Circuit Condensation: Post-Training that Concentrates a Behavior's Causal Circuit

Sai Adith Senthil Kumar

cs.LGAug 27, 2026

Circuit Condensation: Post-Training that Concentrates a Behavior's Causal Circuit

Authors: Sai Adith Senthil Kumar

Organizations: George Mason University

Abstract

One approach in mechanistic interpretability explains model behavior through circuits: the components and connections that carry it. Frozen discovery often returns hundreds of edges, making them hard to inspect, compare, or verify exhaustively. We introduce Circuit Condensation, which post-trains models to concentrate behaviors into smaller causal graphs. Each round prunes low-attribution edges and trains a low-rank adapter to match the original through what remains. It keeps the cut only if task accuracy and held-out perplexity remain within tolerance. Across four behaviors and eight models, condensed circuits are smaller than the strongest published frozen-weight baseline in 30 of 32 settings, by 8.1x on average and up to 316x. Repeating the search without weight updates produces larger circuits in 29 of 32 settings, showing that weight updates drive the reduction rather than search alone. For 19 small circuits, exhaustive subset tests show that 11 are exactly minimal and identify removable edges in the rest. Pair ablations reveal dependencies that single-edge tests miss. On indirect object identification, condensation isolates 24 heads, including 17 with documented roles, whereas a matched frozen circuit needs 61 heads, 36 without documented roles. The result is a sufficient subcircuit of the published mechanism rather than a reconstruction of it. The resulting circuit tracks the original model's next-token distribution and, at matched edge count, predicts its errors better than a frozen circuit in 10 of 11 evaluable settings.

Figures & tables

Figure 1: One round of Circuit Condensation. Rank surviving edges by causal importance, cut the weakest, then train only the adapter to reproduce the original model through what is left. A round is kept only if the target behavior and perplexity on held-out text remain within tolerance; otherwise the previous state is restored and a smaller cut tried.

C_{1}

is the smallest accepted circuit.

Figure 2: Condensation usually retains fewer edges than frozen discovery at comparable accuracy. Panels are behaviors and rows are models, shaded by family. The axis shows the share of the full head-level graph retained on a log scale, so further right means smaller . Stars show

C_{1}

and other markers show frozen baselines (

C_{0}

runs the same search with the weights fixed); markers are means over seeds 11/22/33 and bars span their range.

Arm	vs $C_{1}$	cells	95% CI
Random	$103\times$	32/32	$[56.8,189.7]$
EAP	$18.4\times$	31/32	$[10.7,32.9]$
EAP-GP	$10.6\times$	30/32	$[6.4,18.1]$
EAP-IG	$8.1\times$	30/32	$[4.9,13.7]$
$C_{0}$	$7.4\times$	29/32	$[4.3,12.5]$

Table 1: Frozen arms against

C_{1}

. Ratios are geometric means (frozen/

C_{1}

) over all 96 seeded runs. Cells reports wins among the 32 task–model combinations at the seed median. CIs resample those combinations; all five comparisons remain significant after Holm correction (Appendix D ).

Figure 3: Three causal checks enabled by smaller circuits. (a) Accuracy after removing

C_{1}

or equally many random edges. (b) Smallest faithful subset (open) and

C_{1}

endpoint (filled), 19 enumerable runs. (c) Share of individually necessary edges; solid markers test every edge, faded ones estimate from 120 sampled edges.

Route	$k\!\leq\!16$ ( $n{=}7{}$ )	$17$ – $200$ ( $n{=}10{}$ )	$>\!200$ ( $n{=}15{}$ )	matches $C_{1}$
$C_{1}$ , re-ranks every round	$0.965$	$0.962$	$0.969$	—
Five-stage cut	$0.450$	$0.780$	$0.977$	21
Single cut	$0.026$	$0.533$	$0.823$	15
Random mask	$0.001$	$0.250$	$0.479$	$0$

Table 2: Shorter pruning paths fail at the smallest

C_{1}

endpoints. Every route targets

C_{1}

’s edge count. Each cut receives up to 2,000 healing steps, four times one controller round. Five-stage re-ranks and heals after five cuts, single cut ranks once and prunes directly, and random keeps an equal-size mask. We score task accuracy only and do not apply the capability gate. Columns report seed-11 medians across 32 cells by endpoint size. Matches counts cells within

0.05

C_{1}

(Appendix F ).

