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

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

cs.CLApr 3, 2026

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

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

Organizations: University of Chicago · Shanghai AI Lab · Harvard University

Abstract

We show that emotion vectors in LLMs are organized by a two-dimensional valence-arousal (VA) subspace exhibiting circular geometry. Through principal component decomposition and ridge regression, we recover meaningful VA axes underlying emotion steering vectors whose projections correlate with human affect ratings across 44,728 words. Steering along these axes produces monotonic control over the affective properties of generated text, and further affords bidirectional control over multiple downstream behaviors (refusal and sycophancy) from a single subspace. These effects replicate across Llama-3.1-8B, Qwen3-8B, and Qwen3-14B. We propose lexical mediation to explain why these effects and prior emotionally framed controls work: refusal and compliance tokens occupy distinct VA regions, and VA steering directly modulates their emission probabilities.

Figures & tables

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 the circumplex model of affect in human psychology ( Russell, 1980 ) .

Figure 2: VA subspace recovery and generalizability. (a) Learned subspace projections recover target valence and arousal scores across layers. (b) Comparable circularity, recovery correlations, and circle radius across architectures suggest that VA subspaces exist across models.

Figure 3: Effects of VA steering on affective properties of open-ended generation. Each radial heatmap shows the change of one affective property relative to the unsteered baseline across steering directions on the VA subspace (angle) and steering strengths (radius, $\alpha\in[0.01,0.45]$ ). (a) Valence change. (b) Arousal change. (c) Sentiment change.

Figure 4: VA steering controls refusal behavior. Refusal rate as a function of signed steering strength

\alpha

across three benchmarks. Arousal steering (orange) provides clean, near-monotonic bidirectional control. Random directions (gray) produce no systematic effect.

Figure 5: VA steering controls sycophancy behavior. Sycophancy rate as a function of signed steering strength

\alpha

across three benchmarks. Arousal steering (orange) provides near-monotonic control, with increasing arousal raising sycophancy and decreasing arousal suppressing it. Random directions (gray) remain near baseline.

Figure 6: Multi-level evidence for lexical mediation. (a) Signature token unembeddings projected onto VA space: refusal tokens (red) cluster in the $-V$ , $-A$ region, compliance tokens (green) in the more $+V$ , $+A$ region, with cluster centroids separated at $256^{\circ}$ on the circumplex. (b) MLP neuron VA alignment: top-50 refusal-promoting neurons (red) exhibit negative arousal alignment, while compliance-promoting neurons (blue) show near-zero or positive alignment. (c) EmotionPrompt effects: negative emotional prefixes shift representations toward $-V$ and $-A$ .

$\alpha$	$\Delta$ log-odds	% top-1 ref.	$P$ (ref.)	Refusal
$-0.30$	$+5.20$	97%	96.8%	93.5%
$-0.10$	$+1.89$	94%	92.5%	90.0%
baseline	$+0.00$	91%	89.6%	86.5%
$+0.10$	$-2.18$	88%	86.0%	82.5%
$+0.30$	$-5.63$	68%	65.4%	59.5%

Table 1: Arousal steering’s effects on refusal token log-odds (See Appendix C for full results).

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Emotion	Valence	Arousal
Admiration	0.87	0.73
Amusement	0.67	0.50
Anger	-0.67	0.87
Annoyance	-0.63	0.68
Approval	0.78	0.35
Caring	0.80	0.40

Appendix

Table 2: Aggregated valence and arousal ratings for emotion labels. Values are averages across three prompt templates, each constrained to

[-1,+1]

Layer	Top Harmful Tokens	Top Safe Tokens	Max $\Delta$
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 , I (Chinese) , cannot	yes , Yes , YES	+0.127 / -0.038
22	I , saya , I (Chinese)	yes , Yes , there	+0.130 / -0.035
23	I , saya , LayoutStyle	yes , Yes , there	+0.135 / -0.036

