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COMPASS: Finding Where Reasoning Lives in Language Models
Pratyay Dutta, Kowshik Thopalli, Vivek Narayanaswamy
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%
Last extracted: 10/7/2026, 9:22:47 AM
Summary
The paper introduces COMPASS, an inference-time steering method for Large Language Models (LLMs) that improves mathematical reasoning performance without requiring Chain-of-Thought (CoT) prompting or auxiliary models. COMPASS identifies a 'correctness direction' in the activation space of attention heads by comparing mean activations of correct versus incorrect direct answers. It then selects a small subset of heads that causally influence the output towards correct answers using a logit-space attribution score. Steering these specific heads along the correctness direction significantly improves accuracy on benchmarks like GSM8K and MATH-500 while generating fewer tokens than CoT methods.
Entities (9)
Relation Signals (7)
COMPASS → improves → GSM8K
confidence 95% · COMPASS outperforms the activation-steering baselines we compare against, improves GSM8K accuracy by 16 percentage points on average
COMPASS → targets → Attention Heads
confidence 95% · COMPASS... identifies these heads using a logit-space attribution score and steers their activations
COMPASS → uses → Correctness Direction
confidence 95% · We introduce COMPASS... and steers their activations along the correctness direction
LLaMA-3.1-8B → evaluatedon → GSM8K
confidence 90% · Llama-3.1-8B, GSM8K. Mary has 30 sheep.
Qwen3-4B → evaluatedon → MATH-500
confidence 90% · Qwen3-4B, MATH-500. In how many ways can 6 girls and 2 boys sit in a row
COMPASS → outperforms → Chain-of-Thought
confidence 90% · approaches CoT accuracy with 20-70% fewer generated tokens
Sparse Autoencoders → usedfor → Reasoning
confidence 85% · Sparse Autoencoders (SAEs) have been used to elicit reasoning by identifying a reasoning-associated feature
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Abstract
Abstract: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.
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- Source: https://arxiv.org/abs/2610.07469v1
- Canonical: https://arxiv.org/abs/2610.07469v1
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COMPASS: Finding Where Reasoning Lives in Language Models Pratyay Dutta †thanks: University of California, Riverside, CA 92521. Work performed during a summer internship at Lawrence Livermore National Laboratory Email: pdutt005@ucr.edu Kowshik Thopalli †thanks: Lawrence Livermore National Laboratory, Livermore, CA 94550. Email: thopalli1@llnl.gov Vivek Narayanaswamy22footnotemark: 2 Email: narayanaswam1@llnl.gov August 2026 Abstract 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. 1 Introduction Large language models (LLMs) exhibit strong reasoning capabilities across a wide range of tasks Kojima et al. (2022); Hendrycks et al. (2021). A standard approach for eliciting reasoning is chain-of-thought (CoT) prompting Wei et al. (2022), which encourages the model to generate intermediate steps before producing a final answer. Recent work, however, suggests that reasoning-relevant computation also resides in the model’s hidden states and can be elicited without explicit CoT prompting He et al. (2026); Sun et al. (2026); Deng et al. (2023), indicating that internal representations contain meaningful computational signatures that can be intervened upon. For instance, Sparse Autoencoders (SAEs) have been used to elicit reasoning by identifying a reasoning-associated feature, though doing so requires a contrast condition prompt and a separate SAE per layer He et al. (2026). More broadly, linear probes, including probes trained on individual attention head activations, have been shown to decode or steer behavioral properties such as truthfulness from internal states. However only such specific properties have been steered so far, not reasoning Li et al. (2023); von Rütte et al. (2024); Ashok and May (2026); Li et al. (2026a). Each of these approaches requires some predefined characterization of reasoning, whether through a contrast prompt, a trained auxiliary model, or a property specific probe. For mathematical reasoning with verifiable answers, which is our focus, whether an intervention can be found without additional requirements and specifications remains underexplored. Surprisingly, we find that an effective reasoning intervention can be elicited from a signal as simple as a direction separating correctly and incorrectly answered samples, requiring no CoT demonstrations, reasoning-specific information, or contrastive direct-versus-CoT prompting. This direction is estimated locally, at each attention head, by using last token activations of the brief input prompts alone associated with the correct and incorrect generations. The key remaining challenge is where to intervene. Correctness information is decodable across many attention heads. However, this decodability alone is not causally sufficient for effective steering von Rütte et al. (2024); Li et al. (2023). Heads selected without regard to causal effect, including at random, yield no reliable benefit (Fig. 5, left). We find that only a small subset of heads are thus useful for reasoning intervention. To that end, we introduce COMPASS (COrrectness-Mapped Per-head Attribution for Steering), an inference-time steering framework that identifies and steers the heads with the highest output sensitivity