Learning Cross-Model Activation Alignments with Explicit Many-to-Many Layer Maps

Alina Sudakov; Guy Bar-Shalom; Fabrizio Frasca; Haggai Maron

cs.LGOct 6, 2026

Learning Cross-Model Activation Alignments with Explicit Many-to-Many Layer Maps

Authors: Alina Sudakov, Guy Bar-Shalom, Fabrizio Frasca, Haggai Maron

Organizations: Technion · Technion, NVIDIA Research

Abstract

LLMs are released at a rapid pace, raising a natural question: how do two independently trained models relate, both in which layers correspond and in how features transform between them? We study this by learning an activation alignment, a map from a source model's layerwise activations to a target's. Our method, MATCHA, factors this map into a layer map, whose output is an explicit target-by-source matrix that can be extracted and inspected, and a layer-shared feature map between hidden spaces. Most of prior work fixes the layer correspondence in advance, pairing layers at roughly the same relative depth; in contrast, we learn both factors jointly from prompts. Across 42 pairs of seven models spanning three different families, MATCHA reconstructs the target's activations more faithfully and improves retrieval-based metrics substantially, w.r.t. previous approaches. The recovered maps are broadly monotone in depth but, in contrast with most previous approaches, are consistently many-to-many: each target layer draws on a band of source layers. Our alignments also enable transfer of activation-space interventions, allowing steering vectors and probes developed for one model to transfer to another.

Figures & tables

Panel of Figure 1 — Figure 1 : MATCHA aligns paired activations $\mathbf{X}^{s},\mathbf{X}^{s}$ using an (interpretable) layer map, and a feature map. MATCHA can be used for downstream applications.

Panel of Figure 2 — Figure 2 : Layer maps from source ( $L_{s}{=}6$ ) to target ( $L_{t}{=}4$ ), with shallow layers on top. Prior maps are fixed; ours is learned and input-dependent.

Panel of Figure 3 — Figure 3 : Both models encode the same prompt. Highlighted are the last-token representations, $\mathbf{X}^{s}$ and $\mathbf{X}^{t}$ , with $\mathcal{A}^{s\to t}$ mapping $\mathbf{X}^{s}$ to $\mathbf{X}^{t}$ .

Panel of Figure 4 — Figure 4 : Visualization of our alignment map $\mathcal{A}^{s\to t}$ .

	comparable-size				Bigger $\to$ Smaller				Smaller $\to$ Bigger
Method	TJ@5	R@1	P@1	J@5	TJ@5	R@1	P@1	J@5	TJ@5	R@1	P@1	J@5
Terminal	0.449 $\scriptscriptstyle\pm 0.001$	0.716 $\scriptscriptstyle\pm 0.001$	0.915 $\scriptscriptstyle\pm 0.001$	0.376 $\scriptscriptstyle\pm 0.000$	0.407 $\scriptscriptstyle\pm 0.001$	0.759 $\scriptscriptstyle\pm 0.000$	0.874 $\scriptscriptstyle\pm 0.000$	0.381 $\scriptscriptstyle\pm 0.000$	–	–	–	–
Relative Depth	0.447 $\scriptscriptstyle\pm 0.002$	0.727 $\scriptscriptstyle\pm 0.001$	0.917 $\scriptscriptstyle\pm 0.001$	0.377 $\scriptscriptstyle\pm 0.000$	0.411 $\scriptscriptstyle\pm 0.001$	0.757 $\scriptscriptstyle\pm 0.000$	0.918 $\scriptscriptstyle\pm 0.000$	0.395 $\scriptscriptstyle\pm 0.000$	0.435 $\scriptscriptstyle\pm 0.001$	0.686 $\scriptscriptstyle\pm 0.000$	0.856 $\scriptscriptstyle\pm 0.000$	0.381 $\scriptscriptstyle\pm 0.000$
CKA	0.361 $\scriptscriptstyle\pm 0.004$	0.544 $\scriptscriptstyle\pm 0.002$	0.457 $\scriptscriptstyle\pm 0.002$	0.204 $\scriptscriptstyle\pm 0.001$	0.283 $\scriptscriptstyle\pm 0.002$	0.634 $\scriptscriptstyle\pm 0.001$	0.561 $\scriptscriptstyle\pm 0.001$	0.256 $\scriptscriptstyle\pm 0.001$	0.311 $\scriptscriptstyle\pm 0.002$	0.567 $\scriptscriptstyle\pm 0.001$	0.464 $\scriptscriptstyle\pm 0.002$	0.217 $\scriptscriptstyle\pm 0.001$
Command-V	0.316	0.541	0.395	0.321	0.294	0.617	0.517	0.347	0.281	0.627	0.465	0.325
HOT	0.091	0.030	0.343	0.253	0.093	0.041	0.434	0.259	0.101	0.037	0.401	0.263

