sampling, a difference of 0.089 percentage points with a paired-bootstrap 95% confidence interval of [−0.268, 0.447]. The underlying answer distributions agree nearly as closely: 80.86% have DTV at most 10−7 and 94.36% have DTV at most 0.01, while only 0.36% exceed 10−0.5 . Mean inference time falls from 19.50 to 2.63 seconds per question, a 7.42× reduction.

P ROOFS FOR W EIGHTED TARGET A PPROXIMATION

Proof of Lemma 1. Fix an answer a and write fa (τ ) = 1{A(τ ) = a}. Define N

UN =

1 X rα (Xi )fa (Xi ), N i=1

1 X rα (Xi ). N i=1

(20)

Ep [rα (X)] = Zα .

(21)

RN =

Since qα (τ ) = p(τ )rα (τ )/Zα , Ep [rα (X)fa (X)] = Zα Qqα (a),

The strong law of large numbers therefore gives U N → Zα Qqα (a) and RN → Zα almost surely. Because Zα > 0, their ratio converges almost surely to Qqα (a). Moreover, 0 ≤ U N /RN ≤ 1, so 13

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Figure 6: Full coverage comparison among Base, Power, and SoftSat on LiveAoPSBench. Rows correspond to Nemotron-4B, Qwen3.5-9B, and Ministral-8B; columns follow Figure 4. bounded convergence yields i h b weight (a) = Qq (a). lim E Q α N

N →∞

(22)

b sample (a)] = Qq (a) for For target samples, fa (Yi ) is Bernoulli with mean Qqα (a). Hence E[Q α N every N , proving the result. Proof of Lemma 2. Let ρ(τ ) = qα (τ )/p(τ ) and θ = Qqα (a). Multiplying all weights by the same positive constant does not change the self-normalized estimator, so b weight (a) − θ = H N , Q N ρN N

HN =

1 X ρ(Xi )(fa (Xi ) − θ), N i=1

ρN =

1 X ρ(Xi ). N i=1

(23)

Here Ep [ρ(X)] = 1 and Ep [ρ(X)(fa (X) − θ)] = 0. Put σa2 = Varp [ρ(X)(fa (X) − θ)], which is finite because 0 < ρ ≤ C. Consider the event EN = {ρN ≥ 1/2}. On EN , equation 23 implies 2 σ2 2 b weight (a) − θ ≤ 4H 2N , Q E[H N ] = a . N N 14

(24)

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Figure 7: Budget-dependent SoftSat behavior for Ministral across BigCodeBench, LiveAoPSBench, and PHYSICS. Within each benchmark, the upper panel shows the best β at each budget and the lower panel compares uniform SCN with SoftSat-weighted SCN .

99.55%

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Total-variation distance, DTV

Figure 8: Finite-sample agreement between repeated Power-SMC sampling and its weighted realization on MoreHopQA Case-5. (a) The two procedures return the same normalized answer on 99.55% of questions; the remaining changes are split by whether weighting corrects or introduces an error. (b) Most answer distributions are also close in total variation distance (DTV), although a small tail remains. c On EN , the squared error is at most one. Since 0 ≤ ρ(X) ≤ C and E[ρ(X)] = 1, Hoeffding’s inequality gives N c Pr(EN ) = Pr(ρN − 1 < −1/2) ≤ exp − 2 . (25) 2C

Combining the two events, 2 4σ 2 N weight a b E QN (a) − θ ≤ + exp − 2 . N 2C

(26)

The target-sampling estimator is unbiased with variance θ(1 − θ)/N . It is independent of the Base-pool estimator, so the cross term between their centered errors vanishes. Writing DN (a) = 15

b weight (a) − Q b sample (a), Q N N 4σa2 + θ(1 − θ) N E[DN (a) ] ≤ + exp − 2 N 2C 2

(27)

= O(N −1 ). The Op (N −1/2 ) statement follows immediately from this second-moment bound and Markov’s inequality. Finally, suppose the answer space is finite and a⋆ is the unique mode of Qqα with margin δ = Qqα (a⋆ ) − maxa̸=a⋆ Qqα (a) > 0. Both estimators converge in probability to Qqα (a) for every answer. A union bound over the finite answer space shows that, with probability tending to one, all estimated masses are within δ/3 of their targets. On this event, both estimators select a⋆ , establishing the mode-agreement consequence stated in the main text.

More Correct Mass, Worse Answers: Why Power Sampling Can Fail and How to Fix It

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