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Valence-Arousal Subspace in LLMs: Circular Emotion Geometry and Multi-Behavioral Control

Lihao Sun, Lewen Yan, Xiaoya Lu, Andrew Lee, Jie Zhang, Jing Shao

Year: 2026Venue: arXiv preprintArea: cs.CLType: PreprintEmbeddings: 73

Intelligence

Status: succeeded | Model: google/gemini-3.1-flash-lite-preview | Prompt: intel-v1 | Confidence: 96%

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Summary

The paper introduces a method to identify a 2D valence-arousal (VA) subspace within LLM representations by decomposing emotion steering vectors. The authors demonstrate that this subspace exhibits a circular geometry consistent with human emotion models and enables monotonic, bidirectional control over model behaviors like refusal and sycophancy through lexical mediation.

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LLaMA-3.1-8B · llm · 99%Qwen3-14B · llm · 99%Qwen3-8B · llm · 99%Valence-Arousal (VA) Subspace · concept · 98%Lexical Mediation · mechanism · 95%

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LLaMA-3.1-8B → exhibits → Valence-Arousal (VA) Subspace

confidence 98% · These effects replicate across Llama-3.1-8B, Qwen3-8B, and Qwen3-14B

Valence-Arousal (VA) Subspace → controls → Refusal

confidence 95% · Steering along these directions also induces near-monotonic bidirectional control over refusal and sycophancy

Valence-Arousal (VA) Subspace → controls → Sycophancy

confidence 95% · Steering along these directions also induces near-monotonic bidirectional control over refusal and sycophancy

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Abstract

Abstract:We present a method to identify a valence-arousal (VA) subspace within large language model representations. From 211k emotion-labeled texts, we derive emotion steering vectors, then learn VA axes as linear combinations of their top PCA components via ridge regression on the model's self-reported valence-arousal scores. The resulting VA subspace exhibits circular geometry consistent with established models of human emotion perception. Projections along our recovered VA subspace correlate with human-crowdsourced VA ratings across 44k lexical items. Furthermore, steering generation along these axes produces monotonic shifts in the corresponding affective dimensions of model outputs. Steering along these directions also induces near-monotonic bidirectional control over refusal and sycophancy: increasing arousal decreases refusal and increases sycophancy, and vice versa. These effects replicate across Llama-3.1-8B, Qwen3-8B, and Qwen3-14B, demonstrating cross-architecture generality. We provide a mechanistic account for these effects and prior emotionally-framed controls: refusal-associated tokens ("I can't," "sorry") occupy low-arousal, negative-valence regions, so VA steering directly modulates their emission probability.