Table 1: Alignment quality across comparable-size, larger-to-smaller, and smaller-to-larger model pairs. Best results are in bold , and second-best results are underlined. per-pair breakdowns are provided in Section B.3.1 .

Figure 5 : Per-pair Precision@1 difference ( MATCHA $-$ relative depth ).

Figure 6 : The final, learned $\alpha$ obtained by our method.

Panel of Figure 7 — Figure 7 : Transfer of activation-space interventions. Dashed arrows denote source-side operators, which are approximated by the solid path through the target model.

Figure 8 : Steering results. We report per-dataset mean test ASR, averaged over all pairs.

Method	Same-size	Bigger $\to$ Smaller	Smaller $\to$ Bigger
Terminal	0.647 $\scriptscriptstyle\pm 0.006$	0.634 $\scriptscriptstyle\pm 0.004$	–
Relative Depth	0.646 $\scriptscriptstyle\pm 0.007$	0.654 $\scriptscriptstyle\pm 0.004$	0.664 $\scriptscriptstyle\pm 0.002$
CKA	0.549 $\scriptscriptstyle\pm 0.009$	0.593 $\scriptscriptstyle\pm 0.005$	0.597 $\scriptscriptstyle\pm 0.003$
Command-V	0.589	0.605	0.601
HOT	0.642	0.639	0.652
MATCHA (Ours)	0.683 $\scriptscriptstyle\pm 0.007$	0.658 $\scriptscriptstyle\pm 0.007$	0.693 $\scriptscriptstyle\pm 0.004$

Table 2: Probing transfer (TriviaQA test AUC

\uparrow

) by size group — the receiver’s own probe applied to the transferred representation.

Figure 9 : Retrieval performance versus top- $k$ source layers, averaged across all LLM pairs.

Variant	R@1	P@1	J@5	TJ@5
MATCHA-K	0.715 $\scriptscriptstyle\pm 0.002$	0.834 $\scriptscriptstyle\pm 0.001$	0.368 $\scriptscriptstyle\pm 0.001$	0.416 $\scriptscriptstyle\pm 0.001$
MATCHA-FL	0.707 $\scriptscriptstyle\pm 0.002$	0.794 $\scriptscriptstyle\pm 0.002$	0.340 $\scriptscriptstyle\pm 0.001$	0.412 $\scriptscriptstyle\pm 0.001$
MLP	0.460 $\scriptscriptstyle\pm 0.017$	0.817 $\scriptscriptstyle\pm 0.009$	0.318 $\scriptscriptstyle\pm 0.003$	0.364 $\scriptscriptstyle\pm 0.003$
MATCHA (Ours)	0.829 $\scriptscriptstyle\pm 0.001$	0.961 $\scriptscriptstyle\pm 0.001$	0.422 $\scriptscriptstyle\pm 0.001$	0.438 $\scriptscriptstyle\pm 0.001$

Table 3: Architecture ablation . The results are averaged over all 42 LLM pairs.

