Organizations: University of California, Riverside, CA 92521. Work performed during a summer internship at Lawrence Livermore National Laboratory · Lawrence Livermore National Laboratory, Livermore, CA 94550.
Explicitly eliciting reasoning substantially improves LLM performance. Existing approaches require a predefined characterization of reasoning, whether through CoT prompt design, contrastive CoT directions, or via SAE derived reasoning features. For mathematical reasoning with verifiable answers, we show that a much simpler signal suffices, which is the correctness of the model's own direct answer attempts. This signal yields a latent direction that elicits reasoning. This direction is decodable within the activations of most attention heads, but only a small subset of them can be effectively intervened. We introduce COMPASS, an inference-time steering method that identifies these heads using a logit-space attribution score and steers their activations along the correctness direction, requiring only per-head activation statistics. Across three model families and multiple math benchmarks, COMPASS outperforms the activation-steering baselines we compare against, improves GSM8K accuracy by 16 percentage points on average, and approaches CoT accuracy with 20-70% fewer generated tokens. Interventions transfer without re-fitting to unseen benchmarks, and ablations show that both the correctness direction and the small set of heads carrying it are necessary, with the effect concentrated in remarkably few heads.
Figures & tables
Figure 1 : Left: The per-head correctness direction. In the activation space of a head, activations from correctly and incorrectly answered problems form two populations, with means μ+l,h and μ−l,h . θlh is the unit vector connecting these means, and σlh is the standard deviation of the activations projected onto θlh . Steering shifts an activation on the incorrect side by ασlh along θlh . Right: Head selection by attribution score. Top: the score slh of each head, with the K heads selected within the promoted layer band outlined in blue. Bottom: the outcome contrast Δp measured on the fitting rows. First generated tokens for correct answers concentrate on “To”, and for incorrect answers on “$”.
Figure 2 : Steering elicits a short derivation, not a chain-of-thought transcript. The direct-answer prompt gets the answer wrong. With COMPASS, the same model reaches the correct answer with a short derivation, generating 3.0 – 3.8× lesser tokens than a chain-of-thought prompt.
Llama-3.1-8B
Qwen3-4B
Gemma4-12B
GSM8K
SVAMP
MATH-500
GSM8K
SVAMP
MATH-500
GSM8K
SVAMP
MATH-500
Standard Prompt
66.8 (93)
81.0 (48)
42.4 (345)
76.2 (68)
81.0 (22)
55.2 (250)
49.4 (67)
74.7 (3)
68.4 (347)
ITI Li et al. (2023)
73.8 (132)
68.7 (45)
23.6 (409)
71.9 (45)
44.3 (78)
30.6 (56)
22.3 (52)
64 (46)
30.3 (872)
Fractional Reasoning Liu et al. (2025)
79.9 (214)
81.0 (119)
42.2 (524)
88.3 (220)
89.3 (118)
77.8 (561)
65.4 (55)
85.6 (7)
72.7 (371)
COMPASS (ours)
83.7 (143)
85.7 (81)
45.2 (515)
89.8 (138)
90.7 (68)
77.8 (404)
88.0 (140)
79.3 (51)
79.8 (366)
CoT (oracle)
83.4 (205)
85.0 (157)
47.0 (788)
92.3 (303)
91.7 (224)
83.2 (924)
96.1 (338)
94.3 (261)
91.4 (730)
Table 1 : Accuracy (%) scores with mean generated tokens in gray. The heads selected by COMPASS lift the direct-answer (standard prompt) baseline toward the chain-of-thought and frontier oracles at a fraction of their token cost. ITI and Fractional Reasoning were re-run and we report their optimal performance.
Llama-3.1-8B
Qwen3-4B
GSM-Plus
HARP
GSM-Plus
HARP
Standard Prompt
60.4 (136)
21.5 (366)
65.6 (82)
35.4 (396)
COMPASS (ours)
71.1 (180)
22.7 (750)
80.4 (164)
49.1 (575)
CoT (oracle)
72.7 (247)
27.1 (1020)
84.1 (360)
56.9 (1505)
GPT-5.4-mini (oracle)
88.4 (385)
69.7 (1509)
88.4 (385)
69.7 (1509)
Table 2 : Transfer of steering heads and strength selected on GSM8K (to GSM-Plus) and MATH (to HARP), applied zero-shot, without re-tuning. Accuracy (%) with mean generated tokens in gray.
Figure 3 : Aggregated layer score The sum of the positive head-attribution scores in each layer. Yellow marks the layers from which the steering heads are selected based on the score (Eq 9 ).
Figure 4 : Best development set accuracy per candidate band. Bands are compared at constant α and the comparison isolates which heads are steered rather than how hard. Yellow is the band the scan promoted (Llama 32–32 at α=15.66 ; Qwen3 35–36 at α=25.53 ). The dashed line is the unsteered direct-answer baseline on the same rows.
Figure 5 : Head-level ablations. Left: Head and direction controls, on the full 500-row split. K and α are identical across bars. Only the intervention differs (error bars: sample standard deviation). Middle: Accuracy as only the K best-ranked heads are steered, with band and α frozen. Right: Reverses the sign of the injected direction at the selected heads.