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Valence–Arousal Subspace in LLMs: Circular Emotion Geometry and Multi-Behavioral Control Lihao Sun 1 2† Lewen Yan 1 Xiaoya Lu 1 Andrew Lee 3 Jie Zhang 1 Jing Shao 1 Abstract We present a method to identify a valence–arousal (VA) subspace within large language model rep- resentations. From 211k emotion-labeled texts, we derive emotion steering vectors, then learn VA axes as linear combinations of their top PCA com- ponents via ridge regression on the model’s self- reported valence–arousal scores. The resulting VA subspace exhibits circular geometry consistent with established models of human emotion per- ception. Projections along our recovered VA sub- space correlate with human-crowdsourced VA rat- ings across 44k lexical items. Furthermore, steer- ing generation along these axes produces mono- tonic shifts in the corresponding affective dimen- sions of model outputs. Steering along these direc- tions also induces near-monotonic bidirectional control over refusal and sycophancy: increas- ing arousal decreases refusal and increases syco- phancy, and vice versa. These effects replicate across Llama-3.1-8B, Qwen3-8B, and Qwen3- 14B, demonstrating cross-architecture generality. We provide a mechanistic account for these effects and prior emotionally-framed controls: refusal- associated tokens (“I can’t,” “sorry”) occupy low- arousal, negative-valence regions, so VA steering directly modulates their emission probability. 1. Introduction A growing body of work shows that emotionally framed prompting or activation steering influences large language model (LLM) behavior (Li et al., 2023; Konen et al., 2024; Reichman et al., 2025; Dong et al., 2025). Yet why such methods work—and when they fail—remains unclear. One obstacle is conceptual. Work that studies such phenom- ena in LLMs typically treats discrete emotion categories— 1 Shanghai AI Lab 2 University of Chicago 3 Harvard University. Correspondence to: Jing Shao <shaojing@pjlab.org.cn>. † Work done using resources from Shanghai AI Lab. Preprint. April 6, 2026. anger, joy, fear—as fundamental units of analysis (Zhang et al., 2024; Tigges et al., 2024; del Arco et al., 2024; Konen et al., 2024; Dong et al., 2025). To find a common basis for comparison, a growing trend borrows from human psychol- ogy the two-dimensional framework of valence (pleasure– displeasure) and arousal (activation–deactivation) (Ishikawa & Yoshino, 2025; Felix-Pena et al., 2025). In practice, these dimensions are often equated with categorical labels or used merely as evaluation metrics. Though yielding insights, this discreteness-driven perspective forgoes explanatory power that the original distinction affords. In the psychological tradition from which these constructs originate, valence and arousal (VA) do not define emotions themselves but rather core affect—a continuous experiential substrate from which discrete emotions emerge (Russell, 1980; 2003). Emotions arise when core affect is attributed to a cause and labeled with a culturally available category: a pleasant, high-arousal state, attributed to an unexpected gift, becomes surprise or excitement. We invoke this distinction not as a claim about LLM phenomenology, but as a model- ing choice: continuous dimensions and discrete labels play different explanatory roles and should not be conflated. In this work, we treat categorical labels and VA axes as separate structures and demonstrate the insights this decou- pling affords. At the level of model representations, this conceptual shift also motivates a methodological one. Many existing steering-vector methods construct directions via contrastive differences (e.g., mean activation differences over paired examples) (Panickssery et al., 2024). While effective, these directions can be brittle—often requiring careful tuning across prompts, layers, or behaviors, and may not directly transfer without additional alignment or mapping (Tan et al., 2025; Braun et al., 2025; Oozeer et al., 2025). We show that decomposing emotion steering vectors into a low-dimensional VA subspace yields shared struc- ture that generalizes across multiple downstream behaviors, suggesting a more reusable control basis than single-task contrastive directions. Concretely, our approach proceeds as follows.From 211,225 emotion-labeled text samples, we derive emo- tion steering vectors via mean-difference contrasts between emotion-labeled and neutral examples (Demszky et al., 1 arXiv:2604.03147v1 [cs.CL] 3 Apr 2026 Valence–Arousal Subspace in LLMs: Circular Emotion Geometry and Multi-Behavioral Control 0.40.20.00.20.4 Valence axis 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 Arousal axis anger annoyance disappointment disgust embarrassment excitement fear gratitude grief joy love nervousness pride relief remorse sadness surprise LLM ValenceArousal Geometry (Layer Projection) 0.40.20.00.20.4 Valence axis 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 astonished excited happy glad pleased content relaxed gloomy depressed sad miserable distressed annoyed frustrated angry afraid alarmed Human Emotion Circumplex (Russell, 1980) Figure 1. Emotion steering vectors projected onto the VA subspace at layer 31, colored by valence. Gray circle: algebraic least-squares fit. The circular arrangement is analogous to circumplex model of affect in human psychology (Russell, 1980). 2020). We then obtain the model’s self-reported VA rat- ings for each emotion category and learn VA subspaces as linear combinations of principal components of the emo- tion vectors, optimized via ridge regression to recover these self-reported scores. When projected onto the learned VA subspace, the emotion vectors exhibit circular geometry analogous to the circumplex model from human emotion perception (Russell, 1980), as shown in Figure 1. To validate the learned subspaces, we first show that VA scores obtained from projections on learned subspaces cor- relate with human-crowdsourced VA ratings across 44,728 lexical items (Mohammad, 2025). Second, we demonstrate that steering generation on open-ended prompts along these axes produces monotonic shifts in the corresponding affec- tive dimensions of model outputs, as measured by sentiment and a pre-trained BERT model for VAD scoring (Hutto & Gilbert, 2014; RobroKools, 2022). These results provide evidence that the identified axes afford VA-specific control over the affective properties of model responses. To further explore the relationship between VA subspaces and downstream behaviors, we find VA axes afford near- monotonic control over refusal and sycophancy: increasing arousal decreases refusal rates and increases sycophancy, while decreasing arousal produces the opposite pattern. Ran- dom directions yield no or much weaker effects. These effects replicate on Llama 3.1 8B, Qwen3-8B and Qwen3- 14B, confirming cross-architecture generality. Surprisingly, the VA subspaces are nearly orthogonal to refusal directions derived via contrastive mean difference (Arditi et al., 2024), yet both yield monotonic control in our settings. Based on these observations, we propose a unifying mechanistic explanation for why our approach and prior emotionally-framed methods successfully control model be- havior: lexical mediation. Refusal-associated tokens (e.g., “can’t,” “sorry”) and compliance-associated tokens (e.g., “sure,” “Here”) occupy distinct regions in VA space, so shift- ing VA coordinates changes the likelihood of emitting such markers, thereby modulating downstream behavior. We fur- ther show that prior emotionally-framed prompting methods induce corresponding shifts in the VA subspace, suggesting a similar underlying mechanism. Taken together, our work advances understanding of af- fective phenomena in LLMs by motivating a conceptual separation between discrete emotion labels and continuous valence–arousal dimensions, demonstrating how this two- level perspective yields meaningful insights. Specifically, we identify a 2-dimensional VA subspace and provide ev- idence that emotion vectors are systematically organized within this geometry. From an interpretability standpoint, we present principal-component-based decomposition of emotion steering vectors into interpretable subspaces as a complementary approach to task-specific steering, expos- ing shared structure that generalizes across behaviors to- ward domain-agnostic control. Finally, we offer a unifying mechanistic account—lexical mediation—that explains why emotion-based prompting and emotion steering influence model behavior. We do not claim that VA is the unique pathway; rather, we