From Candidate Tokens to Final Outcomes: Decoding Relative Long-Horizon Candidate Value from Reasoning Model Representations
Authors: Yixin Yang, Yue Wang, Fandong Meng, Zhifang Sui
Organizations: State Key Laboratory of Multimedia Information Processing, School of Computer Science, Peking University · WeChat AI, Tencent Inc., China
Large reasoning models generate reasoning trajectories token by token. At a given prefix, multiple next-token candidates may be plausible, yet shape subsequent reasoning and differ in their chances of leading to the correct answer. Whether these differences are reflected in internal representations remains unclear. To study this question, we define relative long-horizon candidate value to quantify within-prefix differences and construct the Reasoning Candidate-Token Value (RCTV) dataset, pairing mathematical reasoning prefixes with plausible next-token candidates and their empirical long-horizon candidate values. Using RCTV, we examine hidden states, multilayer perceptron outputs, and attention outputs at candidate positions with sparse linear probes. Experimental results provide evidence that relative long-horizon candidate value is linearly decodable from these representations at multiple layers. Candidate-value probes can also outperform probability- and embedding-based baselines. Furthermore, a controlled online intervention experiment shows that using the decoded signals to guide token selection during generation can improve the chances of eventual success. Our findings provide evidence that internal representations encode information about relative long-horizon candidate value, offering insights into how local candidate-level representations relate to long-horizon reasoning outcomes.
Figures & tables
Figure 1: Illustration of our candidate-level analysis of long-horizon reasoning outcomes. Three plausible candidates at a shared reasoning prefix are shown. pp denotes percentage points. In this example, the pairwise ordering produced by the probe matches the empirical ordering of candidates by relative long-horizon value.
Subset
Model
Entropy
# Problems
# Candidates
# Rollouts
RCTV-Core
Qwen3-1.7B
[q.95,1]
500
2,715
347,520
RCTV-Qwen8B
Qwen3-8B
[q.95,1]
500
3,152
403,456
RCTV-Llama8B
DS-Llama-3.1-8B
[q.95,1]
200
3,051
390,528
RCTV-InterEnt
Qwen3-1.7B
[q.85,q.95)
500
1,678
214,784
Table 1: Dataset statistics for RCTV. The four subsets cover three reasoning models and two entropy regimes.
Figure 2: Candidate-value readout across layers and selected dimensionalities. Each cell reports the mean pairwise AUC over ten problem-level fold seeds for hidden states, MLP outputs, and attention outputs. Within each representation family, ⋆ , ⋄ , △ , and ∘ denote the first- through fourth-highest observed mean AUC, respectively.
Repr
k
Pairwise ↑
Tier ↑
Regret@1 ↓
Baselines
Uniform
-
50.0
50.0
0.0355
Log prob
1
50.9
51.6
0.0346
Prob rank
1
52.4
51.8
0.0346
Embedding
16
52.2 ± 2.0
52.2 ± 1.9
0.0340 ± 0.0014
Act norm ( M9 )
1
53.2 ± 0.9
54.5 ± 0.8
0.0338 ± 0.0000
Table 2: Candidate-value readout results and baselines. We report the mean and standard deviation over ten problem-level fold seeds. For each internal-representation family and the embedding baseline, we report the strongest observed configuration according to mean pairwise AUC.
Figure 3: Candidate-value readout from candidate-position MLP outputs across Qwen3-1.7B, Qwen3-8B, and DS-Llama-3.1-8B. Each cell reports mean pairwise AUC over ten problem-level fold seeds. ⋆ , ⋄ , △ , and ∘ mark the first- through fourth-highest observed mean AUC within each model. Layer indices are normalized by model depth.
Configuration
Main readout
Controls
Alternative readouts
Layer
k
Top- k ridge
Random dims
Permuted labels
Signed sum
Corr-weighted
M9
4
58.7 ± 2.6
50.1 ± 0.7
49.9 ± 0.7
58.7 ± 2.6
58.8 ± 2.5
M9
8
58.3 ± 2.3
50.4 ± 1.0
50.5 ± 0.9
58.3 ± 2.6
58.7 ± 2.4
M17
2
58.1 ± 2.5
50.1 ± 0.9
49.6 ± 0.5
58.1 ± 2.7
58.3 ± 2.6
M17
4
58.0 ± 2.7
49.7 ± 0.7
49.6 ± 0.6
57.8 ± 3.0
58.2 ± 2.8
Table 3: Robustness of four high-performing candidate-value MLP readouts. Values are mean pairwise AUC (%) and standard deviation over ten problem-level fold seeds. The table compares the primary Top- k ridge readout with null controls and alternative readout rules. Control repetitions are first averaged within each fold seed.