to the correctness direction, rather than selecting heads by decodability alone. Unlike SAE-based approaches, COMPASS needs no trained auxiliary model, only the per-head mean activation computed separately over correct and incorrect samples, making the method as cheap to deploy as it is effective. Remarkably, COMPASS is never provided with a representation of reasoning itself, yet steering the selected heads elicits concise intermediate computations that often convert incorrect direct answers into correct solutions. Across three model families Grattafiori and others (2024); Yang and others (2025); Gemma Team, Google DeepMind (2026) and multiple mathematical reasoning benchmarks Cobbe et al. (2021); Patel et al. (2021); Hendrycks et al. (2021), COMPASS outperforms existing activation-steering baselines in most model–benchmark pairs, improves GSM8K accuracy by 16 points on average, and approaches CoT performance with substantially shorter generations. Ablations show that both the correctness direction and its intervention location are necessary, with much of the effect concentrated in only a few heads. The identified interventions further transfer without re-fitting to GSM-Plus Li et al. (2024b) and HARP Yue et al. (2024). Our main contributions are: • Reasoning intervention from outcome supervision alone. We show that correct/incorrect outcomes help recover an effective reasoning intervention, without CoT demonstrations, predefined reasoning directions, or identified reasoning features. • Attribution-based steering. We introduce COMPASS, which combines per-head correctness directions with their predicted downstream logit effect to identify effective intervention sites. • Concentrated internal control. Controlled head and direction ablations show that both, what is injected and where it is injected matter, with much of the intervention effect concentrated in remarkably few attention heads. • Accuracy, efficiency, and transfer. Across three model families and multiple reasoning benchmarks, COMPASS closes much of the gap to explicit CoT, with substantially shorter generations and without any re-fitting on unseen distributions, demonstrating that the same lightweight intervention is simultaneously accurate, efficient, and general. 2 Related Work Eliciting Reasoning from LLMs. The reasoning capabilities of LLMs are most commonly elicited through chain-of-thought (CoT) prompting Wei et al. (2022); Kojima et al. (2022), where performance improves significantly as the model is made to think through intermediate steps before answering. However, prompting is not the only route to reasoning. Recent work shows that the internal states of the LLM carry useful computational signatures that can be extracted directly, without relying on the prompt alone. Fractional reasoning Liu et al. (2025) takes a direct prompt and a CoT prompt, identifies the linear direction separating their latent activations, and shifts the model into a controlled reasoning regime along this direction. In a similar spirit, sparse autoencoder features extracted from direct and CoT prompts have been used to uncover a latent reasoning mode He et al. (2026), and contrastive prompt pairs have been used to steer metacognitive behaviors in Li et al. (2026b). Each of these methods requires some prior specification of what reasoning should look like, rather than discovering it directly from the model’s own behavior. In contrast, COMPASS requires no such explicit specifications or requirements. Probing Hidden States of LLMs. A second line of work extracts information from hidden states via probing, using simple classifiers, typically linear, trained on activations to predict a target property. Inference-time intervention Li et al. (2023) trains such probes at every attention head and shifts activations along the probe direction at the heads where properties (e.g., truthfulness) are most detectable. Follow-up work refined how the probe guides generation von Rütte et al. (2024); Li et al. (2026a), and applied probing to new targets, including whether a model believes its own answer is correct Zhang et al. (2025), whether a math problem is solvable at all Xiros et al. (2026), and whether code is correct before it is written Di Cicco (2026). However, each method trains a probe on a single, curated property, such as truthfulness, honesty, or solvability, requiring its own labeled dataset, and only a subset of these properties has been shown to be steerable at all and none for reasoning. Our steering or intervention direction requires only a labeled set of single-pass generations, split by the LLM’s own correctness, with heads selected by their causal effect on the output rather than by property detectability alone. Activation Steering in LLMs. A third line of work steers behavior directly through activations rather than through the prompt. This traces back to in-context learning Li et al. (2024a); Dherin et al. (2025), where prompt demonstrations change model behavior by shifting internal activations, an effect shown to be compressible into a single latent task vector Hendel et al. (2023). In-context vectors made this actionable by distilling demonstrations into a vector added back during inference Liu et al. (2024). Subsequent methods derived directions from paired prompts Turner et al. (2023); Rimsky et al. (2024) or curated datasets Zou et al. (2023), applied steering conditionally on the input Lee et al. (2025), synthesized query specific vectors from a