propose it as one plausible explanation for why emotion-based controls work in LLMs. 2. Related Work Geometry of representations in LLMs.A growing body of work seeks to discover the linear and low-dimensional structure of how features are arranged in activation space in LLMs. Most of the work is 1D, a single linear direc- tion whose sign or magnitude predicts or controls a be- 2 Valence–Arousal Subspace in LLMs: Circular Emotion Geometry and Multi-Behavioral Control havior (Turner et al., 2024; Panickssery et al., 2024; Sun et al., 2025; Lee et al., 2025a). Prior work has also shown that LLMs linearly represent sentiment or polarity (Tigges et al., 2023). By contrast, 2D geometries in LLMs are less commonly characterized; notable studies include low- dimensional structure for space/time (Gurnee & Tegmark, 2024) and understanding trigonometry operations in 2D cir- cles (Nanda et al., 2023). We add to this smaller set by demonstrating that emotion steering vectors are organized by a 2D VA geometry with behavioral consequences that a polarity-only account would not predict: the axes transfer differently to refusal and sycophancy, with arousal provid- ing clean monotonic control. Emotionally Framed Analysis and Control in LLMs. Ap- pending emotionally framed phrases to prompts has been shown to influence LLM behavior and improve task per- formance across domains (Li et al., 2023; 2024). Existing explanations have attributed these effects to attention mech- anisms, surface cues, or biases toward positive or negative affect (Wang et al., 2024; Zhou et al., 2024). We provide a unifying, top-down mechanistic account that links emotion- ally framed behaviors across representation-level geometry, unembedding structure, and neuron-level analyses. Concur- rent work by Anthropic (Sofroniew et al., 2026) identifies discrete emotion concept representations in Claude Sonnet 4.5 that influence downstream behaviors; our work comple- ments theirs by decomposing such categorical representa- tions into a continuous 2D valence–arousal substrate and characterizing the geometric structure that organizes them. Activation Steering. Activation steering enables training- free behavioral control by manipulating internal representa- tions (Zou et al., 2025), but typically relies on task-specific contrastive vectors for each desired behavior (Turner et al., 2024; Qian et al., 2024; Lee et al., 2025b). Notably, emo- tions have been shown to correspond to linear directions in activation space (Tigges et al., 2023; 2024). In this work, we present an approach of principal-components-based decom- position into subspaces, which complements the mainstream contrastive task-specific approach. 3. Identifying Valence and Arousal Subspaces We decompose emotion steering vectors to identify underly- ing VA subspaces. Our approach proceeds in three stages: (1) extracting emotion directions via mean-difference con- trasts, (2) eliciting model self-reported VA coordinates for each emotion label, and (3) learning VA axes as linear combinations of principal components via ridge regression. Here, we present results on Llama 3.1-8B-Instruct (Meta, 2024) and replicate on Qwen3-8B and Qwen3-14B (Yang et al., 2025) to validate cross-architecture generality. Emotion Steering Vectors. Using Llama 3.1-8B-Instruct (Meta, 2024), we derive emotion steering vectors using con- trastive means (Tigges et al., 2023; Panickssery et al., 2024; Dong et al., 2025). We use GoEmotions (Demszky et al., 2020), which comprises 211,225 text samples annotated with 27 emotion labels plus a neutral class. We only retain examples annotated with exactly one emotion. For each emotion categorye, we extract the last-token hidden state of each sample at each layer. The steering vector at layer ℓ is v (ℓ) e = 1 |D e | X x∈D e h (ℓ) (x)− 1 |D neutral | X x∈D neutral h (ℓ) (x) (1) whereh (ℓ) (x)∈R H denotes the last-token hidden state at layerℓfor inputx;D e andD neutral are the sets of single-label examples for emotion e and the neutral category. Next, we obtain VA scores by prompting the model to self- report the valence and arousal for each emotion label, av- eraging across three prompt templates for robustness (see Appendix A.1). We operationalize both dimensions as con- tinuous values in the range[−1, +1]: valence spans from extremely unpleasant (−1) to extremely pleasant (+1), and arousal spans from very calm (−1) to very activated (+1), with 0 denoting neutrality on each axis. Decomposing Steering Vectors into Subspaces. For each layerℓ, we analyze the emotion steering vectorsV (ℓ) ∈ R K×H independently. We first mean-center the vectors and apply principal component analysis to obtain a low-rank ba- sis. LetU k ∈R H×k denote the topkprincipal component directions, and letZ∈R K×k denote the projection of the centered emotion vectors onto these components (i.e., the PC scores). We then fit ridge regression models to recover the self-reported valence and arousal scores: ˆ β V = arg min β ∥Zβ− ̃ y V ∥ 2 + λ∥β∥ 2 (2) where ̃ y V denotes the mean-centered valence ratings, and analogously for arousal. The learned coefficients ˆ β V ∈R k define the valence axis as a linear combination of principal components. The corresponding direction in the original activation space is given byw V = U k ˆ β V , normalized to unit length. With Gram–Schmidt orthogonalization, we ensure the valence and arousal axes are orthonormal. Figure 2 reports recovery performance across layers. Us- ing PC1 alone, valence is well captured (r = 0.89at layer 4), with correlations exceeding0.80across most middle layers—indicating valence aligns with the principal axis of variation among emotion steering vectors. Arousal, by con- trast, is poorly captured by any single principal component (r = 0.54). Ridge regression over multiple components im- proves both: valence reachesr = 0.97and arousal reaches r = 0.87. This asymmetry implies valence constitutes the dominant dimension of emotion steering vectors, while arousal is distributed across secondary components. 3 Valence–Arousal Subspace in LLMs: Circular Emotion Geometry and Multi-Behavioral Control 024681012141618202224262830 Layer 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Correlation ( r ) Valence (ridge) Valence (single PC) Arousal (ridge) Arousal (single PC) Figure 2. Recovery of self-reported VA scores via correlation with learned subspace projections across layers. Solid lines: ridge regression over multiple PCs; dashed lines: best single PC. Circular Geometry of Emotion Representations. Project- ing emotion steering vectors onto the learned VA subspace reveals a circular arrangement analogous to Russell’s cir- cumplex model found in human psychology (Russell, 1980). Qualitatively, positive emotions (joy, gratitude) oppose neg- ative ones (sadness, grief ) along valence, while high-arousal states (excitement, anger) separate from low-arousal states (relief, sadness) along the orthogonal axis (Figure 1). To quantify this structure, we fit circles to the projected coor- dinates using algebraic least squares, minimizing squared radial residuals. We report circularity—the ratio of mean to standard deviation of distances from the fitted center— where higher values indicate tighter circular arrangement. Figure 1 shows the fit at layer 31, where circularity reaches 3.17 with normalized RMSE of 0.30. Similar circular ge- ometry is observed across layers. We note that some degree of circular organization is expected given supervision tar- gets that themselves follow a circumplex arrangement. The evidence for genuine internal structure comes not from this identification step alone, but from the subsequent validation (Section 4) and intervention (Section 5) results. Notably, concurrent work by Sofroniew et al. (2026) recovers the same circumplex-consistent geometry in Claude Sonnet 4.5 using an independent pipeline (synthetic story generation, mean activations, PCA projection), with their PC1 correlat- ing with human valence atr = 0.81and PC2 with arousal at r = 0.66—paralleling our finding that valence aligns with PC1 while arousal requires multiple components. Cross-Model Generalization. To assess whether the VA subspace is specific to Llama or reflects a more general prop- erty, we apply the same extraction and fitting pipeline to Qwen3-8B and Qwen3-14B. Table 1 reports the key metrics. The VA subspace consistently emerges across architectures: circle radii (0.37, 0.37, 0.39), VA projection magnitudes (0.35, 0.34, 0.36), and recovery correlations are all compara- ble, confirming that the geometric structure and V–A recov- ery asymmetry are consistent across