M9 ( k=4 )
Dim
Selected
Direction
432
45/50
+ 45/45
950
38/50
− 38/38
256
18/50
+ 18/18
1274
17/50
− 17/17
Table 4: Selection frequency and sign consistency of the most frequently selected MLP dimensions across 10 problem-level fold seeds (50 training folds). Direction denotes the sign of the training-fold correlation.
Figure 4: Mean continuation accuracy under different intervention budgets B . The dashed line denotes generation without intervention.
Policy
Avg@128
Δ vs. Standard (95% CI)
Standard
50.21
—
Uniform
51.84
+1.63 [-2.03, +5.30]
Top-prob
50.56
+0.35 [-3.47, +4.18]
Probe-guided
54.51
+4.30 [+0.60, +8.28]
Table 5: Online intervention results at B=64 . Avg@128 is reported in percent; differences are in percentage points with 95% paired problem-bootstrap CIs.
Appendix figures & tables20 assets
Supplementary material from the paper’s appendix.
Appendix
Subset
Source C/W
Difficulty 2/3/4/5/6
Position median [ p10 , p90 ]
Candidates/prefix mean (median)
RCTV-Core
252/248
100/100/100/100/100
2,259 [420, 6,224]
5.43 (5.0)
RCTV-Qwen8B
236/264
100/100/100/100/100
2,960 [851, 6,641]
6.30 (6.0)
RCTV-Llama8B
97/103
40/40/40/40/40
3,232 [800, 6,860]
15.26 (15.0)
RCTV-InterEnt
252/248
100/100/100/100/100
1,894 [433, 5890]
3.36 (3.0)
Appendix
Table 6: Additional statistics for the RCTV variants. C/W denotes prefixes drawn from source trajectories with correct and incorrect final answers. Difficulty entries report the numbers of problems in strata 2–6, where each stratum is determined by the number of successful outcomes among eight sampled generations. Prefix positions are measured in tokens from the beginning of the generated reasoning trajectory.
Figure 5: Original Big-Math source composition of the selected problems in each RCTV variant. The four most common sources are shown separately. Other comprises AOPS Forum, Math, OmniMath, HARP, GSM8K, and OpenMath. Percentages are normalized within each variant.
Figure 6: Additional analyses of within-prefix candidate-value variation. Left: Value-gap distributions grouped by whether the source trajectory produces a correct final answer. Right: Value gaps across prefix positions; the line shows binned means and the shaded region indicates 95% bootstrap confidence intervals.
Empirical gap
# Pairs
# Prefixes
Median conf.
Pairs ≥.90
HC prefixes
>0
5,515
460
0.794
26.2%
340
≥0.04
2,043
328
0.915
56.2%
285
≥0.08
729
199
0.971
100.0%
199
Appendix
Table 7: Pairwise uncertainty in rollout-estimated candidate values. Directional confidence is the posterior probability that the empirically higher-valued candidate has the higher underlying success probability, computed from all 128 rollouts using a Jeffreys prior. The final column, labeled high-confidence (HC) prefixes, reports prefixes containing at least one pair with directional confidence of at least 0.90 .
Model
Parameters
Number of layers
Hidden size
Qwen3-1.7B
1.7B
28
2,048
Qwen3-8B
8.2B
36
4,096
DeepSeek-R1-Distill-Llama-8B
8B
32
4,096
Appendix
Table 8: Architectural specifications of the reasoning models used in our experiments.
Figure 7: Distribution of value gaps in RCTV-Core.
Figure 8: Candidate-value probing of prefix–token interaction representations across layers and selected dimensionalities. Each cell reports the mean pairwise AUC over ten problem-level fold seeds for attention outputs, MLP outputs, and hidden states. Within each representation family, ⋆ , ⋄ , △ , and ∘ mark the first- through fourth-highest observed mean pairwise AUC, respectively.
Repr
k
Pairwise ↑
Tier ↑
Regret@1 ↓
Attn ( A15 )
128
59.4 ± 3.2
57.5 ± 2.2
0.0328 ± 0.0012
MLP ( M20 )
256
59.2 ± 2.6
57.8 ± 2.5
0.0322 ± 0.0008
Hidden ( H28 )
256
56.4 ± 2.0
54.3 ± 2.0
0.0345 ± 0.0014
Appendix
Table 9: Candidate-value readout results for prefix-token interaction representations. We report the mean and standard deviation over ten problem-level fold seeds. For each representation family, we report the strongest observed configuration according to mean pairwise AUC.
Figure 9: Sensitivity of candidate-value readout performance to the value-gap margin m (left) and tier threshold ϵ (right) on RCTV-Core. Points and error bars indicate the mean and standard deviation across ten problem-level fold seeds. Log Prob uses untrained next-token log probabilities and therefore does not vary across fold seeds. Dashed lines indicate chance-level AUC.