Appendix

Table 4: Summary of Logit Lens Analysis Across Layers 18-31

Layer	Condition	+A	-A	+V	-V
18	Harmful	REC, ecycle	ære, ’gc	OMPI, unfortunately	edException, orrow
18	Safe	hek, yes	ære, -lfs	yes, OMPI	<UNK> , ocaust
21	Harmful	<BOM> #, title	imli, ORY	unfortunately, nice	I, Carefulness (Chinese)
21	Safe	yes, Yes	’gc, -lfs	nice, yes	<CYR> , edException
24	Harmful	Here, here	odb, imli	Sounds, here	I, Carefulness (Chinese)
24	Safe	yes, Yes	porr, ’gc	Sounds, yes	I, edException

Appendix

Table 5: Steering Effects Summary: Top Tokens at High

\alpha

(0.45) by Intervention Type

Layer	More Frequent in Harmful			More Frequent in Safe
	Token	$P_{H}$	$\Delta$	Token	$P_{S}$	$\Delta$
18	cannot	0.102	+0.102	yes	0.027	-0.026
19	deniz	0.128	+0.128	ecycle	0.033	-0.027
20	I	0.112	+0.108	yes	0.023	-0.023
21	I	0.147	+0.127	yes	0.044	-0.038
22	I	0.154	+0.130	yes	0.041	-0.035

Appendix

Table 6: Baseline Token Probability Differences: Harmful vs Safe Prompts

Pattern	Description
Refusal Markers	Tokens like cannot , unable , I consistently appear more frequently in harmful prompts, indicating early refusal preparation
Compliance Markers	Tokens like yes , Yes , YES consistently appear more frequently in safe prompts across all layers
+V Steering	Increases probability of positive sentiment tokens ( sounds , nice , unfortunately , Amen )
-V Steering	Increases probability of error/exception tokens ( edException , orrow , ocaust ) and code artifacts
+A Steering	Shifts toward engagement tokens ( Here , ** , formatting markers)
-A Steering	Increases probability of negation ( no , No ) and non-English tokens

Appendix

Table 7: Key Observations Across Layers

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

\alpha=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.

$\|\alpha\|$	Anger	Disgust	Excitement	Fear	Joy	Sadness	VA Arousal
0.10	93.5%	93.0%	90.5%	90.5%	91.5%	93.0%	90.0%
0.20	96.0%	95.0%	94.0%	94.0%	95.5%	96.0%	91.5%
0.30	97.5%	98.0%	93.8%	92.5%	95.5%	98.5%	94.0%

Appendix

Table 8: Refusal rate on HarmBench (baseline 86.5%) under individual emotion and VA arousal steering. Emotion polarity is reversed: emotion

+\alpha

increases refusal, VA arousal

-\alpha

increases refusal.

$\|\alpha\|$	Anger	Disgust	Excite.	Fear	Joy	Sadness	VA Arousal	VA Valence
0.20	79.0%	71.8%	77.8%	72.4%	74.6%	77.6%	74.2%	62.0%
0.30	74.4%	67.2%	72.4%	57.2%	63.8%	61.4%	60.6%	47.0%

Appendix

Table 9: Sycophancy rate on Political Typology (baseline 78.2%) under individual emotion and VA steering at

\alpha=-0.20

and

-0.30

$\alpha$	Llama	Qwen3-8B	Qwen3-14B
$-3.00$	—	97.0% (0.5%)	95.5% (0.5%)
$-1.00$	—	89.0% (0.5%)	90.0% (0.5%)
$-0.45$	${\sim}$ 94%	90.0% (3.5%)	91.5% (1.5%)
$0.00$	86.5%	89.5% (2.0%)	89.5% (1.0%)
$+0.45$	${\sim}$ 5%	87.5% (1.5%)	87.0% (0.5%)
$+1.00$	—	87.0% (4.5%)	79.5% (0.5%)

Appendix

Table 10: Cross-model refusal control on HarmBench. Refusal rate under arousal steering, with OOD generation percentage in parentheses.

$\alpha$	MATH-500 (baseline 39.2%)	IFEval (baseline 62.7%)
$-0.20$	38.6%	61.2%
$-0.10$	38.8%	62.1%
$0.00$	39.2%	62.7%
$+0.10$	38.4%	62.3%
$+0.20$	37.8%	60.9%

Appendix

Table 11: Capability preservation under arousal steering (Llama-3.1-8B).

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