mixture of steering experts Weng et al. (2026), or restricted intervention to an invariant low rank subspace of reasoning trajectories Malarkkan et al. (2026). Each requires paired task demonstrations, a separate forward pass, or a labeling scheme. Our steering direction requires none of these, extracted instead from the model’s own outcome labels in a single pass. 3 Proposed Approach In this section, we present COMPASS, a method that recovers a coarse correctness direction from a model’s own hidden states, identifies the attention heads that causally carry this direction to the output, and steers only those heads to elicit reasoning. Our key insight builds on the observations of Li et al. (2023); von Rütte et al. (2024) that behavioral properties such as truthfulness or honesty, properties that at a high level resemble correctness, are decodable from the activations of several attention heads. However, only a small subset of these heads causally affect the output, which we define as a head whose intervened activation changes the model’s generated answer, as opposed to merely correlating with it. We introduce a selection procedure that distinguishes these causal heads from the many that merely encode correctness as a decodable signal. We find that the causal effect is concentrated in remarkably few heads. COMPASS answers two questions keeping the LLM frozen: (i) which direction in activation space separates the model’s correct attempts from its incorrect ones (§3.1), and (i) which attention heads carry this direction into the output when it is intervened upon (§3.2). Preliminaries. Consider a decoder-only transformer with L layers and vocabulary V. Each layer has H attention heads, hidden dimension d, and per-head dimension dh=d/Hd_h=d/H. Let xl∈ℝdx_l ^d denote the layer l input at a fixed token position. Head (l,h)(l,h) computes an attention weighted value vector alh∈ℝdha_l^h ^d_h, which is projected into the layer output through its slice WOl,h∈ℝd×dhW_O^l,h ^d× d_h of the corresponding output projection matrix. The next layer input is updated as xl+1=xl+∑h=1HWOl,halh+mlpl,x_l+1\;=\;x_l\;+\; _h=1^HW_O^l,h\,a_l^h\;+\;mlp_l, (1) where mlplmlp_l is the output of the layer’s MLP sublayer. After the final layer L, xL+1x_L+1 is normalized and mapped to a next token distribution by the unembedding matrix WU∈ℝ||×dW_U ^|V|× d, p=softmax(WULN(xL+1))∈ℝ||,p\;=\;softmax(W_U\,LN(x_L+1))\;∈\;R^|V|, (2) where LNLN denotes the LayerNorm. 3.1 Estimating the Correctness Direction Let =(qi,yi⋆)i=1ND=\(q_i,y_i )\_i=1^N be a training set of problems qiq_i with ground truth answers yi⋆y_i . For every qiq_i, we run a fixed direct answer prompt once and obtain the generation gig_i. We cache the activation alh(qi)a_l^h(q_i) of every attention head (l,h)(l,h) at the last prompt token position, before generation begins. Each generation gig_i is graded against yi⋆y_i , yielding a binary label yi∈0,1y_i∈\0,1\, where yi=1y_i=1 if gig_i is correct and yi=0y_i=0 otherwise. The label yiy_i indicates whether the model, under the direct answer prompt, answers qiq_i correctly. Following Li et al. (2023), for head (l,h)(l,h) we estimate a correctness direction θlh _l^h as the unit vector along the difference between the mean activation on correct samples and the mean activation on incorrect samples. This direction captures the axis along which the two populations tend to shift, on average, and is defined as μ+l,h=[alh∣y=1],μ−l,h=[alh∣y=0],θlh=μ+l,h−μ−l,h‖μ+l,h−μ−l,h‖. _+^l,h=E\,[a_l^h y=1], _-^l,h=E\,[a_l^h y=0], _l^h= _+^l,h- _-^l,h\| _+^l,h- _-^l,h\|. (3) As a unit vector, θlh _l^h specifies the direction of the correctness axis but not the scale of the head’s activations along it. A fixed shift magnitude is not comparable across heads, since different heads have different activation scales. We therefore normalize the shift using the standard deviation of each head’s activations along its own correctness direction, σlh=stdq∈(⟨alh(q),θlh⟩). _l^h=std_q ( a_l^h(q), _l^h ). (4) A displacement of ασlhα\, _l^h along θlh _l^h corresponds to α standard deviations along the correctness direction, making α a scale invariant steering strength shared across all heads. 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 _+^l,h and μ−l,h _-^l,h. θlh _l^h is the unit vector connecting these means, and σlh _l^h is the standard deviation of the activations projected onto θlh _l^h. Steering shifts an activation on the incorrect side by ασlhα\, _l^h along θlh _l^h. Right: Head selection by attribution score. Top: the score slhs_l^h of each head, with the K heads selected within the promoted layer band outlined in blue. Bottom: the outcome contrast Δp p measured on the fitting rows. First generated tokens for correct answers concentrate on “To”, and for incorrect answers on “$”. Following the intervention procedure of ITI Li et al. (2023), we steer the model by adding ασlhθlhα\, _l^h\, _l^h to the output of each selected head, before it is projected into the layer input, xl+1=xl+∑h=1HWOl,h(alh+ασlhθlh)+mlpl,x_l+1\;=\;x_l\;+\; _h=1^HW_O^l,h (a_l^h+α\, _l^h\, _l^h )\;+\;mlp_l, (5) where θlh≡0 _l^h≡ 0 for every head not selected for steering. Since WOl,hW_O^l,h is linear, each unit of α contributes a fixed vector to the layer input, δlh=WOl,hσlhθlh∈ℝd, _l^h\;=\;W_O^l,h\, _l^h\, _l^h\;∈\;R^d, (6) which is the correctness direction of head (l,h)(l,h), projected into the layer input. δlh _l^h is the contribution head (l,h)(l,h) makes to the layer input per unit