Control	What changes	Cells	Seeds	Outcome	Reference
Shorter path (Tab. 2 )	route to the endpoint	32	1	0/7 match at $k\leq 16$	$C_{1}$ at equal size
Random order	edge ranking	32	3	$50\times$ larger	$C_{1}$ endpoint
Reverse order	edge ranking	32	1	0 edges removed	full graph
No-prune heal	pruning removed	32	3	PR ratio $0.99$	before healing
Target test	behavior condensed	32	3	target $16\times$ , rest $1.0\times$	untouched model
Sibling swap	head identity	32	3	30/32 to chance	intact circuit

Table 3: Controls , in discussion order. Each row changes one ingredient; Reference gives its comparison. Outcomes are medians over runs. The target-test row reports IOI (other targets shrink

2

–

8\times

). PR is the EAP-IG participation ratio after versus before healing (Appendix F ).

Appendix figures & tables22 assets

Supplementary material from the paper’s appendix.

Appendix

Data
Setting	Value
Data per task and model	2000 train / 2000 validation ( eval_big.jsonl ) / 1000 sealed test
Validation read per controller decision	full 2000-example split
Paper seeds	11, 22, 33
Ranking and pruning
Edge-ranking method	EAP-IG, re-ranked after every accepted cut

Appendix

Table 4: Controller and run configuration. Values are shared across all 96 runs and 32 cells unless a control states otherwise.

Task	Runs	Ratios available	Reduction (geom. mean)	$C_{1}$ fewer edges	Max reduction
IOI	24	24	$17.7{\times}$	24/24	$209.2{\times}$
Agreement	24	24	$5.1{\times}$	21/24	$30.6{\times}$
Induction	24	24	$18.2{\times}$	24/24	$316.0{\times}$
Docstring	24	24	$2.7{\times}$	19/24	$35.5{\times}$
Overall	96	96	$8.1{\times}$	88/96	$316.0{\times}$

Appendix

Table 5: Circuit-size reduction relative to frozen EAP-IG, by task. Each task has 24 runs (eight models, three seeds). We select the smallest circuit within

0.05

of each arm’s own full-graph validation accuracy (§ 4 ). Reductions are geometric means of per-run ratios; the maximum is the largest single-run reduction.

Quantity	Ranked on	Selected on	Reported on	Examples
Primary sizes, every arm	train	validation (own full graph $-0.05$ )	sealed test	2000 / 1000
Matched frozen circuit (§ 5.3 )	train	matched to $C_{1}$ test endpoint	sealed test	1000
Subset enumeration	–	circuits fixed above	sealed-test subsample	200
Single-edge ablations	–	circuits fixed above	sealed-test subsample	96
Whole-set ablation, pairs, roles, specificity	–	circuits fixed above	sealed-test subsample	200

Appendix

Table 6: Data used for selection and reporting. One rule determines the primary sizes. The analyses of § 5.3 re-evaluate those circuits on the sealed test split, using a subsample when they require thousands of masked circuits. We verify each sample size from the stored accuracy denominator rather than the configuration.

Task	Runs	Median reduction	Wins	$\geq 3\times$
IOI	24	$14.2{\times}$	24/24	21/24
Agreement	24	$3.8{\times}$	21/24	16/24
Induction	24	$8.7{\times}$	24/24	21/24
Docstring	24	$1.7{\times}$	19/24	8/24
Overall	96	$5.2{\times}$	88/96	66/96

Appendix

Table 7: Edges needed to recover

90\%

of full-model accuracy (32 cells, three seeds). We compare

C_{1}

with the smallest frozen alternative among EAP, EAP-IG, random ranking, ACDC, and

C_{0}

. ACDC covers 8 cells on the two smallest graphs, where runs take

3.4

–

5.9

hours; we do not run it on larger graphs. It uses the attribution arms’ answer-slot KL metric.

C_{0}

is best in 78 of 96 runs and ACDC in one. Ratios above

1\times

favor condensation.

C_{1}

loses eight runs, all on agreement or docstring, where the capability gate stops it early (§ 6 ).

