In-Context Learning Amplifies a Latent Symbolic Circuit

Melissa Wessel

cs.LGSep 28, 2026

In-Context Learning Amplifies a Latent Symbolic Circuit

Authors: Melissa Wessel

Organizations: Independent Researcher.

Abstract

Large language models can learn abstract rules from just a few in-context examples, but how their internal mechanisms activate as examples accumulate is not well understood. We trace a three-stage symbolic reasoning circuit (abstraction, induction, retrieval) across shot counts in three model families and find it is detectable and functional well before the model achieves high accuracy. Per-head causal contribution grows up to 8x from 1- to 10-shot, and cross-shot activation patching raises accuracy from 1% to 56% at 0-shot and 17% to 88% at 1-shot. Function vectors scaled and injected at 0-shot rescue accuracy up to 86%, largely substituting for the induction stage but depending critically on an intact downstream retrieval stage. The infrastructure for abstract rule-following is present in the weights before any demonstrations; in-context examples, function vectors, and related interventions appear to supply input to the same latent circuit.

Figures & tables

Figure 1 : The Symbolic Reasoning Circuit in Gemma 2-2B at 1 through 10-Shot. CMA Scores by layer and head by circuit stage across shot counts.

Panel of Figure 2 — Figure 2 : Cross-Shot Patching Results (ABA). Accuracy of Gemma 2-2B when patching 10-shot donor activations into 1-shot (a) and 0-shot (b) target prompts. Full results for all models and both rules are in Appendix Figures 9 – 9 .

Figure 3 : Function-vector rescue at 0-shot, ABA.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Model	Layers	Q heads	KV heads	$d_{\text{model}}$	$d_{\text{head}}$	BOS
Gemma 2-2B ( Gemma Team, 2024 )	26	8	4	2304	256	yes
Llama 3.1-8B ( Grattafiori et al., 2024 )	32	32	8	4096	128	yes
Qwen 3-4B ( Yang et al., 2025a )	36	32	8	2560	128	no

Appendix

Table 1: Model architectures. All three use grouped-query attention (GQA). All experiments use bfloat16 precision and load the models via TransformerLens, which materializes one

W_{K}

W_{V}

per query head so attention-head indexing is uniform across architectures.

Stage	Patch position(s)	CMA condition
Symbolic Abstraction (SA)	all $C$ positions: $\{b+6i+4:0\leq i<n\}$	abstract
Symbolic Induction (SI)	final query position: $b+6n+3$	abstract
Retrieval (Ret)	final query position: $b+6n+3$	token

Appendix

Table 2: Patch positions per stage. SA writes to the

C

tokens of the demonstrations (so no SA evaluation is possible at

n{=}0

). SI and Retrieval write to the same query position but are distinguished by the contrastive condition.

Panel of Figure 4 — Figure 4 : Cross-shot activation patching. Activations from 10-shot donors (middle) are patched into 1-shot (left) and 0-shot (right) targets (bottom), steering outputs to the correct rule-following completion. Yellow highlights indicate the SA head patch position (C tokens); blue indicates the SI and Retrieval head patch position (query token).

Condition	ABA	ABB
Baseline (no inject)	1.0	11.0
FV inject	86.0	88.5
FV inject + ablate SI	65.0	89.0
FV inject + ablate Ret	13.0	28.5
Random vector (norm-matched)	0.0	0.5

Appendix

Table 3: FV

\times

per-stage ablation factorial at

L{=}12

s{=}25

. Accuracy (%),

n{=}200

per cell.

Figure 5 : The Symbolic Reasoning Circuit in Llama 3.1-8B at 1 through 10-Shot. CMA Scores by layer by head, by circuit stage across shot counts.

Figure 6 : The Symbolic Reasoning Circuit in Qwen 3-4B at 1 through 10-Shot. CMA Scores by layer by head, by circuit stage across shot counts.