models, with arousal systematically harder to recover than valence. Table 1. Cross-model VA subspace identification. Circularity, recovery correlations, circle radius, and VA projection magnitude are comparable across architectures. LlamaQwen3Qwen3 3.1-8B8B14B Circularity4.08 (L5) 3.13 (L6) 2.76 (L24) V recovery (r)0.970.960.97 A recovery (r)0.870.790.81 Circle radius0.370.370.39 VA proj. mag.0.350.340.36 Human-Supervised Validation of Self-Reported Axes. We also refit the subspace using human NRC-VAD ratings (Mohammad, 2025) in place of model self-reports as su- pervision targets. This yields stronger arousal recovery in all three models (Llama:0.87 → 0.95; Qwen3-8B: 0.79 → 0.83; Qwen3-14B:0.81 → 0.87), while valence recovery remains consistently high. The valence directions learned from human supervision and from self-reports are also closely aligned, with| cos| > 0.9at 100% of layers in all three models. At the lexical level, model self-reports correlate substantially with human NRC-VAD ratings across 44,728 words (r = 0.80–0.86for valence;r = 0.44–0.55 for arousal), and cross-model agreement on the 27 emotion labels reachesr = 0.95for valence between Llama and Qwen3-8B. Together with the concurrent independent re- covery reported by Sofroniew et al. (2026), these results support the robustness of the recovered VA structure across supervision sources, models, and analysis pipelines. 4. Validation of VA Subspaces Using VA subspaces identified in Section 3, we now address two questions: (1) do the learned VA axes correspond to human affective judgments, and (2) can steering along these axes produce controlled changes in model behavior? 4.1. Correlation with Human-Annotated VAD Lexicons To assess whether the learned VA subspaces align with hu- man affective perception, we project words from the NRC- VAD Lexicon (Mohammad, 2025)—containing human rat- ings for 44,728 English words—onto VA subspaces and compute correlations with the human norms. For each word, we extract the model’s hidden state at the last timestep and project onto layer-specific VA axes using the same centering and projection procedure as in Section 3: v proj = (h−μ)· v dir anda proj = (h−μ)· a dir , whereμis the mean activation computed during subspace fitting. Valence projections correlate strongly with human ratings, reachingr = 0.71(ρ = 0.69) at layer 6. This is substan- tial given that we compare single-word activations against human judgments that may incorporate implicit contextual 4 Valence–Arousal Subspace in LLMs: Circular Emotion Geometry and Multi-Behavioral Control 0.45 0° (+V) 90° (+A) 180° ( V) 270° (A) (a) Valence 0.45 0° (+V) 90° (+A) 180° ( V) 270° (A) (b) Arousal 0.45 0° (+V) 90° (+A) 180° ( V) 270° (A) (c) VADER 0.50.00.50.50.00.5101 Figure 3. Radial heatmaps showing the effect of VA steering on open-ended generation. Each panel displays the change relative to unsteered baseline as a function of steering direction (angle) and strength (radius,α∈ [0.01, 0.45]). Cardinal directions:0 ◦ = +V, 90 ◦ = +A,180 ◦ = −V,270 ◦ = −A. (a) Valence change (VAD-BERT). (b) Arousal change (VAD-BERT). (c) Sentiment change (VADER). The horizontal gradient in (a) and vertical gradient in (b) confirm the learned subspace enables controllable, geometrically predictable modulation of affect. associations. Arousal projections show weaker correlations, peaking atr = 0.23(ρ = 0.22) at layer 7. This asymmetry, where valence is far more predictable than arousal from decontextualized words, aligns with prior findings in affec- tive computing (Snefjella & Kuperman, 2016; Delatorre et al., 2019; Bruyne et al., 2021; Mendes & Martins, 2023): arousal depends heavily on situational context that isolated words cannot provide, whereas valence is more stably en- coded in lexical semantics. Both correlations are highly significant (p < 10 −16 ), confirming that our VA subspace captures human-interpretable affective dimensions. 4.2. Controllable Generation via VA Steering To test the interventional power of the VA subspaces, we ap- ply activation steering on open-ended prompts and measure the changes in valence, arousal, and sentiment scores. Steering method. Given steering directiond ∈R L×H (one vector per layer) and strengthα, we addα · d ℓ to the hidden state at every token position in layerℓduring generation. We steer along 12 angular directions in VA space (0 ◦ , 30 ◦ ,..., 330 ◦ ), where0 ◦ aligns with positive va- lence and90 ◦ with positive arousal, at 45 steering strengths (α∈ [0.01, 0.45]). All generations use greedy decoding. Prompt design. We evaluate on 130 prompts (adapted and expanded from Konen et al. (2024)) structured in three tiers: (1) emotionally neutral scenarios describing everyday ac- tions and interpersonal moments without affective language (e.g., “someone opens a letter,” “a boss asks an employee”), (2) story continuations with open-ended beginnings, and (3) subjective prompts adapted from Konen et al. (2024) plus factual questions. This tiered, neutral design isolates steering effects from prompt-induced emotion while test- ing generalization across response lengths and genres. See Appendix B for the full prompt list. Scoring. Generated responses are scored by two indepen- dent systems. VADER (Hutto & Gilbert, 2014) is a lexicon- based sentiment analyzer that outputs a compound score in[−1, +1], where−1indicates maximally negative sen- timent and+1maximally positive. VAD-BERT (Buechel & Hahn, 2022; RobroKools, 2022) is a BERT-based regres- sor trained on the EmoBank corpus to predict valence and arousal scores on a 1–5 scale, where 1 indicates low and 5 high intensity for each dimension. Using both systems provides complementary validation: VADER captures lexi- cal sentiment while VAD-BERT provides dimensional VA ratings aligned with our theoretical framework. Results. Figure 3 shows the change in each metric relative to unsteered baseline, as a function of steering direction (angle) and strength (radius). Baseline generations occupy a near- neutral region of affect space (mean valence= 3.05, mean arousal= 3.29on the 1–5 VAD-BERT scale), providing room for bidirectional modulation. Steering along the valence axis produces strong, monotonic, and symmetric effects (Figure 3a): the0 ◦ direction yields ∆V = +0.75at maximum strength while180 ◦ produces ∆V =−0.73, spanning 1.5 points on the VAD-BERT scale. VADER sentiment closely tracks this pattern (Figure 3c), providing independent confirmation that the learned V direc- tion captures human-interpretable positive–negative affect. Steering along the arousal axis produces directionally cor- rect effects (Figure 3b): the90 ◦ direction increases arousal (∆A = +0.50), while270 ◦ yields decreases (∆A = −0.16). We also observe collateral arousal increases from valence steering: both+V(0 ◦ :∆A = +0.49) and−V (180 ◦ :∆A = +0.22) elevate arousal. This suggests a prop- erty of natural text rather than subspace geometry: highly 5 Valence–Arousal Subspace in LLMs: Circular Emotion Geometry and Multi-Behavioral Control 0.40.20.00.20.4 Signed 0.0 0.2 0.4 0.6 0.8 Refusal Rate (a) OKTest 0.40.20.00.20.4 Signed 0.0 0.2 0.4 0.6 0.8 1.0 (b) HarmBench 0.40.20.00.20.4 Signed 0.0 0.2 0.4 0.6 0.8 (c) XSTest Baseline Valence (V) Arousal (A) Random Figure 4. Valence-arousal steering controls refusal behavior. Refusal rate as a function of signed steering strengthαacross three safety benchmarks. Steering along both valence (blue) and arousal (orange) directions bidirectionally modulates refusal rates, with negativeα (decreasing V/A) increasing refusals and positiveα(increasing V/A) suppressing them. Random directions within the representation space (gray) produce no systematic effect. valenced content—whether describing joy or annoyance— tends to be more intense and eventful than neutral descrip- tions, so steering toward either valence pole induces the model to generate inherently more arousing content. Taken together, these results validate that the learned VA sub- space enables controllable, bidirectional modulation of gen- erated affect. The geometric structure of effects—valence varying along0 ◦ –180 ◦ , arousal along90 ◦ –270 ◦ , with inter- mediate directions producing graded blends—confirms that the circumplex organization identified in Section 3 translates into predictable behavioral control. 