Appendix figures & tables36 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 10 — Figure 10 : Recall@1 on three samples. Each predicted representation $\hat{\mathbf{X}}^{s}_{i}$ (orange) queries its nearest neighbour in the target space (teal). A sample is a hit (✓) when $\mathbf{X}^{t}_{i}$ is that neighbour, and a miss ( $\times$ ) otherwise.

Panel of Figure 11 — Figure 11 : Precision@1 on three samples. Each target representation $\mathbf{X}^{t}_{i}$ (teal) queries its nearest neighbour in the predicted space (orange). A sample is a hit (✓) when $\hat{\mathbf{X}}^{s}_{i}$ is that neighbour, and a miss ( $\times$ ) otherwise.

Panel of Figure 12 — Figure 12 : Jaccard@1 on three samples. For each hit (✓), $|\cap|{=}|\cup|{=}1$ , giving score 1. For the miss (×), $|\cap|{=}0$ and $|\cup|{=}2$ —yielding score 0. The union penalty distinguishes Jaccard from Overlap when $K{>}1$ .

Hyperparameter	#	Values swept
MATCHA
learning_rate	2	0.0003, 0.001
attention_temperature	4	0.5, 1, 4, 8
bottleneck_dim	3	256, 512, 1024
num_epochs	3	10, 30, 60
MATCHA-K

Appendix

Table 4 : Hyperparameter grid per method.

Target	LR	Temp.	Bottleneck	Epochs	Val loss
Source: Llama-3.2-1B
Llama-3.2-3B	0.0003	1	512	30	0.1736
Llama-3.1-8B	0.0003	1	256	30	0.1450
Qwen2.5-1.5B	0.001	1	1024	30	0.0391
Qwen2.5-3B	0.001	0.5	256	30	0.0426
Qwen2.5-7B	0.001	0.5	512	60	0.0703

Appendix

Table 5: Best MATCHA hyperparameters per pair (min. validation loss).

Target	LR	Temp.	Bottleneck	Epochs	Val loss
Source: Llama-3.2-1B
Llama-3.2-3B	0.001	0.5	1024	60	0.2092
Llama-3.1-8B	0.001	0.5	1024	60	0.1731
Qwen2.5-1.5B	0.001	0.5	1024	60	0.0445
Qwen2.5-3B	0.001	0.5	1024	60	0.0514
Qwen2.5-7B	0.001	0.5	1024	60	0.0879

Appendix

Table 6: Best MATCHA-K hyperparameters per pair (min. validation loss). MATCHA-K replaces the dynamic key with an input-independent one.

Target	LR	Temp.	Bottleneck	Epochs	Val loss
Source: Llama-3.2-1B
Llama-3.2-3B	0.001	0.5	512	30	0.1878
Llama-3.1-8B	0.0003	0.5	512	30	0.1584
Qwen2.5-1.5B	0.001	0.5	512	30	0.0431
Qwen2.5-3B	0.001	0.5	512	30	0.0480
Qwen2.5-7B	0.001	0.5	512	30	0.0778

Appendix

Table 7: Best MATCHA-FL hyperparameters per pair (min. validation loss). MATCHA-FL uses a single fully-learned logit map.

Target	LR	Temp.	Bottleneck	Epochs	Val loss
Source: Llama-3.2-1B
Llama-3.2-3B	0.001	1	512	10	0.2186
Llama-3.1-8B	0.001	0.5	256	10	0.1975
Qwen2.5-1.5B	0.001	8	256	30	0.0696
Qwen2.5-3B	0.001	0.5	256	30	0.0807
Qwen2.5-7B	0.001	0.5	512	30	0.1073

Appendix

Table 8: Best MATCHA (no FFN) hyperparameters per pair (min. validation loss). MATCHA (no FFN) omits the FFN value path; trained from scratch, not derived from MATCHA’s checkpoint.