Task	Median KL	KL IQR	Median circuit acc.	Median original acc.	Sealed pass
IOI	0.048	[0.045, 0.063]	0.978	0.996	24/24
Induction	0.081	[0.060, 0.124]	0.951	0.965	19/24
Docstring	0.089	[0.074, 0.138]	0.994	1.000	24/24
Agreement	0.254	[0.145, 0.343]	0.777	0.764	24/24
All	0.088	–	–	–	91/96

Appendix

Table 8: Sealed-test faithfulness across the head grid (96 runs, medians with interquartile ranges). Agreement is the only task where circuit accuracy exceeds original-model accuracy at the median (

0.777

versus

0.764

). Interchange ablation can sharpen the margin by removing distractor pathways outside the circuit, which also explains logit-difference recoveries above

1

Seed	Median ratio	Worst ratio	Median KL	Median $\Delta$ LAMBADA	Cells $>1.05$
11	1.088	1.672	0.278	$-0.068$	19/32
22	1.091	1.742	0.255	$-0.057$	19/31
33	1.087	1.621	0.262	$-0.052$	18/32

Appendix

Table 9: General-capability retention of

C_{1}

. We report the WikiText-2 perplexity ratio (adapted/original), per-token KL from the original, and change in LAMBADA last-token accuracy. The mixed-domain gate corpus is separate from both datasets. Values are medians across 32 cells for each seed. Seed 22 has 31 cells, giving 95 of 96 runs in total.

Task	Model	EAP-IG (ours)	Random	Anti-ranked
		edges @ acc	edges @ acc	edges @ acc
IOI	GPT-2	58 @ 0.94	1484 @ 0.94	2041 @ 0.98 (full)
Agreement	Llama-3.2-1B	263 @ 0.71	7644 @ 0.60	8993 @ 0.70 (full)
Induction	Gemma-3-1B	3 @ 0.90	1197 @ 0.92	3537 @ 0.89 (full)
Docstring	Llama-3.2-1B	519 @ 1.00	7128 @ 1.00	8993 @ 1.00 (full)

Appendix

Table 10: Ranking controls. Only the edge order changes. Values are seed medians except for deterministic anti-ranking, which retains the full graph. Random and anti-ranked accuracies are measured on validation, while EAP-IG accuracy is measured on the sealed test split. These controls do not use a

C_{1}

endpoint.

Task	Model	PR before	PR after (no-prune heal)	Ratio
IOI	GPT-2	50.7	50.5	1.00
IOI	Llama-3.2-1B	331.0	328.3	0.99
IOI	Qwen3-4B	1017.7	876.1	0.86
Agreement	Llama-3.2-1B	267.9	261.7	0.98
Agreement	Llama-3.2-3B	479.3	475.2	0.99
Induction	Gemma-3-1B	47.1	46.7	0.99

Appendix

Table 11: Healing-only control at head granularity. Representative cells are shown, with EAP-IG participation ratios averaged over seeds 11/22/33. Higher PR means attribution is spread across more edges. The grid-wide median ratio is

0.99

across 32 cells. Runs start from fresh checkpoints rather than a

C_{1}

endpoint.

Task	Model	Edges	Swapped	Kept	All off	Chance	Swapped acc.	To chance
IOI	Gemma-3-4B	4	2	1.000	0.555	0.50	0.550	yes
	Llama-3.2-3B	10	3	0.985	0.510	0.50	0.490	yes
	Gemma-3-1B	12	5	0.985	0.510	0.50	0.435	yes
	Llama-3.2-1B	18	7	0.980	0.515	0.50	0.580	no
	GPT-2	58	17	0.970	0.485	0.50	0.475	yes
	Qwen3-1.7B	94	23	0.975	0.510	0.50	0.560	yes

Appendix

Table 12: Sibling-swap control (head granularity, all 32 cells). Values are three-seed medians on 200 examples. Swapped counts re-pointed head

\to

logits edges. All off and Kept give accuracy with every edge closed and with the circuit intact. Chance is

0.50

for the two-choice tasks,

0.25

for docstring, and the reciprocal of the induction pool size. To chance marks swapped circuits no better than guessing.

Task	Model	W3 edges	$C_{1}$ edges	$C_{1}$ /W3	Admissible configs
IOI	GPT-2	5	58	11.6 $\times$	2 of 16
	Llama-3.2-1B	22	18	0.8 $\times$	2 of 5
	Llama-3.2-3B	none	10	—	0 of 5
	Gemma-3-1B	3	12	4.0 $\times$	3 of 5
	Gemma-3-4B	3	4	1.3 $\times$	4 of 5
	Qwen3-0.6B	none	169	—	0 of 5

Appendix

Table 13: Joint gates and adapter across all 32 cells.

C_{1}

/W3 above

1

means the baseline returns fewer edges; below

1

, the controller does. Dashes mean no tested configuration passes the accuracy and capability criteria. The final column gives admissible configurations out of those scored.