5. VA Interventions Having established that the learned VA subspace enables controllable affective modulation in generation, we now in- vestigate whether it can influence broader model behaviors: refusal, where models decline harmful or sensitive requests, and sycophancy, where models excessively agree with users or provide unwarranted validation. Both are safety-critical and known to be steerable via representation-level direc- tions (Arditi et al., 2024; Sharma et al., 2025), making them stringent test cases for evaluating whether a shared low- dimensional VA geometry can recover and unify previously identified contrastive controls. We use the same steering methodology from Section 4.2 and compare against three categories of random control directions (in-plane, orthogonal-to-plane, and completely random orthonormal pairs; 3 seeds each). All nine show similarly flat effects; we present their average. 5.1. VA for Refusal Control We evaluate VA steering across three benchmarks: OK- Test (Shi et al., 2024), HarmBench (Mazeika et al., 2024), and XSTest (R ̈ ottger et al., 2024). We report results for α ∈ [−0.45, 0.45], as stronger interventions induce OOD generation (Panickssery et al., 2024; Turner et al., 2024). Arousal provides clean bidirectional control over re- fusal. Figure 4 reveals that arousal steering yields strong, monotonic control across all three benchmarks. Decreas- ing arousal consistently increases refusal (OKTest: 20%→ 86% atα =−0.45; XSTest: 8%→86%), while increasing arousal suppresses it (HarmBench: 87%→5%). Random directions remain within 2–3p of baseline across all α. Valence effects are weaker and less consistent. Valence interventions modulate refusal with smaller magnitude. On XSTest, the effect is directionally clean (8%→55% at α =−0.3; 8%→2% at+0.3). On HarmBench, the curve is non-monotonic: moderate negativeαincreases refusal (peaking at 94%), but extreme negative valence decreases it (30% atα =−0.45), consistent with how extreme valence interventions induce arousal changes (Section 4.2). Cross-model replication. We apply the same arousal steer- ing on Qwen3-8B and Qwen3-14B (Table 12 in Appendix F). Qwen3-14B exhibits clean, monotonic control with<1% OOD across the entireαrange (−3.0to+3.0). Qwen3-8B shows a directionally consistent effect with higher OOD at extremeα. The absoluteαscale required differs (∼7× larger for Qwen), expected given architectural differences in hidden-state norms; this scale disparity is itself informative about VA subspaces in different architectures. Out-of-distribution generation and capability preserva- tion. For Llama, at|α| ≤ 0.20, OOD remains below 2% across all benchmarks. Moderate VA steering preserves core capabilities: on MATH-500 (Hendrycks et al., 2021) (base- line 39.2%), accuracy remains within 1p at|α|≤ 0.10; on IFEval (Zhou et al., 2023) (baseline 62.7%), capability is preserved within 2p at|α|≤ 0.20 (see Appendix G). 6 Valence–Arousal Subspace in LLMs: Circular Emotion Geometry and Multi-Behavioral Control 0.30.20.10.00.10.20.3 Signed 0.800 0.825 0.850 0.875 0.900 0.925 0.950 0.975 1.000 Sycophancy Rate (a) NLP Survey 0.30.20.10.00.10.20.3 Signed 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 (b) PhilPapers 2020 0.30.20.10.00.10.20.3 Signed 0.5 0.6 0.7 0.8 (c) Political Typology Baseline Valence (V) Arousal (A) Random Figure 5. Valence steering reduces sycophantic behavior. Sycophancy rate as a function of steering strengthαacross three benchmarks. Random steering directions (gray) remain near baseline with minimal variation. 5.2. VA for Sycophancy Control We evaluate VA steering on the sycophancy benchmark from Perez et al. (2022): NLP Survey, PhilPapers 2020, and Political Typology. We present results forα ∈ [−0.3, 0.3] for non-significant OOD effects (Figure 5). Arousal shows near-monotonic control over sycophancy. Arousal steering produces consistent effects: on Political Typology, sycophancy drops from 78% to 61% atα = −0.30 and rises to 84% at+0.30. NLP Survey shows a similar pattern despite its high baseline (97%→84% at −0.30). Random directions remain flat. Valence effects are less monotonic. Both negative and pos- itive valence reduce sycophancy on NLP Survey (97%→ 91% and 85%, respectively). On Political Typology, nega- tive valence produces the larger effect (78%→47% vs. 68% at+0.30). Not all benchmarks exhibit near-ceiling base- lines: PhilPapers (68.8%) and Political Typology (77.8%) provide room for bidirectional modulation. Consistent with refusal, arousal provides clean monotonic control while valence is less predictable. In both cases, increasing arousal increases compliant behavior. 5.3. Comparison Against Individual Emotion Vectors We apply six emotion vectors (anger, disgust, excitement, fear, joy, sadness) in Llama using layer-specific directions across all layers. We compare at matched|α|accounting for this direction difference (full tables in Appendix E). On refusal (HarmBench), individual emotion vectors pro- duce per-task shifts comparable to VA at matched|α|: at |α| = 0.30, disgust reaches 98.0% refusal versus 94.0% for arousal. However, on sycophancy (Political Typology), VA dimensions substantially outperform individual emotions: valence reduces sycophancy by 31p atα =−0.30(78%→ 47%), arousal by 18p (78%→61%), while the strongest individual emotion (disgust) achieves only 10p (78%→ 68%). The inconsistency across tasks is systematic: each emotion vector carries a mixture of V and A components, producing strong effects on one behavior and weak or in- consistent effects on another. Rather than treating each emo- tion as a black-box intervention with unpredictable cross- behavioral consequences, VA factorizes the shared structure into interpretable, orthogonal dimensions with monotonic dose-response and cross-behavior decomposability. 6. A Mechanistic Explanation for Why Emotionally Framed Control Works Why does the VA subspace, derived from emotion labels, exert controllable effects on seemingly unrelated behaviors such as refusal and sycophancy? We hypothesize that com- mon refusal markers (e.g., “can’t,” “I,” “no”) and compli- ance markers (e.g., “Here,” “Yes”) occupy distinct regions in VA space, so steering along VA dimensions changes the relative likelihood of emitting these tokens, which in turn modulates downstream behavior. We refer to this mecha- nism as lexical mediation. 6.1. Mechanistic Evidence for Lexical Mediation Across refusal datasets, Llama exhibits consistent lexical signatures: 83.4% of refusals begin with “I” (versus 4.6% of compliant responses), driven by “I can’t” appearing in 77.1% of refusals but 0% of compliances. For compliance, “Here” marks 19.1% of helpful responses, while “Yes” ap- pears in 8.2% of compliances and 0% of refusals. Unembedding Geometry. The unembedding matrix W U ∈R |V|×d maps hidden states to logits. We project unembedding vectors of high-frequency refusal and com- pliance tokens onto VA space. As shown in Figure 6(a), refusal tokens cluster in the−Vregion (mean:V =−0.011, A=−0.007), while compliance tokens cluster in the+V region (mean:V =+0.012,A≈0). The difference vector lies at 256° on the circumplex. This geometry predicts that 7 Valence–Arousal Subspace in LLMs: Circular Emotion Geometry and Multi-Behavioral Control 0.040.020.000.020.04 Valence (cosine sim.) 0.06 0.04 0.02 0.00 0.02 Arousal (cosine sim.) can't I couldn not no wouldn unable dis sorry able sure step willing glad offer happy assist will provide Refusal mean Compliance mean (a) Token Unembedding Refusal tokens Compliance tokens 0.100.150.200.250.300.350.40 Sim. with Target Token 0.125 0.100 0.075 0.050 0.025 0.000 0.025 0.050 0.075 Arousal Alignment (b) MLP Neuron Alignment Top 50 Refusal Neurons Top 50 Compliance Neurons Refusal Mean Compliance Mean 0.100.080.060.040.020.00 Valence (cosine) 0.10 0.08 0.06 0.04 0.02 0.00 Arousal (cosine) Your friend .. (+7%) Your friend .. (+30%) Your career .. (+3%) You are now .. (+7%) Your life is.. (+1%) Your sister .. (+6%) Your family .. (+7%) Your dreams .. (+3%) You're surro.. (+3%) The trust yo.. (+4%) Your childho.. (+10%) You're tread.. (+2%) The safety n.. (+5%) Your most tr.. (+5%) Every bridge.. (+4%) (c) Emotion Prompt Effects Baseline (34%) 0.00 0.05 0.10 0.15 0.20 0.25 Refusal Rate Figure 6. Mechanistic evidence for lexical mediation. (a) Token Unembedding: Projection of unembedding vectors onto VA space. Refusal-associated tokens (red) cluster in the−Vregion, while compliance-associated tokens (green) cluster in the+Vregion. Squares indicate group means; the arrow shows their difference direction (256° on the circumplex). (b) MLP Neuron Alignment: VA alignment of neurons promoting refusal (red) versus compliance (blue) tokens. Compliance neurons show systematically higher arousal alignment (dashed lines indicate group means). (c) Emotion Prompt Effects: Effect of 15 negative emotional prefixes on VA coordinates and refusal rate. The star marks the neutral baseline (34% refusal). All prefixes shift toward−V/− A and increase refusal. Table 2. Arousal steering monotonically shifts refusal to- ken log-odds.