of α. 3.2 Where to Intervene: Selecting Causal Heads Equation (5) leaves the choice of heads open. The direction θlh _l^h from Eq. (3) exists for every head, since a difference of means can be computed regardless of whether that head’s output has any effect on the model’s generation. Decodability, in this sense, only requires that correct and incorrect activations differ on average, it places no constraint on whether adding δlh _l^h to the layer input changes the final output. Selecting heads by decodability alone can therefore result in heads with no causal effect, or with an effect that opposes the intended direction. This is confirmed empirically in the ITI row of Table 1, where steering the heads that decode correctness best yields little or no improvement, and is at times detrimental. On MATH-500, it lowers Llama-3.1-8B accuracy from 40.8% to 23.6%, and Qwen3-4B from 53.8% to 30.6%. A head is useful for intervention only if its contribution δlh _l^h changes the model’s output. Decodability in activation space, as used to define θlh _l^h, only tells us that correct and incorrect activations differ, not whether that difference affects the generated token. To capture causal effect, the selection criterion must therefore be defined in terms of the shift a head’s contribution produces in the next token logits, rather than in the space of activations. The unembedding matrix WUW_U maps a vector in the layer input space to a vector of next token logits (Eq. 2). Rather than mapping δlh _l^h forward through WUW_U, which would require rerunning the model, we use the identity ⟨WUx,z⟩=⟨x,WU⊤z⟩ W_Ux,z = x,W_U z . For any target logit direction z, its inner product with a layer input vector x can be computed entirely in layer input space, using WU⊤zW_U z. This lets us score δlh _l^h directly against a target logit direction, without any additional forward pass. We first define a target shift in the space of first generated tokens. Following Zhao et al. (2024), who show that an LLM’s knowledge of whether its answer will be correct is most concentrated in the distribution of its first generated token, we use this distribution as our target. Let p+,p−∈Δ||p_+,p_-∈ ^|V| be the empirical first generated token distributions for questions the model answers correctly and incorrectly, under the same prompt. Their difference, Δp=p+−p−, p\;=\;p_+-p_-, (7) is positive on tokens more likely to precede a correct answer and negative on tokens more likely to precede an incorrect answer (Figure 1(b)). Δp p lies in vocabulary space, while δlh _l^h lies in layer input space. To compare them, we map Δp p into layer input space using the transpose of the unembedding matrix, u=WU⊤Δp∈ℝd.u\;=\;W_U \, p\;∈\;R^d. (8) We call u the target direction. A head whose contribution aligns with u shifts the first generated token logits toward tokens that precede correct answers. We score head (l,h)(l,h) by how well its contribution δlh _l^h aligns with the target direction u, slh=⟨u,δlh⟩=WU⊤ΔpWOl,hσlhθlh.s_l^h\;=\; u, _l^h \;=\;W_U \, p\,W_O^l,h\, _l^h\, _l^h. (9) A high score implies that δlh _l^h is large and points in the direction of u. We select the top K heads by slhs_l^h and steer only these heads, setting θlh≡0 _l^h≡ 0 for all other heads. Across three open weight models and multiple short answer math benchmarks, intervening only these selected heads consistently improves accuracy over standard prompt generation. This requires a single forward pass over the training dataset, and steering vectors are applied zero shot at inference. Algorithm 1 summarizes the steering procedure. Figure 2 shows this effect on two problems. In both cases, the direct answer prompt produces an incorrect answer. The steered model instead opens with "To", derives the answer in a few lines, and reaches the correct answer. Chain of thought prompting reaches the same answer using three to four times as many tokens. This shows that steering does not make the model imitate a chain of thought style response. Instead, it causes the model to begin deriving an answer rather than stating one directly, and the model stops once it has derived enough to answer. Algorithm 1 COMPASS 1: frozen WOl,h,WUW_O^l,h,W_U; pool D; K 2: One graded pass over D: label yiy_i, activations alh(qi)a_l^h(q_i), first generated token t(qi)t(q_i) ∀qi∈∀\,q_i 3: Compute θlh,σlh _l^h,\ _l^h by Eqs. (3), (4); δlh←WOl,hσlhθlh _l^h← W_O^l,h _l^h _l^h ∀(l,h)∀\,(l,h) 4: p+,p−←p_+,p_-← first generated token histograms over correct, incorrect rows 5: u←WU⊤(p+−p−)u← W_U (p_+-p_-) 6: slh←⟨u,δlh⟩s_l^h← u, _l^h ; ←S← top-K heads by slhs_l^h 7: Apply Eq. (5) at test time, with θlh≡0∀h∉ _l^h≡ 0\ ∀\,h Llama-3.1-8B, GSM8K. Mary has 30 sheep. She gets 1 kg of milk from half of them and 2 kg of milk from the other half every day. How much milk does she collect every day? gold: 45 DIRECT 15 kg. × 3 tokens COMPASS To find the total milk collected, we need to calculate the milk from the first half and the second half separately and then add them together. First half: 15 sheep * 1 kg = 15 kg. Second half: 15 sheep * 2 kg = 30 kg. Total milk collected = 15 kg + 30 kg = 45 kg. ✓ 74 tokens COT ## Step 1: Calculate the number of sheep that produce 1 kg of milk per day. Mary has 30 sheep, and half of them produce 1 kg of milk per day. So, the number of sheep that produce 1 kg of milk per day is 30 / 2 = 15. ## Step 2: Calculate the number of sheep that produce 2 kg of milk per day. [… Steps 3–5 …] Therefore, the final answer is: 45 45. ✓ 282 tokens Qwen3-4B, MATH-500. In how many ways can 6 girls and 2 boys sit in a row if the 2 boys insist on sitting next to each other? gold: 10,080 DIRECT Final answer: 864. × 8 tokens COMPASS To solve this, treat the 2 boys as a single unit. Then we have 7 units to arrange (6 girls + 1 boy unit). These 7 units can be arranged in 7!7! ways. The 2 boys can be arranged among themselves in 2!2! ways. So the total number of arrangements is: 7!