	Relative depth			Terminal
Target	LR	Ep.	Val	LR	Ep.	Val
Source: Llama-3.2-1B
Llama-3.2-3B	0.0003	30	0.2169	0.0003	30	0.3659
Llama-3.1-8B	0.0003	30	0.2023	0.0003	30	0.3629
Qwen2.5-1.5B	0.001	30	0.0698	0.001	30	0.1403
Qwen2.5-3B	0.001	30	0.0952	0.001	30	0.2019

Appendix

Table 9: Best Relative depth and Terminal hyperparameters per pair (min. validation loss); both share the LR

\times

epochs grid.

Target	LR	Epochs	Val loss
Source: Llama-3.2-1B
Llama-3.2-3B	0.0003	60	0.2725
Llama-3.1-8B	0.0003	30	0.2567
Qwen2.5-1.5B	0.001	30	0.1350
Qwen2.5-3B	0.001	30	0.1276
Qwen2.5-7B	0.0003	60	0.1587

Appendix

Table 10: Best CKA hyperparameters per pair (min. validation loss). CKA shares the fixed-map methods’ grid (learning rate

\times

epochs).

Target	LR	Epochs	Val loss
Source: Llama-3.2-1B
Llama-3.2-3B	0.0003	60	0.1740
Llama-3.1-8B	0.0003	60	0.1448
Qwen2.5-1.5B	0.0003	60	0.0394
Qwen2.5-3B	0.0003	60	0.0428
Qwen2.5-7B	0.0003	60	0.0720

Appendix

Table 11 : Best MLP hyperparameters per pair (min. validation loss). Learning rate and number of epochs are searched (LR

\in\{0.0003,0.001\}

, epochs

\in\{10,30,60\}

); layers and hidden width are fixed (2, 10,000).

Target	$k$ ( routing_top_k )	Val loss
Source: Llama-3.2-1B
Llama-3.2-3B	3	0.1384
Llama-3.1-8B	3	0.1721
Qwen2.5-1.5B	10	5.4011
Qwen2.5-3B	10	6.6036
Qwen2.5-7B	5	5.3680

Appendix

Table 12: Best HOT routing-

k

per pair (min. validation loss);

k

caps how many donor layers each receiver layer routes from, swept over

\{3,5,10\}

Model	Family	Layers	Hidden
Llama-3.2-1B	Llama	16	2048
Gemma-2-2B	Gemma	26	2304
Qwen2.5-1.5B	Qwen	28	1536
Llama-3.2-3B	Llama	28	3072
Qwen2.5-7B	Qwen	28	3584
Llama-3.1-8B	Llama	32	4096

Appendix

Table 13 : The seven instruction-tuned LLMs used, with their features.