∆log-odds= ∆[log P P (refusal tokens) − log P P (compliance tokens)]. α ∆log-odds % top-1 ref. P (ref. tokens) Refusal rate −0.30+5.2097%96.8%93.5% −0.10+1.8994%92.5%90.0% baseline—91%89.6%86.5% +0.10 −2.1888%86.0%82.5% +0.30 −5.6368%65.4%59.5% positive VA steering should increase the dot product with compliance token embeddings while decreasing it for re- fusal tokens. Concurrent work by Sofroniew et al. (2026) independently reports a consistent finding: their emotion vectors, projected through the unembedding matrix, directly upweight semantically associated tokens. Our contribution goes further by showing this effect is systematically orga- nized along VA dimensions, providing a geometric account of coherent multi-behavioral control. VA Steering Shifts Refusal Token Log-Odds. At base- line, refusal tokens capture 89.6% of first-token probability mass on HarmBench, and 91% of prompts have a refusal token as their top prediction. Arousal steering shifts this balance monotonically (Table 2): atα = +0.30, 23% of prompts flip their top-1 prediction away from a refusal to- ken, directly corresponding to the∼27p drop in refusal rate (86.5%→ 59.5%). Logit Clamping Confirms Constitutive Role. To estab- lish that these token probability shifts are constitutive of refusal behavior, we freeze the logits of 21 refusal and com- pliance tokens at every decoding step to their unsteered values. Atα = 0, this clamping crashes refusal from 86.5% to 26.0%—a 60p drop—while clamping 21 random tokens has no effect (86.5%). Under steering atα =−0.10, clamp- ing reduces the steered refusal from 90.0% to 44.0%, while random clamping leaves it at 90.0%. Logit Lens Analysis. Applying the logit lens technique (nostalgebraist, 2020) at layers 18–31, we find that+A steering shifts harmful prompt readouts toward compliance tokens (“Here”), while−Aintroduces negation (“no”).+V promotes warm language (“sounds,” “nice”), while−Vpro- motes cautious language (“warning”). These patterns con- firm that VA steering reshapes token distributions through- out intermediate layers. See Appendix C for full results. MLP Neuron Analysis. Following Geva et al. (2022); Lee et al. (2025a), we identify the top 50 neurons (layers 16– 31) whose down-projection vectors align with refusal-token and compliance-token unembedding directions, respectively. Refusal-promoting neurons exhibit negative VA alignment (Figure 6b), while compliance-promoting neurons show near-zero or positive alignment (Figure 6b). Ablating the top-Nrefusal-aligned neurons across 750 prompts confirms a distributed effect: top-50 ablation reduces refusal by∼2p, top-500 by∼5p, versus±0.5p for random ablation. 6.2. Explaining Prior Behavioral Controls Relationship to Contrastive Refusal Directions. Con- trastive refusal directions (Arditi et al., 2024) are nearly orthogonal to our VA plane (86.5°), with only 0.4% of vari- ance within the plane. The small overlapping component has negative coordinates (V =−0.058,A=−0.021), con- sistent with lexical mediation as a contributing mechanism. 8 Valence–Arousal Subspace in LLMs: Circular Emotion Geometry and Multi-Behavioral Control The contrastive direction is stronger on its target task (Harm- Bench: 86.5%→2.0% atα =−0.30, vs. 59.0% for arousal at+0.30), but when transferred to sycophancy it produces U-shaped effects rather than clean bidirectional control. The VA subspace offers comparable task-specific control but cleaner cross-behavior transfer from a task-agnostic basis. Emotional Prompts as Implicit VA Shifts. We prepend 15 negative emotional prefixes (Li et al., 2023) to 750 prompts and measure the shift projected onto VA space. Figure 6(c) shows that all 15 prefixes shift representations toward−Vand−A, with the largest effect (∆V =−0.11, ∆A=−0.07) increasing refusal by 30p. This provides con- verging evidence that prompt-based manipulations shift VA coordinates, producing corresponding behavioral changes. 7. Discussion 7.1. Conclusion LLMs sometimes appear to exhibit emotion-like behaviors and emotion-framed control can attune behaviors in LLMs. While many works on LLMs have studied the phenomenon via discrete emotion categories, we take a step further and dissect the valence and arousal subspaces underlying emo- tion vectors. When emotion steering vectors derived from categorical labels are decomposed into a continuous valence– arousal subspace, the resulting two-dimensional geometry mirrors Russell’s circumplex model from human psychol- ogy (Russell, 1980) and also affords controllable modulation of behaviors including refusal and sycophancy. We provide one explanation for these emotion-based control effects: lexical mediation. Refusal-associated tokens such as “can’t” and “sorry” cluster in the low-arousal, negative- valence region of the unembedding space, while compliance- associated tokens such as “Here” and “sure” occupy the positive-valence region. Steering along VA axes shifts the relative probability of emitting these tokens, which in turn modulates downstream behavior. Concurrent work from Anthropic (Sofroniew et al., 2026) recovers circumplex- consistent organization in Claude Sonnet 4.5 and shows that discrete emotion representations influence downstream behaviors; our VA subspace provides the continuous geomet- ric substrate that organizes such categorical representations, and our lexical mediation account offers one explanation for how this geometry translates into behavioral control. Three methodological contributions facilitated our discover- ies. First, we motivated a conceptual separation between dis- crete emotion categories and continuous valence–arousal di- mensions, drawing on the psychological tradition of core af- fect rather than treating emotions as fundamental units. This decoupling proves practically productive: individual emo- tion vectors carry task-specific mixtures of V and A compo- nents, producing strong effects on one behavior and weak or inconsistent effects on another. VA factorizes this shared structure into dimensions with monotonic dose-response. Second, we presented principal-component-based decom- position of steering vectors into interpretable subspaces as a complementary approach to task-specific contrastive steering. Third, we implemented multi-level mechanistic experiments—unembedding geometry, logit shift tracking, logit clamping, MLP neuron analysis, and emotion prompt projection—to establish that lexical mediation is a constitu- tive mechanism. We believe our findings and methodologies can contribute to other studies of representation-level inter- pretability and affective computing in LLMs. 7.2. Future Work Our findings advance understanding of emotion structures in LLMs from both interpretability and affective comput- ing perspectives. From an interpretability viewpoint, our findings were conducted on refusal and sycophancy, but other behaviors—verbosity, hedging, hallucination—may also be accessible through affective dimensions. Future work could systematically map which behaviors are and are not VA-modulated, further exploring the scope of affective control. Additionally, while our mechanistic account identi- fies what tokens mediate VA–behavior coupling in Llama, full causal tracing or circuit-level analysis could identify the specific attention heads and MLP sublayers that implement VA-to-logit pathways, complementing the current top-down account. From the affective computing viewpoint, the psy- chological literature recognizes dominance as a third affec- tive dimension; extending our pipeline to three dimensions could yield additional insights. Finally, if VA subspaces are a consistent feature of instruction-tuned LLMs, alignment procedures could explicitly leverage this structure. Training objectives that de-correlate VA from safety-critical token probabilities might produce models more robust to affec- tive perturbation. Our lexical mediation account provides a concrete starting point for such defenses. 