Appendix figures & tables1 asset
Supplementary material from the paper’s appendix.
Appendix
Dataset
Description
GSM8K [ 14 ]
Grade-school word problems with integer answers; 7,473 train and 1,319 test problems.
SVAMP [ 15 ]
1,000 one-step arithmetic word problems built by perturbing existing problems so that surface cues do not determine the operation; 700/300 train/test split of the public release (ChilleD/SVAMP). Train and test share no question text.
MATH-500 [ 34 ]
The 500-problem subset of the MATH test set [ 2 ] , seven subjects (prealgebra to precalculus) at five difficulty levels, LaTeX answers. No fitting is done on these 500 problems: the Llama selection is fitted on the 7,500-problem MATH train split, and the Qwen3 selection on the remaining 4,500 MATH test problems (MATH test minus MATH-500, matched on normalized problem text and verified disjoint).
GSM-Plus [ 16 ]
10,552 adversarial perturbations of GSM8K test problems across eight perturbation types. The 1,319 critical-thinking rows are unanswerable by construction (gold “None”) and are dropped, leaving 9,233 evaluated problems.
HARP [ 17 ]
4,780 short-answer problems from US national mathematics competitions (AMC, AIME, and others), six difficulty levels, LaTeX answers. No train split by design; steered with the MATH selection.
Appendix
Table 3 : Benchmarks. Train splits are fitting pools only; test splits are evaluated in full. GSM-Plus and HARP are used for transfer only.
Mathematical reasoning benchmarks are typically organized by topic, but language models may organize their internal computation by reusable reasoning approach instead. In this paper, we investigate whether open math-capable LLMs organize internally by topical sub-skill or by reasoning approach, and we present evidence that the approach is the key. We introduce a generation-replay protocol: a model first generates a solution, after which we replay the exact prompt-plus-generation trajectory and extract activation-importance signatures over the reasoning tokens. We cluster these signatures without supervision across eight models and five mathematical reasoning sources, then evaluate the recovered structure with structural, semantic, and intervention tests. Across all 40 model-source cells, the recovered clusters outperform matched-size random baselines. Two independent frontier-LLM judges find approach-level coherence in 77-82% of real clusters versus 6-11% in within-source controls, and topic-pure clusters usually receive labels finer than the topic itself. In approach-controlled prompting, changing the requested reasoning approach shifts cluster assignment in seven of eight model conditions, whereas paraphrases largely preserve it. These results indicate that math-capable LLMs organize internal mathematical computation by reasoning approach rather than benchmark topic. The implication is that topic-stratified benchmarks and topic-balanced training corpora can still miss the axis that matters: even deliberately topic-balanced corpora may remain imbalanced over reasoning approaches.
Chain-of-Thought (CoT) prompting remains the standard baseline for evaluating models' reasoning abilities. Originally, this technique was introduced to elicit step-by-step reasoning from large language models (LLMs), which would otherwise tend to directly output the final answer. However, many modern LLMs produce CoT-style responses \textit{natively} when presented with reasoning tasks, which made us revisit the effectiveness of standard CoT prompting. We evaluate several modern mid-sized language models on a math problem-solving task and find that models specialized for reasoning achieve better performance in a simple zero-shot setting than when using few-shot CoT examples - significantly surpassing officially reported results at no additional cost (e.g., from ∼77% to ∼84% for Mathstral on GSM8K). For the tested general-purpose model, a zero-shot CoT prompt is also sufficient to outperform a few-shot CoT baseline. We attribute this to a `guidance-distraction' tradeoff: standard CoT prompting also demands style adaptation, formatting compliance, and potentially undesired contextualization, which can distract models from the core reasoning task. Our findings suggest that using standard CoT prompting increasingly acts as a source of distraction as models grow stronger.
Large language models frequently produce errors in reasoning tasks despite possessing the underlying knowledge required for correct reasoning. One possible approach to improve reasoning consistency is through activation steering. However, existing activation steering approaches apply fixed, pre-computed correction vectors, ignoring where the model currently sits along its generation trajectory; the result is indiscriminate perturbation that disrupts already-correct steps as freely as erroneous ones. We propose Manifold-Guided Attention Steering (MAGS), a trajectory-aware inference-time intervention grounded in a geometric observation: the output activations of specific attention heads diverge from a low-dimensional correctness manifold at the point of error, and this deviation compounds through subsequent steps. For each identified attention head, we learn a low-dimensional subspace from contrastive pairs of correct and incorrect traces that capture the directions along which error behavior deviates from correct behavior. During inference, we monitor each head's proximity to this manifold and apply a targeted projection correction when deviation exceeds a learned threshold, steering the attention output back toward the correct subspace before the error propagates. MAGS consistently outperforms both unsteered baselines and static steering approaches across benchmarks spanning mathematical reasoning (MATH-500, GSM8K), code generation (HumanEval, MBPP), and molecular generation (SMILES), suggesting that correctness manifolds are a general feature of LLM attention geometry.