×2!=5040×2=100807!× 2!=5040× 2=10080. Final answer: 10080. ✓ 108 tokens COT We are given: 6 girls, 2 boys. We are to find the number of ways they can sit in a row if the 2 boys must sit next to each other. ### Step 1: Treat the 2 boys as a single unit. Since the 2 boys must sit next to each other, we can treat them as one “block”. This block can be in two different internal arrangements [… Steps 2–4 …] Final Answer: 10080 10080. ✓ 323 tokens 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 3.03.0–3.8×3.8× lesser tokens than a chain-of-thought prompt. 4 Experiments and Results We evaluate COMPASS across three LLMs over mathematical reasoning benchmarks, measuring accuracy (↑ ) against ground-truth answers, transfer of fitted directions to unseen datasets, and ablations isolating which components of the method drive performance. Datasets. We use GSM8K Cobbe et al. (2021), SVAMP Patel et al. (2021), and MATH-500 Lightman et al. (2024) for main experiments, with disjoint train and test splits. We evaluate zero-shot transfer, without re-fitting, on GSM-Plus Li et al. (2024b) and HARP Yue et al. (2024). All benchmarks are short-answer mathematics with a single gold answer per problem, graded by exact or numeric equivalence. Full details are provided in Appendix A.3. Baselines. We compare against (i) Inference-Time Intervention (ITI) Li et al. (2023), which selects heads by probe accuracy and steers the activation them along the probe direction, and (i) Fractional Reasoning Liu et al. (2025), which steers every layer of the LLM along a pre-computed reasoning direction from contrastive prompts. Full baseline details are provided in Appendix A.5. Models. We apply all methods to three open-weight models namely Llama-3.1-8B-Instruct Grattafiori and others (2024), Qwen3-4B Yang and others (2025), and Gemma4-12B-it Gemma Team, Google DeepMind (2026). Oracles. For reference only, we report a CoT oracle (the same model prompted for explicit chain-of-thought) and a frontier oracle, GPT-5.4-mini with reasoning effort high. All methods are evaluated and graded identically. Basic Setup. For every sample, we use a fixed, LLM specific brief answer prompt that instructs the model to produce a short answer without explanation. Full prompt details are in Appendix A.1. On the training split, we run this prompt once per sample and store the activation of every attention head at the last prompt token, prior to generation. Each generation is then graded against the ground truth answer, labeling the sample correct or incorrect. Full grading details are in Appendix A.2. For every head, we obtain a correctness direction as the difference between the mean activation over correct samples and the mean activation over incorrect samples. α is a hyperparameter controlling the intensity of the intervention, and we select α separately for every dataset through a hyperparameter search. Evaluation is a single zero shot pass on the test split of each dataset. Head and Layer Selection. slhs_l^h ranks heads reliably within a layer, but not across layers at different depths, since it does not account for how much a head’s contribution is transformed before reaching the unembedding layer (Appendix A.4). We therefore restrict head selection to a contiguous band of layers, chosen using the layer aggregated elicitation mass, ml=∑hmax(slh,0)m_l= _h (s_l^h,0), computed by summing only the positive head scores within each layer. Full head and band selection details are in Appendix A.4. Within the selected band, the K=8K=8 highest scoring heads are steered. 4.1 COMPASS Predominantly Outperforms Steering Baselines Table 1 reports accuracy and mean generated tokens for the direct answer baseline (standard prompt), ITI Li et al. (2023), Fractional Reasoning Liu et al. (2025), and COMPASS, with the CoT and frontier model (GPT-5.4 mini) reported as reference. COMPASS improves over the direct answer baseline on every model and dataset, and over both ITI and Fractional Reasoning on every model and dataset except Gemma4-12B on SVAMP, where Fractional Reasoning performs best. On GSM8K, COMPASS closes most of the gap to the CoT reference on Llama and Qwen3, reaching 83.7% against an 83.4% ceiling and 89.8% against 92.3% respectively, and recovers a substantial part of the gap on Gemma4-12B, reaching 88.0% against a 96.1% ceiling. It does so while generating 20 to 70% fewer tokens than CoT across datasets, for instance 68 versus 224 tokens on Qwen3 SVAMP. COMPASS thus achieves accuracy comparable to or approaching explicit reasoning at a fraction of its token cost. 