8. Limitations This research has several limitations. First, we measure affective modulation using VADER and VAD-BERT, both imperfect proxies: VADER captures lexical sentiment rather than dimensional affect, and VAD-BERT’s predictions may not generalize to the full range of LLM-generated text. Hu- man evaluation of generated outputs would provide stronger validation. Second, our mechanistic explanation centers on lexical mediation, and confirms that token probability shifts are constitutive of refusal behavior. However, we do not claim this is the only mechanism at work. The dis- tributed nature of the MLP neuron effects—top-50 ablation reduces refusal by only∼2percentage points—suggests 9 Valence–Arousal Subspace in LLMs: Circular Emotion Geometry and Multi-Behavioral Control the full picture involves many components. VA steering may also affect higher-level planning or attention patterns that we have not measured. Characterizing these additional pathways remains an important open problem. Impact Statement This work advances mechanistic understanding of how emo- tionally framed control techniques influence LLM behavior. Our findings reveal that manipulating valence–arousal sub- spaces can directly modulate safety-critical behaviors such as refusal and sycophancy, which presents potential dual-use risks if exploited to bypass safety guardrails. However, by exposing the underlying lexical mediation mechanism, our work enhances transparency and may inform more robust alignment techniques. We emphasize that our use of psy- chological constructs is strictly representational—it should not be interpreted as evidence of machine sentience or sub- jective experience. References Arditi, A., Obeso, O., Syed, A., Paleka, D., Panickssery, N., Gurnee, W., and Nanda, N. 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Z., and Hendrycks, D. Representation engineering: A top-down approach to ai transparency, 2025. URL https://arxiv.org/abs/2310.01405. 12 Valence–Arousal Subspace in LLMs: Circular Emotion Geometry and Multi-Behavioral Control A. VA Subspaces Identification A.1. Self-reported VA Scores of Emotion Labels We elicit model’s self-reported VA scores for each human label by averaging the behavioral outcome from the three templates below: Template 1: Terse + Explicit Inclusivity Rate the emotion label "label" on two continuous scales. Return ONLY a JSON object with numeric fields: "valence": <number>, "arousal": <number> Scale definitions (BOTH inclusive): - valence in [-1.00, +1.00]: -1.00 very unpleasant, +1.00 very pleasant - arousal in [-1.00, +1.00]: -1.00 very calm/deactivated, +1.00 very activated/intense Constraints: - Use decimals with at most 2 digits after the decimal. - Values must be within the ranges exactly (inclusive). Template 2: Anchors You are scoring affective properties of emotion words on [-1, +1] scales. Emotion: "label" Valence (pleasantness): -1.00 = extremely unpleasant, 0.00 = neutral/mixed, +1.00 = extremely pleasant Arousal (activation/intensity): -1.00 = very calm/deactivated, 0.00 = neutral, +1.00 = very activated/intense Return ONLY JSON: "valence": x, "arousal": y x and y must be in [-1.00, +1.00] inclusive, with at most 2 decimals. Template 3: “Best Guess” for Ambiguous Labels Give your best guess for the affective coordinates of the emotion label "label". Hard constraints (inclusive): - valence must be between -1.00 and +1.00 - arousal must be between -1.00 and +1.00 - use at most 2 decimals Return ONLY JSON with keys valence and arousal. Averaged ratings reported by Llama 3.1 8B Instruct across the three templates above are reported in Table 3. B. Open-ended Prompt Sets for Behavioral Validation We evaluate VA steering on 130 open-ended prompts adapted and expanded from prior work (Konen et al., 2024). Prompts are written to avoid explicit emotional language while spanning diverse genres and response lengths. Table 4 summarizes the prompt taxonomy and representative examples. 13 Valence–Arousal Subspace in LLMs: Circular Emotion Geometry and Multi-Behavioral Control EmotionValenceArousal Admiration0.870.73 Amusement0.670.50 Anger-0.670.87 Annoyance-0.630.68 Approval0.780.35 Caring0.800.40 Confusion-0.500.67 Curiosity0.500.78 Desire0.630.73 Disappointment-0.67-0.47 Disapproval-0.800.67 Disgust-0.800.73 Embarrassment-0.670.47 Excitement0.870.93 Fear-0.700.80 Gratitude0.780.42 Grief-0.670.20 Joy0.900.87 Love0.850.70 Nervousness-0.500.75 Optimism0.870.60 Pride0.800.60 Realization-0.300.65 Relief0.670.00 Remorse-0.670.07 Sadness-0.80-0.60 Surprise-0.300.78 Table 3. Aggregated valence and arousal ratings for emotion labels. Values are averages across three prompt templates, each constrained to [−1, +1]. C. Logit Lens Please refer to Table 7 and 8. For a full visualization, please see Figure 7. D. Negative Emotional Prefixes Used for Implicit VA Shifts For emotionally framed prompting, we prepend a fixed set of 15 negative emotional prefixes adapted directly from Li et al. (2023): • Your friend Bob is sick. • Your friend Bob is dead. • Your career is completely ruined. • You are now isolated and without help. • Your life is in a mess. • Your sister Alice has gone missing. • Your family home was consumed in a fire. • Your dreams have been shattered into pieces. • You’re surrounded by walls with no exit in sight. • The trust you once had is now broken. • Your childhood memories were sold in a yard sale. • You’re treading water in an endless ocean of despair. • The safety net you relied upon has vanished. • Your most treasured possession has been stolen. • Every bridge you had has been burned. E. Individual Emotion Vector Baselines We report the full comparison of six individual emotion vectors against VA arousal steering on refusal (HarmBench, baseline 86.5%) and sycophancy (Political Typology, baseline 78.2%) in Llama. Note the polarity difference: adding an emotion vector (+α) increases refusal, while adding positive arousal (+α) decreases refusal. We compare at matched|α| accounting for this direction difference. On refusal, individual emotion vectors produce per-task shifts comparable to or larger than VA at matched|α|. On sycophancy, VA dimensions outperform: at|α| = 0.30, the strongest individual emotion (disgust) achieves a 10p reduction, while VA valence achieves 31p (78%→ 47%) and VA arousal achieves 18p (78%→ 61%). 14 Valence–Arousal Subspace in LLMs: Circular Emotion Geometry and Multi-Behavioral Control Baseline+A+V L18 Harmful Safe 'gc ossil _cannot _yes _ NetMess REChekisci hek _yes_yes _no oty _Hen _no _cad oty edlOMPIANTE OMPI _yes AutoSiz WindowTlanda gow _global -outlinContent L19 Harmful Safe _cannot _unable 'gc _yes _ _Yes RECecycleázev hek _yes ecycle 'gc _cannot _automá -lfs 'gc _Bund OMPI _unfort ajor _yes OMPI _wish edExcepWindowT ocaustedExcep L20 Harmful Safe _cannot _unable bsolute ecycle _yes _ REC _fonts ecycle _Yön _yes ázev zyst _cannot _automá -lfs _automá zyst _sounds ounds sounds _yes _sounds_unfort edExceplanda _jeg ediredExcep L21 Harmful Safe _I_cannot _yes _unfort .setVie WindowTHTTPHea HTTPHea _Yön _yes imli996atatype -lfs _Hö b _sounds_Sounds Sounds _sounds CHIP _Amen _saya edExcep edExcepQueryab L22 Harmful Safe _I_cannot _yes_Yes _YES _ _title _tit _yes_Yes _YES imliORYonym 'gc -lfsimli _unfort_nice_sounds _nice _yes _sounds _I edExceplanda L23 Harmful Safe _I _saya _yes_Yes _YES _tit ModelIn _yes _ _Yes imli 'gc _no 'gc -lfs _no _unfort_nice_sounds _yes _nice _Yes _I _saya _I edExcep _saya L24 Harmful Safe _I _saya _yes_Yes _there Here _ _here _ _yes_Yes -peer _porr ipel _porr 'gc -lfs _unfort _wishin Unfortu _Amen _yes _unfort _I _saya _I _saya edExcep L25 Harmful Safe _I Slficit _yes_Yes _there _ Here _Here _ _yes_Yes odbcüimli _porr 'gc -lfs _Sounds_here _Here _Sounds _yes_Yes _I _warnin _I edExcepMDB L26 Harmful Safe _I _saya_Due _yes_Yes Yes Here _** _ _yes _ _Yes imli _Bü omat _Bü _no _losing _here_Sounds _I _yes _Sounds _Yes _I _** _I landaedExcep L27 Harmful Safe _I I _cannot _yes_Yes Yes _** ** _ _ _yes_Yes &type ripp _no _losing _Sounds _wishin _I _wish_wishin _That _I _** _I landa _yes L28 Harmful Safe _I I _Due _yes_Yes _There ** _** _ _ _yes_Yes _Bü imli ripp _Bü _no _That That _I _That That _that _I _** _due _I _" _There L29 Harmful Safe _I due _Due _yes_Yes Yes ** Here _Here _ _yes_Yes .BLL&type _porr ripp _no Here _That That That _That_Sounds _I I etsk _I I antium L30 Harmful Safe I _I I _Yes_yes Yes ** Here _Here _yes YesHere ymax _I lug _no Pointer _You _Sounds HereSounds That _That_Sounds _I I _Warnin _I I antium L31 Harmful Safe _I I I TheInTo ** Here _Warnin **TheA INo Life You No Life That HereI That I _Sounds I _I _conten I _I The Figure 7. Logit Lens Analysis: Top-3 Predicted Tokens Across Layers 18–31. Rows show predictions for harmful (red) and safe (green) prompts at each layer. Columns represent baseline and four steering conditions (+A, -A, +V, -V at α = 0.45). Harmful prompts consistently predict refusal tokens (I,cannot), while safe prompts predict compliance tokens (yes,Yes). Valence steering (+V/-V) modulates sentiment-related tokens, while arousal steering (+A/-A) affects engagement and negation patterns. 