Target	Term	Rel-dep	CKA	MATCHA	MATCHA-K	MATCHA-FL	MATCHA-avg	MATCHA-no FF	CmdV	HOT	MLP
Source: Llama-3.2-1B
Llama-3.2-3B	0.453 $\scriptscriptstyle\pm 0.008$	0.455 $\scriptscriptstyle\pm 0.002$	0.327 $\scriptscriptstyle\pm 0.002$	0.455 $\scriptscriptstyle\pm 0.002$	0.427 $\scriptscriptstyle\pm 0.005$	0.411 $\scriptscriptstyle\pm 0.006$	0.457 $\scriptscriptstyle\pm 0.003$	0.433 $\scriptscriptstyle\pm 0.001$	0.431	0.188	0.418 $\scriptscriptstyle\pm 0.003$
Llama-3.1-8B	0.461 $\scriptscriptstyle\pm 0.003$	0.442 $\scriptscriptstyle\pm 0.008$	0.259 $\scriptscriptstyle\pm 0.004$	0.461 $\scriptscriptstyle\pm 0.002$	0.413 $\scriptscriptstyle\pm 0.007$	0.410 $\scriptscriptstyle\pm 0.012$	0.464 $\scriptscriptstyle\pm 0.002$	0.413 $\scriptscriptstyle\pm 0.005$	0.393	0.182	0.400 $\scriptscriptstyle\pm 0.001$
Qwen2.5-1.5B	0.474 $\scriptscriptstyle\pm 0.008$	0.468 $\scriptscriptstyle\pm 0.004$	0.378 $\scriptscriptstyle\pm 0.003$	0.493 $\scriptscriptstyle\pm 0.009$	0.490 $\scriptscriptstyle\pm 0.004$	0.484 $\scriptscriptstyle\pm 0.003$	0.495 $\scriptscriptstyle\pm 0.002$	0.443 $\scriptscriptstyle\pm 0.005$	0.315	0.020	0.466 $\scriptscriptstyle\pm 0.009$
Qwen2.5-3B	0.431 $\scriptscriptstyle\pm 0.007$	0.435 $\scriptscriptstyle\pm 0.001$	0.339 $\scriptscriptstyle\pm 0.006$	0.455 $\scriptscriptstyle\pm 0.002$	0.428 $\scriptscriptstyle\pm 0.002$	0.431 $\scriptscriptstyle\pm 0.007$	0.455 $\scriptscriptstyle\pm 0.002$	0.421 $\scriptscriptstyle\pm 0.009$	0.317	0.052	0.402 $\scriptscriptstyle\pm 0.001$
Qwen2.5-7B	0.436 $\scriptscriptstyle\pm 0.003$	0.435 $\scriptscriptstyle\pm 0.004$	0.322 $\scriptscriptstyle\pm 0.005$	0.428 $\scriptscriptstyle\pm 0.004$	0.422 $\scriptscriptstyle\pm 0.004$	0.425 $\scriptscriptstyle\pm 0.007$	0.434 $\scriptscriptstyle\pm 0.003$	0.412 $\scriptscriptstyle\pm 0.008$	0.297	0.104	0.416 $\scriptscriptstyle\pm 0.008$

Appendix

Table 14 : Token-Jaccard@5 (

\uparrow

): full per-pair results, seed mean

\pm

std. Best bold , second underlined ; grouped by source (donor) LLM.

Target	Term	Rel-dep	CKA	MATCHA	MATCHA-K	MATCHA-FL	MATCHA-avg	MATCHA-no FF	CmdV	HOT	MLP
Source: Llama-3.2-1B
Llama-3.2-3B	0.397 $\scriptscriptstyle\pm 0.001$	0.783 $\scriptscriptstyle\pm 0.001$	0.737 $\scriptscriptstyle\pm 0.002$	0.857 $\scriptscriptstyle\pm 0.003$	0.745 $\scriptscriptstyle\pm 0.019$	0.758 $\scriptscriptstyle\pm 0.005$	0.853 $\scriptscriptstyle\pm 0.002$	0.744 $\scriptscriptstyle\pm 0.009$	0.904	0.063	0.728 $\scriptscriptstyle\pm 0.006$
Llama-3.1-8B	0.338 $\scriptscriptstyle\pm 0.001$	0.800 $\scriptscriptstyle\pm 0.001$	0.735 $\scriptscriptstyle\pm 0.002$	0.862 $\scriptscriptstyle\pm 0.002$	0.777 $\scriptscriptstyle\pm 0.002$	0.779 $\scriptscriptstyle\pm 0.002$	0.863 $\scriptscriptstyle\pm 0.004$	0.730 $\scriptscriptstyle\pm 0.028$	0.898	0.053	0.745 $\scriptscriptstyle\pm 0.008$
Qwen2.5-1.5B	0.445 $\scriptscriptstyle\pm 0.001$	0.612 $\scriptscriptstyle\pm 0.001$	0.498 $\scriptscriptstyle\pm 0.004$	0.762 $\scriptscriptstyle\pm 0.006$	0.735 $\scriptscriptstyle\pm 0.005$	0.679 $\scriptscriptstyle\pm 0.014$	0.770 $\scriptscriptstyle\pm 0.002$	0.593 $\scriptscriptstyle\pm 0.026$	0.500	0.037	0.610 $\scriptscriptstyle\pm 0.007$
Qwen2.5-3B	0.330 $\scriptscriptstyle\pm 0.001$	0.591 $\scriptscriptstyle\pm 0.000$	0.435 $\scriptscriptstyle\pm 0.001$	0.765 $\scriptscriptstyle\pm 0.007$	0.640 $\scriptscriptstyle\pm 0.017$	0.630 $\scriptscriptstyle\pm 0.023$	0.758 $\scriptscriptstyle\pm 0.002$	0.488 $\scriptscriptstyle\pm 0.039$	0.451	0.040	0.554 $\scriptscriptstyle\pm 0.013$
Qwen2.5-7B	0.412 $\scriptscriptstyle\pm 0.001$	0.668 $\scriptscriptstyle\pm 0.001$	0.573 $\scriptscriptstyle\pm 0.003$	0.856 $\scriptscriptstyle\pm 0.003$	0.619 $\scriptscriptstyle\pm 0.025$	0.693 $\scriptscriptstyle\pm 0.009$	0.862 $\scriptscriptstyle\pm 0.002$	0.620 $\scriptscriptstyle\pm 0.012$	0.557	0.046	0.632 $\scriptscriptstyle\pm 0.008$