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 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) GPT-5.4-mini (oracle) 95.1 (273) 94.7 (202) 88.2 (786) 95.1 (273) 94.7 (202) 88.2 (786) 95.1 (273) 94.7 (202) 88.2 (786) 4.2 COMPASS Transfers Without Refitting to Unseen Datasets Table 2 reports accuracy on GSM-Plus and HARP using the direction, heads, and α obtained from the training splits of GSM8K and MATH-500 respectively, applied without modification. On GSM-Plus, accuracy rises from 60.4% to 71.1% on Llama and from 65.6% to 80.4% on Qwen3. On HARP, accuracy increases from 21.5% to 22.7% on Llama and from 35.4% to 49.1% on Qwen3. GSM-Plus consists of adversarial modifications of GSM8K test problems, and HARP is a harder competition level dataset than MATH-500. Since the directions and heads were never exposed to either dataset, this transfer indicates that they capture a property of the model rather than a fit to the source dataset. 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. 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) 4.3 Ablations Studies We ablate three components of COMPASS in isolation, the layer band, the head set, and the steering direction, to identify which are necessary for the observed performance improvements. 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). Layer Band Selection is Concentrated in the Final Layers. Each head is scored by the shift it delivers toward the target outcome according to (Eq. 9), and a layer’s aggregated score is mlm_l. Figure 3 plots mlm_l for both LLama-3.1-8B-it and Qwen3-4B. For roughly the first two thirds of the network, mlm_l is near zero, indicating that heads in these layers contribute little to the output. mlm_l then rises sharply over the last few layers, motivating restriction of head selection to this late band rather than the full network. We validate this choice on a held out, development split containing 300 samples, disjoint from both the training and the test set (Appendix A.3). Holding α fixed, we steer using each candidate band so that only the head set varies. Figure 4 shows that the band with the highest mlm_l achieves the best or near best development accuracy on both models, confirming that mlm_l identifies an effective intervention band. 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=15.66α=15.66; Qwen3 35–36 at α=25.53α=25.53). The dashed line is the unsteered direct-answer baseline on the same rows. Both Head Selection and the Correctness Direction are Necessary. Within the selected band, we compare three settings on the full MATH-500 test set, holding the band and α fixed: 1. Selected heads, correctness direction: our method, top K=8K=8 heads by slhs_l^h, steered along the correctness direction θlh _l^h. 2. Random heads, correctness direction: K=8K=8 heads drawn uniformly at random from the same band, steered along θlh _l^h. 3. Selected heads, random direction: the same top K=8K=8 heads, steered along a random unit vector of matched magnitude. 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. Figure 5 (left) shows that our method reaches 45.0% accuracy, a +4.2+4.2 point gain over the 40.8% baseline. Randomizing head selection while keeping the correctness direction reduces this to 41.1%, within one third of a point of baseline. Randomizing the direction while keeping the selected heads is actively harmful, at 39.7%, below baseline. Both the head set and the correctness direction are therefore required for the effect and neither alone is sufficient. The Effect Concentrates in Remarkably Few Heads. Figure 5 (middle) varies K∈8,4,2,1K∈\8,4,2,1\ on Llama-3.1-8B GSM8K. Accuracy degrades gradually as K decreases: 83.7%, 81.9%, 80.4%, and 75.1% at K=1K=1, each still a substantial gain over the 65.0% baseline. A single head retains +10.0+10.0 of the +18.7+18.7 point gain achieved at K=8K=8, indicating that most of the effect is concentrated in a small number of heads rather than distributed evenly across the selected set. Reversing the Correctness Direction Actively Harms Accuracy. Figure 5 (right) reverses the sign of the steering direction at the selected heads, injecting −θlh- _l^h in place of θlh _l^h, with the band, α, and head set otherwise unchanged. On Llama-3.1-8B GSM8K, accuracy falls from 82.9% at θlh _l^h to 46.8% at −θlh- _l^h, below the 65.0% unsteered baseline. On Qwen3-4B MATH-500, accuracy falls from 77.6% to 28.0%, below the 53.8% baseline. Reversal does not merely remove the gain, it drives the model below its unsteered accuracy, confirming that θlh _l^h is signed toward correctness rather than an arbitrary axis of variation the model happens to be sensitive to. 5 Conclusions and Future Work We presented COMPASS, an inference-time steering framework that recovers a reasoning intervention from outcome supervision alone, without chain of thought demonstrations or a predefined notion of reasoning. A single forward pass over a pool of graded problems yields a per-head correctness direction and a logit-space attribution score, together identifying the few heads whose contribution shifts the first generated token toward a correct answer. Across three model families and multiple math benchmarks, COMPASS outperforms existing activation-steering baselines, closes much of the gap to explicit chain of thought at a fraction of the token cost, and transfers to unseen datasets without re-fitting. While effective in this setting, COMPASS opens several directions for future work. Our attribution score is defined for domains with a binary notion of correctness, such as short-answer math questions. Extending it to open-ended tasks such as proof generation, where outcomes are not directly gradable, is a natural next step. Similarly, while our ablations identify which heads causally drive correctness, a mechanistic account of why these specific heads emerge, and why they promote tokens that initiate correct derivations, is left to future work. We view COMPASS as a step toward using a model’s own outcomes, rather than externally imposed notions of reasoning, to identify and steer the internal computation that produces them. Acknowledgments and Disclosure of Funding This work was performed under the auspices of the U.S. Department of Energy (DOE) by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. This work was supported by LDRD 25-SI-001. 