Gemma 2-2B
Stage	Shot	Top-5 (CMA score)
SA	1	L6H1 (0.50)	L8H1 (0.45)	L6H2 (0.38)	L3H1 (0.14)	L3H0 (0.14)
	2	L6H1 (1.05)	L8H1 (0.88)	L6H2 (0.63)	L10H4 (0.47)	L3H1 (0.22)
	4	L6H1 (1.73)	L8H1 (1.33)	L10H4 (1.01)	L6H2 (0.87)	L12H3 (0.58)
	10	L6H1 (2.07)	L8H1 (2.03)	L10H4 (1.72)	L12H3 (1.30)	L6H2 (1.22)
SI	1	L14H0 (1.40)	L12H1 (0.73)	L15H0 (0.61)	L14H1 (0.31)	L13H0 (0.25)

Appendix

Table 4 : Top-5 attention heads per stage and shot count, ranked left-to-right by mean CMA score (in parentheses; averaged across pairs and across ABA/ABB rule directions). Bold marks persistent-core heads (top-5 at every shot count). Reading down a column shows reranking among the leading heads.

Stage	Top-5 core size	Top-10 core size	Top-5 core members
Gemma 2-2B
SA	3	6	L6H1, L6H2, L8H1
SI	4	6	L12H1, L14H0, L14H1, L15H0
Ret	3	7	L18H6, L20H7, L22H4
Llama 3.1-8B
SA	3	5	L2H27, L10H1, L14H19

Appendix

Table 5: Persistent cores: heads in the top-

K

at every shot count

n\in\{1,2,4,10\}

. Qwen’s top-5 is noisier at low shot counts but recovers at top-10.

Gemma 2-2B
	1 $\leftrightarrow$ 2	1 $\leftrightarrow$ 4	1 $\leftrightarrow$ 10	2 $\leftrightarrow$ 4	2 $\leftrightarrow$ 10	4 $\leftrightarrow$ 10
SA	0.77	0.71	0.66	0.86	0.79	0.91
SI	0.81	0.77	0.72	0.86	0.81	0.91
Ret	0.84	0.79	0.78	0.94	0.86	0.87
Llama 3.1-8B
SA	0.85	0.58	0.54	0.67	0.62	0.81

Appendix

Table 6: Rank-Biased Overlap (RBO,

p=0.9

, full-depth extrapolated) between per-head CMA rankings at every pair of shot counts. Each ranking averages across the 200 CMA pairs and across the ABA and ABB rule directions. Higher is more similar; for full rankings of unrelated lists RBO would tend to

\approx 1-p=0.1

Figure 7 : Cross-shot patching, Gemma 2-2B. Rows: 1-shot, 0-shot targets. Columns: ABA, ABB. Dashed line: baseline.

Gemma 2-2B
Shot	Rule	Baseline	SA-only	SI-only	SI-shuffled	Ret-only	All stages	Random
0	ABA	1.0	1.0	31.0	23.0	17.5	56.0	0.5
0	ABB	7.5	7.5	41.5	36.5	42.5	72.5	8.5
1	ABA	16.5	19.5	70.5	54.0	67.0	87.5	9.0
1	ABB	57.0	57.5	84.0	78.0	81.5	93.0	47.0
Llama 3.1-8B

Appendix

Table 7: Cross-shot patching accuracy (%). Patching from a 10-shot donor into a 0/1-shot target, by stage. SA-only, SI-only, and Ret-only patch only the heads at that stage’s positions; All stages patches the union of all three. SI-shuffled uses a donor with the same rule but entirely disjoint tokens, patched at SI heads only. Random patches a matched-count random selection of non-circuit heads.

n{=}200

prompts per cell. SA-only is identical to baseline at 0-shot because there are no in-context C positions to patch. Bold marks the All-stages column. Mean answer probability

p

tracks accuracy and is omitted for compactness.

Model	Shot	Succ.	Pred.
Gemma 2-2B	2	89.0%	17.6%
Gemma 2-2B	10	99.8%	76.4%
Llama 3.1-8B	2	98.4%	64.4%
Llama 3.1-8B	10	100%	100%

Appendix

Table 8: LSA accuracy (all three answer letters correct), 500 prompts per rule.

Figure 10 : LSA CMA heatmaps, Gemma 2-2B at 1, 2, and 10 shot, by stage. Same three-stage topology as ABA/ABB; L14H0 (SI) and L18H6 (Ret) dominate at all shot counts.

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