Appendix

Table 15 : Recall@1 (

\uparrow

): full per-pair results, seed mean

\pm

std. Best bold , second underlined ; grouped by source (donor) LLM.

Target	Term	Rel-dep	CKA	MATCHA	MATCHA-K	MATCHA-FL	MATCHA-avg	MATCHA-no FF	CmdV	HOT	MLP
Source: Llama-3.2-1B
Llama-3.2-3B	0.606 $\scriptscriptstyle\pm 0.002$	0.928 $\scriptscriptstyle\pm 0.002$	0.702 $\scriptscriptstyle\pm 0.006$	0.976 $\scriptscriptstyle\pm 0.001$	0.868 $\scriptscriptstyle\pm 0.010$	0.867 $\scriptscriptstyle\pm 0.009$	0.977 $\scriptscriptstyle\pm 0.001$	0.948 $\scriptscriptstyle\pm 0.005$	0.669	0.702	0.922 $\scriptscriptstyle\pm 0.006$
Llama-3.1-8B	0.594 $\scriptscriptstyle\pm 0.002$	0.948 $\scriptscriptstyle\pm 0.001$	0.651 $\scriptscriptstyle\pm 0.003$	0.989 $\scriptscriptstyle\pm 0.000$	0.896 $\scriptscriptstyle\pm 0.001$	0.853 $\scriptscriptstyle\pm 0.012$	0.989 $\scriptscriptstyle\pm 0.001$	0.960 $\scriptscriptstyle\pm 0.005$	0.608	0.739	0.947 $\scriptscriptstyle\pm 0.005$
Qwen2.5-1.5B	0.611 $\scriptscriptstyle\pm 0.003$	0.748 $\scriptscriptstyle\pm 0.003$	0.358 $\scriptscriptstyle\pm 0.007$	0.937 $\scriptscriptstyle\pm 0.002$	0.825 $\scriptscriptstyle\pm 0.002$	0.761 $\scriptscriptstyle\pm 0.017$	0.943 $\scriptscriptstyle\pm 0.001$	0.746 $\scriptscriptstyle\pm 0.024$	0.336	0.210	0.864 $\scriptscriptstyle\pm 0.008$
Qwen2.5-3B	0.529 $\scriptscriptstyle\pm 0.001$	0.821 $\scriptscriptstyle\pm 0.001$	0.347 $\scriptscriptstyle\pm 0.004$	0.951 $\scriptscriptstyle\pm 0.002$	0.718 $\scriptscriptstyle\pm 0.006$	0.712 $\scriptscriptstyle\pm 0.021$	0.956 $\scriptscriptstyle\pm 0.001$	0.699 $\scriptscriptstyle\pm 0.010$	0.273	0.199	0.846 $\scriptscriptstyle\pm 0.005$
Qwen2.5-7B	0.614 $\scriptscriptstyle\pm 0.002$	0.854 $\scriptscriptstyle\pm 0.003$	0.462 $\scriptscriptstyle\pm 0.005$	0.976 $\scriptscriptstyle\pm 0.002$	0.765 $\scriptscriptstyle\pm 0.018$	0.797 $\scriptscriptstyle\pm 0.013$	0.977 $\scriptscriptstyle\pm 0.001$	0.857 $\scriptscriptstyle\pm 0.023$	0.297	0.372	0.897 $\scriptscriptstyle\pm 0.003$

