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.

Explore similar work

LLMs Learn Better In-Context from Rules than from Examples
Sep 2, 2026Xiang Fu, Seungmin Cho, Yukyung Lee +1In-Context Learning In-Context
Boston University
Test, then Route: How Language Models Execute In-Context Conditional Rules Across Models and Languages
Aug 4, 2026Luxshan Thavarasa, Sivasuthan SukumarFlips Switch
Independent Researcher, Colombo, Sri Lanka · Department of Computer Science and Engineering, University of Moratuwa, Sri Lanka
Compile, Don't Memorize: A Context Compilation Architecture (CCA) for In-Context Learning
Sep 1, 2026Jinhu Qi, Minda Hu, Wentao Zhang +4In-Context Learning Skills
The Chinese University of Hong Kong · Macao Polytechnic University · Xi’an Jiaotong University +1

Sep 2, 2026, cs.CLJ/K move · Enter open · S save

LLMs Learn Better In-Context from Rules than from Examples

Xiang Fu, Seungmin Cho, Yukyung Lee, Najoung Kim

Boston University

In-Context Learning In-Context Instruction-Tuned Models Learn Instruction Tuning

Open paper AskPDF arXiv

Large language models (LLMs) exhibit in-context learning capabilities, where they can learn new tasks from prompt contexts without weight updates. We compare the learning efficacies of two prominent modes of in-context learning: (1) learning from descriptions of rules (instruction following); and (2) learning from examples of input-output demonstrations (few-shot prompting). Through five learning tasks that cover diverse domains (games, arithmetic, linguistic inferences), we compare two modes of learning (rules vs. examples) specifying the same underlying task. We furthermore explore model and task properties that modulate the learning efficacies. We find that models generally learn more reliably from rules than from examples alone, and additional examples on top of rules or simply scaling up the number of examples do not lead to consistent and significant gains. Instruction tuning amplifies the benefit of rule-based learning while keeping example-based learning capacities intact. Surprisingly, we find no privileged effect of example-based learning in base models, and rules still lead to gains in algebraic task domains. Overall, the comparative efficacy of rules over examples is larger when the task recruits algebraic abstractions and computations, and smaller when the task requires distributional sensitivity and/or recruits parametric knowledge.

Abstract

Explore similar work

LLMs Learn Better In-Context from Rules than from Examples

Test, then Route: How Language Models Execute In-Context Conditional Rules Across Models and Languages

Compile, Don't Memorize: A Context Compilation Architecture (CCA) for In-Context Learning