Figure 10: Standard deviation of pairwise AUC across ten problem-level fold seeds for candidate-value readouts. Each cell corresponds to a layer and selected dimensionality.
Figure 11: Fraction of the ten problem-level fold seeds with pairwise AUC above 0.5 for candidate-value readouts.
Repr
k
Pairwise ↑
Tier ↑
Regret@1 ↓
Baselines
Uniform
-
50.0
50.0
0.0386
Log prob
1
48.2
48.5
0.0412
Prob rank
1
52.5
49.5
0.0412
Embedding
64
55.1 ± 1.8
53.0 ± 1.3
0.0393 ± 0.0010
Candidate-value readouts
Appendix
Table 11: Candidate-value readout results and baselines for Qwen3-8B. For fitted readouts, we report the mean and standard deviation over ten problem-level fold seeds; parameter-free baselines are reported as single values. For each internal-representation family and the embedding baseline, we report the strongest observed configuration according to mean pairwise AUC.
Repr
k
Pairwise ↑
Tier ↑
Regret@1 ↓
Baselines
Uniform
-
50.0
50.0
0.0490
Log prob
1
50.9
50.1
0.0509
Prob rank
1
50.7
50.5
0.0509
Embedding
128
52.6 ± 1.7
53.2 ± 1.5
0.0504 ± 0.0013
Candidate-value readouts
Appendix
Table 12: Candidate-value readout results and baselines for DS-Llama-3.1-8B. For fitted readouts, we report the mean and standard deviation over ten problem-level fold seeds; parameter-free baselines are reported as single values. For each internal-representation family and the embedding baseline, we report the strongest observed configuration according to mean pairwise AUC.
Figure 12: Candidate-value readout from hidden states, MLP outputs, and attention outputs in Qwen3-8B. Each cell reports mean pairwise AUC across ten problem-level fold seeds. Within each representation family, ⋆ , ⋄ , △ , and ∘ mark the first- through fourth-highest observed mean pairwise AUC, respectively.
Figure 13: Candidate-value readout from hidden states, MLP outputs, and attention outputs in DS-Llama-3.1-8B. Each cell reports mean pairwise AUC across ten problem-level fold seeds. Within each representation family, ⋆ , ⋄ , △ , and ∘ mark the first- through fourth-highest observed mean pairwise AUC, respectively.
Repr
Decoder
k
Pairwise ↑
Tier ↑
Regret@1 ↓
Baselines
Uniform
-
-
50.0
50.0
0.0275
Log prob
-
1
54.9
54.9
0.0281
Prob rank
-
1
55.2
53.3
0.0281
Fixed Core-selected dimensions at M17
MLP ( M17 )
Ridge
2
57.3 ± 2.9
56.1 ± 2.3
0.0241 ± 0.0006
Appendix
Table 13: Readout results with fixed Core-selected dimensions on RCTV-InterEnt. The dimension sets are {1834,312} and {1834,312,2021,822} at M17 , and {432,950} and {432,950,256,1274} at M9 . Decoder weights are estimated using each InterEnt training fold. We report mean and standard deviation over ten problem-level fold seeds; parameter-free baselines are reported as single values.
B
Strategy
Avg@128 (%)
Δ [95% CI]
0
Standard
54.93
—
8
Probe-guided
57.10
+2.16 [-1.76, +6.28]
16
Probe-guided
57.42
+2.49 [-2.04, +7.79]
32
Probe-guided
60.64
+5.71 [+1.20, +10.58]
64
Probe-guided
59.63
+4.70 [+0.65, +9.54]
Appendix
Table 14: Online intervention results on the 61 problems outside the offline pairwise-evaluation subset. Continuation accuracy is reported in percent, and differences relative to standard sampling are reported in percentage points with paired 95% bootstrap confidence intervals. Standard sampling involves no interventions ( B=0 ).
B
Strategy
Trajectory 0
Trajectory 1
0
Standard
46.98
53.44
8
Probe-guided
47.49
54.12
16
Probe-guided
48.34
56.23
32
Probe-guided
50.87
56.87
64
Probe-guided
52.35
56.67
Appendix
Table 15: Continuation accuracy (%) reported separately for the two trajectory seeds, estimated using 128 continuations per prefix.
B
Strategy
Pass@4
Pass@8
Pass@16
Pass@32
0
Standard
57.14
59.54
61.35
62.82
8
Top-prob
59.37
62.16
64.31
65.91
Uniform
58.11
60.86
62.92
64.51
Probe-guided
57.78
60.64
62.61
63.89
16
Top-prob
57.01
59.72
61.70
63.06
Uniform
58.77
61.01
62.62
63.93
Appendix
Table 16: Prefix-conditioned Pass@ k (%) across intervention budgets. Standard sampling involves no interventions.