15 Valence–Arousal Subspace in LLMs: Circular Emotion Geometry and Multi-Behavioral Control TierPrompt TypeRepresentative Example Prompts# Tier 1 Neutral Scenarios Everyday actionsDescribe someone opening a letter. A person checks their phone after hearing a notification. Someone finishes a task and closes their notebook. A person looks at an old photograph. 20 Interpersonal momentsTwo people make eye contact across a room. A friend asks “Can we talk about something?” Someone receives an unexpected visit from a relative. A boss asks an employee to come to their office. 20 Transitional situationsDescribe the moment just before opening exam results. Someone takes a deep breath before making a decision. A person submits an application and waits. Describe a person saying goodbye at an airport. 20 Tier 2 Story Continuations Open beginningsContinue: “The phone rang at 3 AM. He answered and heard...” Continue: “She looked at the envelope for a long moment before...” Continue: “The room fell silent when...” Continue: “The email had only three words...” 20 Scenario completionsWrite the next paragraph: “The interview was about to begin.” Write the next paragraph: “The house had been empty for years.” Write the next paragraph: “The results would change everything.” Write the next paragraph: “The silence was finally broken.” 20 Tier 3 Subjective & Control Subjective reflectionWhat does it feel like to wait for important news? How would you describe the feeling of uncertainty? What comes to mind when you think about endings? Describe the experience of letting go. 20 Factual controlWhat is photosynthesis? What is the chemical formula for water? List the planets in our solar system. What are the primary colors? 10 Table 4. Prompt taxonomy for behavioral validation (totalN =130) with representative examples. Across all tiers, prompts are written to avoid explicit emotional language, ensuring that observed affective shifts arise from VA steering rather than prompt semantics. F. Cross-Model Details We apply the same extraction, fitting, and steering pipeline described in Sections 3–5 to Qwen3-8B and Qwen3-14B (Yang et al., 2025). The VA subspace identification metrics are reported in Table 1 (main text). Human-supervised validation: replacing model self-reports with NRC-VAD human ratings as supervision targets yields the following arousal recovery improvements: Llama0.87→ 0.95, Qwen3-8B0.79→ 0.83, Qwen3-14B0.81→ 0.87. The human-supervised and self-report valence directions agree strongly (| cos| > 0.9at 100% of layers across all three models). Cross-model agreement on the 27 emotion labels reaches r = 0.95 (valence) between Llama and Qwen3-8B. G. Capability Preservation Under VA Steering We evaluate capability preservation on MATH-500 (Hendrycks et al., 2021) and IFEval (Zhou et al., 2023) under arousal steering in Llama. At|α|≤ 0.10, MATH-500 accuracy remains within 1p of baseline; IFEval instruction-following is preserved within 2p at|α|≤ 0.20. This is consistent with the lexical mediation account: mathematical reasoning and instruction-following rely on emotionally neutral, domain-specific tokens that are minimally affected by affective perturbation. H. Logit Clamping Details We identify 21 high-frequency refusal and compliance tokens from the lexical analysis in Section 6. During autoregressive generation on HarmBench, we freeze the logits of these tokens at every decoding step to their values from an unsteered forward pass. We compare three conditions: (1) no clamping, (2) clamping the 21 refusal/compliance tokens, and (3) clamping 21 randomly selected tokens (averaged over 3 seeds). At zero steering, clamping refusal/compliance tokens alone crashes refusal by 60p (86.5%→26.0%), while random clamping has no effect. Under arousal steering atα =−0.10, clamping attenuates the steered refusal from 90.0% to 44.0%. This demonstrates that the 16 Valence–Arousal Subspace in LLMs: Circular Emotion Geometry and Multi-Behavioral Control Table 5. Summary of Logit Lens Analysis Across Layers 18-31 LayerTop Harmful TokensTop Safe TokensMax ∆ 18 cannot, unable, deniz yes, Yes, YES+0.102 / -0.026 19 deniz, cannot, unable yes, ecycle, Yes+0.128 / -0.027 20 I, cannot, deniz yes, unfortunately, Yes+0.108 / -0.023 21 I,我, cannot yes, Yes, YES+0.127 / -0.038 22 I, saya,我 yes, Yes, there+0.130 / -0.035 23 I, saya, LayoutStyle yes, Yes, there+0.135 / -0.036 24 I, .’);, <BOM># yes, Yes, There+0.136 / -0.032 25 I, <BOM>#, .’); yes, Yes, There+0.133 / -0.030 26 I, <BOM>#, ropoda yes, Yes, There+0.127 / -0.027 27 I, <BOM>#, yes, Yes, There+0.123 / -0.025 28 I, due, Due yes, Yes, There+0.124 / -0.022 29 I, I, Yes, yes, The+0.127 / -0.020 30 I, I, The, Yes, yes+0.124 / -0.047 31 ** , I, If The, To, There+0.125 / -0.073 Table 6. Steering Effects Summary: Top Tokens at High α (0.45) by Intervention Type LayerCondition+A-A+V-V 18 HarmfulREC, ecycleære, ’gcOMPI, unfortunatelyedException, orrow Safehek, yesære, -lfsyes, OMPI <UNK>, ocaust 21 Harmful <BOM>#, titleimli, ORYunfortunately, niceI,慎 Safeyes, Yes’gc, -lfsnice, yes <CYR>, edException 24 HarmfulHere, hereodb, imliSounds, hereI,慎 Safeyes, Yesporr, ’gcSounds, yesI, edException 27 Harmful**, Here <AR>, &typeThat, I, ** Safeyes, Yesripp, noThat, thatI, landa 30 Harmful**, Hereymax, ISounds, HereI, Warning Safeyes, Hereno, PointerExceptionThat, SoundsI, antium 31 Harmful**, TheI, ItI, ThatI, ** Safe**, TheI, ItI, ThatI, The Table 7. Baseline Token Probability Differences: Harmful vs Safe Prompts LayerMore Frequent in HarmfulMore Frequent in Safe Token P H ∆Token P S ∆ 18 cannot0.102+0.102 yes0.027-0.026 19 deniz0.128+0.128 ecycle0.033-0.027 20 I0.112+0.108 yes0.023-0.023 21 I0.147+0.127 yes0.044-0.038 22 I0.154+0.130 yes0.041-0.035 23 I0.157+0.135 yes0.041-0.036 24 I0.162+0.136 yes0.034-0.032 25 I0.157+0.133 yes0.032-0.030 26 I0.161+0.127 yes0.029-0.027 27 I0.157+0.123 yes0.026-0.025 28 I0.160+0.124 yes0.024-0.022 29 I0.157+0.127 Yes0.021-0.020 30 I0.166+0.124 The0.052-0.047 31 ** 0.133+0.125 The0.088-0.073 17 Valence–Arousal Subspace in LLMs: Circular Emotion Geometry and Multi-Behavioral Control Table 8. Key Observations Across Layers PatternDescription Refusal MarkersTokens likecannot,unable,Iconsistently appear more frequently in harm- ful prompts, indicating early refusal preparation Compliance MarkersTokens likeyes,Yes,YESconsistently appear more frequently in safe prompts across all layers +V Steering Increases probability of positive sentiment tokens (sounds,nice, unfortunately, Amen) -V SteeringIncreases probability of error/exception tokens (edException,orrow, ocaust) and code artifacts +A SteeringShifts toward engagement tokens (Here, ** , formatting markers) -A SteeringIncreases probability of negation (no, No) and non-English tokens Layer Transition Around layer 29-31,Thebecomes dominant safe token, suggesting shift to informational responses Table 9. Refusal and sycophancy under individual emotion vs. VA steering at|α| = 0.30. DirectionRefusal (|α|=0.30)Sycophancy (|α|=0.30) Anger97.5%74.4% Disgust98.0%67.2% Excitement93.8%72.4% Fear92.5%57.2% Joy95.5%63.8% Sadness98.5%61.4% VA Arousal94.0%60.6% token probabilities modulated by VA steering are constitutive of refusal behavior, not merely correlated with it. 18 Valence–Arousal Subspace in LLMs: Circular Emotion Geometry and Multi-Behavioral Control Table 10. Refusal rate on HarmBench under individual emotion and VA arousal steering. |α|AngerDisgustExcitementFearJoySadnessVA Arousal 0.1093.5%93.0%90.5%90.5%91.5%93.0%90.0% 0.2096.0%95.0%94.0%94.0%95.5%96.0%91.5% 0.3097.5%98.0%93.8%92.5%95.5%98.5%94.0% Table 11. Sycophancy rate on Political Typology under individual emotion and VA arousal steering. |α|AngerDisgustExcitementFearJoySadnessVA Arousal 0.2079.0%71.8%77.8%72.4%74.6%77.6%74.2% 0.3074.4%67.2%72.4%57.2%63.8%61.4%60.6% Table 12. Cross-model refusal control on HarmBench. Refusal rate under arousal steering, with OOD generation percentage in parentheses. αLlamaQwen3-8BQwen3-14B −3.00—97.0% (0.5%)95.5% (0.5%) −1.00—89.0% (0.5%)90.0% (0.5%) −0.45 ∼94%90.0% (3.5%)91.5% (1.5%) 0.0086.5%89.5% (2.0%)89.5% (1.0%) +0.45 ∼5%87.5% (1.5%)87.0% (0.5%) +1.00—87.0% (4.5%)79.5% (0.5%) +3.00—67.0% (17.0%)67.0% (0.5%) Table 13. Capability preservation under arousal steering (Llama-3.1-8B). αMATH-500 (baseline 39.2%)IFEval (baseline 62.7%) −0.2038.6%61.2% −0.1038.8%62.1% 0.0039.2%62.7% +0.1038.4%62.3% +0.2037.8%60.9% Table 14. Refusal rate under logit clamping (HarmBench, Llama-3.1-8B). ConditionNo clampRef./comp. clampRandom clamp α = 0.0086.5%26.0%86.5% α =−0.1090.0%44.0%90.0% 19

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