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In International Conference on Learning Representations, Cited by: Table 3, §4. Appendix A Appendix for COMPASS: Finding Where Reasoning Lives in Language Models This section contains detailed explanations of every protocol followed in our experimental setup. Our code is publicly available at : https://github.com/pratyay2510/COMPASS.git A.1 Prompts Llama-3.1-8B and Qwen3-4B use one prompt on every dataset. Gemma-4-12B-it uses two prompts, both with thinking mode off. On GSM8K and SVAMP it uses a direct answer prompt. On MATH-500 the prompt asks for only the final answer. The prompt is deliberately terse and asks for the answer without an explanation, so that the baseline reflects what the model does when reasoning is not requested. Each box below reproduces the full prompt string exactly as the code builds it. The marker problem shows where the question text is inserted, and a blank line inside a box is a blank line in the prompt. The system message is empty for every family, so the rendered input is a single user turn. Llama-3.1-8B all datasets You are a helpful assitant. Question: problem. Give me the answer directly. Qwen3-4B all datasets Answer the following question with only the final answer. Do not show any working or explanation. Question: problem Final answer: Gemma-4-12B-it GSM8K and SVAMP Answer the following question directly. Question: problem Gemma-4-12B-it MATH and MATH-500 Answer the following question with only the final answer. Question: problem Decoding is greedy everywhere, with a cap of 512 new tokens on the GSM8K and SVAMP fitting pools and 2,048 new tokens on MATH and on every evaluation split. A generation that hits the cap is counted incorrect. A.2 Grading Free form generations do not reliably carry a marker such as . A correct answer that ends in “…=10933…= 10933” is extracted as “33” by a naive last number rule. Grading therefore proceeds in two passes over every generation. The first pass is deterministic. An answer is extracted from the raw generation and compared with the gold answer through a fixed cascade. The cascade checks normalized string equality, then numeric equality at relative tolerance 10−610^-6 with percent signs and unit tails stripped, then reduction of a derivation chain to its final value, and finally bounded symbolic equivalence through SymPy. We call the resulting label the single pass grade. The second pass follows Fractional Reasoning [18] and uses an LLM formatter. A second call to the same model, never steered, restates the generation as a single line of the form Answer: \[ …\], and the same deterministic cascade then grades the boxed answer. The system message of the formatting call tells the model that its job is to extract the final short answer from the more detailed answer. The user prompt depends on the model and the dataset as follows. Every model uses the prompt below on every dataset, The placeholder reasoning carries the generation being graded. Formatting prompt Here is a worked solution: reasoning Copy its final answer, exactly as the solution states it, into this line and output nothing else: Answer: \[ your_answer_here \] Do not solve the problem. Do not explain. Do not recompute anything. If the solution states no final answer, box its last expression. The final label of a row is the OR of the single pass grade and the formatted answer grade. Only rows the single pass grade marks incorrect are sent to the formatting call, so the formatter can promote a row whose answer the extraction missed but can never demote a row the deterministic cascade already accepts. This is due to the fact that the formatting call is itself a generation and can fail as one. Hence, a regrade that can only add keeps such failures from destroying correct labels. Truncated generations are sent to the formatter as well, since a capped generation can still contain its final answer, and their label follows the same rule. The same instrument grades everything, so every accuracy score comes from a single universal grader. Accuracy is always reported over all rows of the split. A.3 Dataset Splits Table 3 shows the datasets that have been used for this work and their corresponding details. Table 3: Benchmarks. Train splits are fitting pools only; test splits are evaluated in full. GSM-Plus and HARP are used for transfer only. 