Task	Containment	Edge Jaccard	$\times$ null
IOI	0.93	0.53	$205\times$
Agreement	0.89	0.68	$59\times$
Induction	0.95	0.83	$243\times$
Docstring	0.94	0.76	$45\times$

Appendix

Table 14: Seed stability of

C_{1}

across all 32 cells (head granularity, seeds 11/22/33). Values are per-task medians over seed pairs.

\times

null is the observed edge Jaccard divided by its expectation for independent random subsets of the same sizes. High containment and above-null Jaccard indicate consistent, often nested circuits across seeds.

Measured on an earlier 70-edge IOI/GPT-2 endpoint; the reported endpoint is 58 edges (Table 20 ), so this understates the saving.
Candidate set	Condensed	Original	Passes	Time	Speedup
Head edges	70	256	1280 $\to$ 350	137.02s $\to$ 37.66s	$3.6{\times}$

Appendix

Table 15: Exact-patching cost on IOI/GPT-2. Forward-pass and wall-clock reductions after restricting candidates to the condensed graph.

Aligned variable	$C_{1}$ condensed	Frozen (matched acc.)	Unmodified model
Output (name-mover $\rightarrow$ logits)	0.995	0.985	0.995
Intermediate (subject inhibition)	0.930	0.735	0.645
per-seed range	$0.910$ – $0.945$	$0.715$ – $0.740$	$0.640$ – $0.650$
Random-edge control (intermediate)	0.03	0.04	0.01
Sham hypothesis (different task)	0.00	0.01	0.01

Appendix

Table 16: Interchange intervention accuracy under Wang et al.’s IOI algorithm (GPT-2, three seeds, 200 prompt pairs each). Every arm uses the same hypothesis and edge indices. We assign the output variable to named name movers and the intermediate variable to subject-inhibition heads feeding them. Higher accuracy means the intervention more often produces the predicted answer. “Random” uses unnamed heads; “Sham” tests a task without a published role map. The coarse all-in-circuit variant saturates and serves only as a control.

Task	Model	Edges	Acc.	Match( $M$ )	$\mathrm{KL}(M\\|\hat{C})$	$\mathrm{KL}(M^{\prime}\\|\hat{C})$	LD rec.
IOI	GPT-2	13	0.96	0.95	0.079	0.025	0.58
	Llama-3.2-1B	15	0.98	0.99	0.217	0.047	0.95
	Llama-3.2-3B	12	0.955	0.95	0.067	0.040	0.58
	Gemma-3-1B	4	0.995	0.99	0.118	0.046	1.23
	Gemma-3-4B	4	0.995	0.99	0.072	0.043	0.91
	Qwen3-0.6B	71	0.98	0.96	0.157	0.030	1.03

Appendix

Table 17: Original-model faithfulness and logit-difference recovery. Results combine the block-level replication with the head-level IOI anchor. Match(

M

) is the fraction of sealed-test examples where the circuit and original model select the same candidate.

\mathrm{KL}(M\|\hat{C})

and

\mathrm{KL}(M^{\prime}\|\hat{C})

compare the circuit with the original and adapted full models. LD recovery is

(m(\text{circuit})-m(\text{ablated}))/(m(\text{full})-m(\text{ablated}))

, where

1

reproduces the full model’s margin. The head-level IOI row gives three-seed medians and has no LD value because that evaluation exists only at block granularity.

Model	Err. rate	$k$	$C_{1}$ (matched)	Frozen (matched)	Frozen (full)
Agreement
GPT-2	$19\%$	453	0.983	0.985	0.992 (600e)
Llama-3.2-1B	$35\%$	263	0.943	0.927	0.965 (1400e)
Llama-3.2-3B	$34\%$	314	0.936	0.635	1.000 (20329e)
Gemma-3-1B	$22\%$	125	0.829	0.639	0.931 (600e)
Gemma-3-4B	$37\%$	180	0.823	0.675	0.981 (2000e)

Appendix

Table 18: Predicting original-model errors in 11 cells. AUROC uses each circuit’s restricted-choice margin to rank the original model’s errors. Values are per-cell medians over three seeds. The

C_{1}

and frozen columns use the same edge budget

k

; the final column uses the full frozen circuit. Agreement/Qwen3-0.6B ties at the shown precision, with

C_{1}

ahead by less than

0.001

. The original model’s own margin defines the error labels and would score perfectly. IOI and docstring lack enough suitable errors or a fixed contrast, respectively.

Figure 4: Fixed-budget IOI legibility. GPT-2 accuracy after retaining each arm’s top EAP-IG-ranked heads, median over three seeds.