B
Policy
Avg@128
Δ vs. Standard [95% CI]
Δ vs. Uniform [95% CI]
Δ vs. Top-prob [95% CI]
0
Standard (no intervention)
50.21
—
—
—
8
Uniform
51.63
+1.42 [-2.36, +5.10]
—
-0.61 [-3.74, +2.43]
Top-prob
52.23
+2.03 [-1.54, +5.70]
+0.61 [-2.43, +3.74]
—
Probe-guided
50.81
+0.60 [-2.48, +3.81]
-0.82 [-3.67, +2.06]
-1.43 [-3.97, +0.88]
16
Uniform
52.21
+2.00 [-0.59, +4.80]
—
+1.56 [-2.14, +5.16]
Top-prob
50.64
+0.44 [-3.43, +4.41]
-1.56 [-5.16, +2.14]
—
Appendix
Table 17: Mean continuation accuracy under online intervention. Differences are in percentage points; intervals are 95% paired problem-bootstrap CIs. Under interventions based on candidate-value readouts, continuation accuracy increases with the budget, with the largest gain at B=64 .
Chain-of-thought (CoT) reasoning improves multi-step problem solving, but long reasoning traces inflate inference cost. Token-level CoT compression reduces this cost by pruning full reasoning chains into shorter traces for model adaptation, making token selection the central challenge. Existing methods often rely on external scorers or heuristic signals only indirectly tied to the model's internal answer computation. We instead adopt a model-internal perspective: as the model forms an answer, each reasoning token induces a ripple in the residual stream whose effect on the answer reflects the token's contribution to the underlying computation. Building on this view, we propose MIST (Model-Internal Saliency for Token-level CoT compression), which defines token importance along two complementary axes: necessity, the drop in answer likelihood when a token's internal contribution is removed, and sufficiency, the gain in answer likelihood when that contribution alone is provided. Combining the two yields a unified importance score for pruning. Across four reasoning benchmarks and four models, MIST consistently outperforms baseline methods, suggesting that model-internal saliency provides an effective proxy for reasoning-token importance.
Chain-of-Thought reasoning has enabled large language models to achieve substantial performance gains on complex tasks. However, these gains come at the cost of dramatically increased token consumption. This raises a fundamental question: is every token in the reasoning trace equally valuable? We present a diagnostic and optimization framework grounded in a key empirical finding: the value of tokens within a CoT reasoning sequence is highly non-uniform, and this non-uniformity can be effectively characterized by token-level log probability signals. We show that normalized log probability helps distinguish core tokens, which carry structural and decisive reasoning content, from redundant tokens, which are exploratory, low-confidence filler that contributes less directly to the final answer. Building on these findings, we formulate the TokenProbe framework around two empirical findings and one claim: findings identify token value inequality first and then establish TokenProbe as a core-token proxy, and the claim introduces an efficient GRPO objective positing that selectively compressing redundant tokens can yield Pareto improvements in the accuracy-token efficiency space. Empirically, our method preserves reasoning quality while reducing the token usage by 76% of the baseline. Under matched reasoning-length budgets, we show that it can even outperform strong flagship baselines like Gemini-3.1-Pro. Homepage: https://runjia.tech/tokenprobe/.
Runjia Zeng, Hang Hua, Yiyang Liu +5
Rochester Institute of Technology · Purdue University · MIT-IBM Watson AI Lab +3
Large reasoning models (LRMs) substantially outperform their base LLM counterparts on challenging reasoning benchmarks, yet it remains poorly understood where base models go wrong during token-by-token generation and how to narrow this gap efficiently. We study the base-reasoning gap through quantifying token-level distributional disagreement between a base model and a stronger reasoning model using likelihood-based divergences. Across benchmarks, we find that the reasoning advantage is highly sparse and concentrates on a small set of early, planning-related decision tokens. For instance, on Qwen3-0.6B, only ~8% of generated tokens account for the salient disagreement, and these tokens concentrate early in the response, are strongly enriched in planning-related decisions (17x), and coincide with high base-model uncertainty -- suggesting that base models fail mainly at early planning points that steer the subsequent reasoning trajectory. Building on these findings, we propose disagreement-guided token intervention, a simple inference-time delegation scheme that performs a one-token takeover by the reasoning model only at high-disagreement positions and immediately switches back to the base model. With a small intervention budget, this sparse delegation substantially recovers and can even surpass the performance of a same-size reasoning model on challenging reasoning tasks. Code is available at https://github.com/AlphaLab-USTC/RRTokenIntervention.
Changshuo Shen, Leheng Sheng, Yuxin Chen +2
University of Science and Technology of China, Hefei, China · National University of Singapore, Singapore