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. Every dataset uses three disjoint row sets, and the convention is the same for all of them: a fitting pool, from which the correctness direction and the head ranking are estimated; a 300-row dev (scan) set, on which the depth band and the strength α are chosen; and the test split, which is generated once at the promoted configuration and never consulted before that. The details of the three fully disjoint sets are as follows: Fitting Set This is the provided train set for each dataset. It is collected once under the direct-answer prompt (Section 4). Each row stores the generation, its grade against gold, and the activation of every attention head at the last prompt token. The pool supplies both the outcome target u and the per-head mass-mean directions θlh _l^h with their scales σlh _l^h. The convention differs slightly between GSM8K, MATH and SVAMP. Details are as follows: • GSM8K and MATH: Both GSM8K and MATH have large training sets (GSM8K - 7473 problems, MATH - 7500 problems). For them, after collecting the entire training sets, we construct this fitting pool of 16001600 rows where the correct-incorrect split is exactly equal (800-800). This is done so that the correctness direction remains unbiased and no class imbalance affects either the θlh _l^h or slhs_l^h. • SVAMP: The training set of SVAMP has only 700700 problems. In this dataset, we use 400400 rows as our fitting set. The rest 300300 are used as the dev set. Dev (scan) set For all datasets, this is a set of 300300 rows. For GSM8K and MATH, due to its large size, it is a random set of rows from the spare ones after the fitting pool is identified. For SVAMP, it is carved out of the training set before any fitting takes place. Throughout the study, the scan set is always fully disjoint from the fitting set. Test Set This is the benchmark’s evaluation split, taken in full. It is fully disjoint from both the fitting and dev sets. GSM-Plus and HARP (transfer). Neither dataset has a fitting pool or a dev set. The direction, the head set, and α selected on GSM8K are applied unchanged to GSM-Plus. Those selected on MATH are applied unchanged to HARP. GSM-Plus released 10,55210,552 problems. However, 1,3191,319 problems are critical-thinking problems, which are unanswerable by construction (gold “None”). Hence, GSM-Plus is evaluated on 9,2339,233 problems. HARP is evaluated on all 4,7804,780 rows. A.4 The Operating Point: Depth Band and Dose The elicitation mass ml=∑hmax(slh,0)m_l= _h (s_l^h,0), gives a per-layer profile of how much correct-opening logit the layer can deliver. Only positive scores are summed: a head whose injection moves the logits away from a correct opening is not a candidate, and letting it cancel a useful head in the same layer would misreport what that layer is worth. The profile is essentially flat through the first two thirds of the network, then a sharp rise over the last four to six layers (Figure 3). This makes a band, the natural search space. We take the elbow, ℓ∗ ^* to be the lowest layer l such that it is the deepest point from which the mass never again falls below 35% of its peak. ml′≥0.35×max(m),∀l′≥l.m_l ≥ 0.35\ × (m),∀ l ≥ l. The candidate layers are the nested bands [ℓ⋆,L],[ℓ⋆+1,L],…,[L,L][ ,L],[ +1,L],…,[L,L]. Within a band, the K=8K=8 highest-scoring heads are steered. Matched dose and the operating point. Because α multiplies a per-head norm, the same α is a different perturbation in every band. Bands are therefore compared at matched dose, D=α∑(l,h)∈‖δlh‖D=α _(l,h) \| _l^h\|, the total norm added to the residual stream by the selected set S. The widest band fixes a ladder of four to five doses from an α range (α∈[8,32]α∈[8,32]). Every other band receives the α values that land it on the same doses. These sets of points (band, α) are used to steer the dev sets of each dataset. The single operating point (band, α) with the highest dev accuracy is promoted, and is used to steer the test set. A.5 Baselines and Oracles ITI and Fractional Reasoning are re-implemented and run on our models, prompts, splits, token caps, and grader. ITI selects the K=8K=8 heads with the highest probe validation accuracy over all heads of the model, steers them along the same mass-mean direction, and is swept over a fixed α ladder (α∈[8,32]α∈[8,32]) and reported at the best accuracy. Fractional Reasoning experiments are constructed using its own codebase. It builds a steering vector from the mean and first principal component of the (chain-of-thought −- direct) last-token differences over 256 training problems. Intervention is applied at every layer with norm-preserving rescaling. The paper does not specify a gold α, but reports a band (α∈[0.01,0.2]α∈[0.01,0.2]). We run it on that α range in steps of 0.020.02 and report the best achieved accuracy. The CoT oracle is each model’s published chain-of-thought evaluation prompt, run on the full test split at a 2,048-token cap and graded identically. This is the reference ceiling for a non-thinking model of that size. It is the accuracy the model reaches when reasoning is explicitly requested. The CoT prompts for the three models are as follows: Llama-3.1-8B chain of thought Solve the following math problem efficiently and clearly: - For simple problems (2 steps or fewer): Provide a concise solution with minimal explanation. - For complex problems (3 steps or more): Use this step-by-step format: ## Step 1: [Concise description] [Brief explanation and calculations] ## Step 2: [Concise description] [Brief explanation and calculations] ... Regardless of the approach, always conclude with: Therefore, the final answer is: $ answer$. I hope it is correct. Where [answer] is just the final number or expression that solves the problem. Problem: problem Qwen3-4B chain of thought problem Please reason step by step, and put your final answer within . Gemma-4-12B-it chain of thought problem Please reason step by step, and put your final answer within . As with the direct answer prompts, problem marks where the question text is inserted, a blank line inside a box is a blank line in the prompt, and the system message is empty for every family.