	$C_{1}$	Frozen
Edges	58	256
Heads	24	61
KL to original	0.056	0.50
No published role	7	36
Precision	0.71	0.41
Recall	$0.65$	$0.96$

Appendix

Table 19: Condensed and frozen IOI endpoints (GPT-2, matched accuracy

0.950

). Values are medians over seeds 11/22/33. Recall is the share of Wang’s 26 heads in a circuit, and precision is the share of its heads with a published role. For both arms, KL is

\mathrm{KL}(\text{orig}\,\|\,\text{circuit})

against the unmodified model. Since entries are per-seed medians, the reported

F_{1}

need not equal the value computed from median precision and recall. The frozen circuit is re-matched for each seed, giving its

256

–

400

edge range.

		IOI		Agreement		Induction		Docstring
Model	Full	Edges	Acc	Edges	Acc	Edges	Acc	Edges	Acc
GPT-2 small	2041	58 (54–71)	0.94	453 (317–595)	0.79	160 (148–160)	0.91	144 (118–175)	0.44
Llama-3.2-1B	8993	18 (17–27)	0.97	263 (205–279)	0.71	104 (104–119)	0.97	519 (480–519)	1.00
Llama-3.2-3B	20329	10 (8–267)	0.98	314 (254–316)	0.72	91 (77–97)	0.97	697 (545–720)	1.00
Gemma-3-1B	3537	12 (4–16)	0.99	125 (51–129)	0.83	3 (3–3)	0.90	3 (3–4)	0.97
Gemma-3-4B	10745	4 (3–6)	1.00	180 (176–226)	0.71	4 (4–6)	0.90	7 (6–7)	0.98

Appendix

Table 20: Condensed endpoints at head granularity, all 32 cells. Median edges over seeds 11/22/33 with the seed range beneath, and median accuracy on the sealed 1000-example test split. “Full” is the complete head-level graph for that model.

Figure 5: Condensed circuit size against full graph size. One point per run (93 total; low-accuracy docstring/GPT-2 is excluded). On log–log axes, the solid line fits the non-Gemma runs and the dashed line shows proportional growth. Hollow Gemma points are excluded from the fit because several endpoints have only 3–4 edges.

Task	$C_{1}$	random edges	$C_{1}$	frozen	subset
	Accuracy after cutting		Load-bearing alone		No smaller
IOI	$0.52$	$0.98$	$15\%$	$0\%$	2 of 7
Agreement	$0.22$	$0.62$	$4\%$	$0\%$	none enumerable
Induction	$0.00$	$0.96$	$12\%$	$0\%$	6 of 6
Docstring	$0.00$	$0.97$	$3\%$	$0\%$	3 of 6

Appendix

Table 21: Verification checks by task (§ 5.3 ). Values are medians over eight models and three seeds. Accuracy after cutting compares removal of all

C_{1}

edges with an equal-size random removal. Load-bearing alone is the share of edges that matter individually. No smaller subset counts enumerable cells with no smaller faithful subset among all

2^{k}

possibilities.

Model	Adapted matrices	LoRA params	% of base	Top-quartile share
GPT-2 small	48	2.36M	1.90%	37.6%
Llama-3.2-1B	112	11.27M	0.91%	50.0%
Gemma-3-1B	182	13.05M	1.30%	47.9%
Qwen3-1.7B	196	17.43M	1.01%	61.3%

Appendix

Table 22: LoRA accounting for one representative endpoint per family (IOI, seed 22). “Top-quartile share” is the fraction of total squared update norm carried by the top quarter of layers.

Arm	Cells	PR ratio	Concentrates	Task acc.	Probe
SFT	$32$	$0.76$	$23/32$	$0.99\to 1.00$	$1.37\times$
DPO ( Rafailov et al., 2023 )	$16$	$0.51$	$15/16$	$0.89\to 1.00$	$12.9\times$
GRPO ( Shao et al., 2024 )	$10$	$0.72$	$10/10$	$0.76\to 0.82$	$1.0\times$
REINFORCE ( Williams, 1992 )	$10$	$1.88$	$3/10$	$0.76\to 0.49$	$6{,}336\times$

Appendix

Table 23: Post-training arms. Ratios are after/before, with values below

1

indicating concentration. Probe is general-corpus perplexity relative to the original model, and the controller’s limit is

1.05

. DPO covers fewer cells because induction and docstring do not provide its required contrast answer.

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