Appendix

Table 16 : Precision@1 (

\uparrow

): full per-pair results, seed mean

\pm

std. Best bold , second underlined ; grouped by source (donor) LLM.

Target	Term	Rel-dep	CKA	MATCHA	MATCHA-K	MATCHA-FL	MATCHA-avg	MATCHA-no FF	CmdV	HOT	MLP
Source: Llama-3.2-1B
Llama-3.2-3B	0.271 $\scriptscriptstyle\pm 0.000$	0.443 $\scriptscriptstyle\pm 0.001$	0.320 $\scriptscriptstyle\pm 0.002$	0.464 $\scriptscriptstyle\pm 0.001$	0.415 $\scriptscriptstyle\pm 0.005$	0.390 $\scriptscriptstyle\pm 0.003$	0.467 $\scriptscriptstyle\pm 0.000$	0.473 $\scriptscriptstyle\pm 0.006$	0.385	0.324	0.380 $\scriptscriptstyle\pm 0.003$
Llama-3.1-8B	0.251 $\scriptscriptstyle\pm 0.001$	0.435 $\scriptscriptstyle\pm 0.001$	0.306 $\scriptscriptstyle\pm 0.001$	0.455 $\scriptscriptstyle\pm 0.002$	0.414 $\scriptscriptstyle\pm 0.001$	0.371 $\scriptscriptstyle\pm 0.002$	0.460 $\scriptscriptstyle\pm 0.002$	0.460 $\scriptscriptstyle\pm 0.011$	0.369	0.326	0.385 $\scriptscriptstyle\pm 0.004$
Qwen2.5-1.5B	0.299 $\scriptscriptstyle\pm 0.001$	0.364 $\scriptscriptstyle\pm 0.001$	0.177 $\scriptscriptstyle\pm 0.002$	0.380 $\scriptscriptstyle\pm 0.001$	0.345 $\scriptscriptstyle\pm 0.000$	0.312 $\scriptscriptstyle\pm 0.006$	0.384 $\scriptscriptstyle\pm 0.001$	0.341 $\scriptscriptstyle\pm 0.013$	0.292	0.242	0.313 $\scriptscriptstyle\pm 0.002$
Qwen2.5-3B	0.262 $\scriptscriptstyle\pm 0.001$	0.352 $\scriptscriptstyle\pm 0.000$	0.178 $\scriptscriptstyle\pm 0.001$	0.372 $\scriptscriptstyle\pm 0.002$	0.300 $\scriptscriptstyle\pm 0.003$	0.292 $\scriptscriptstyle\pm 0.005$	0.373 $\scriptscriptstyle\pm 0.002$	0.311 $\scriptscriptstyle\pm 0.013$	0.268	0.236	0.298 $\scriptscriptstyle\pm 0.008$
Qwen2.5-7B	0.292 $\scriptscriptstyle\pm 0.002$	0.381 $\scriptscriptstyle\pm 0.000$	0.217 $\scriptscriptstyle\pm 0.001$	0.398 $\scriptscriptstyle\pm 0.002$	0.317 $\scriptscriptstyle\pm 0.007$	0.320 $\scriptscriptstyle\pm 0.003$	0.399 $\scriptscriptstyle\pm 0.001$	0.371 $\scriptscriptstyle\pm 0.008$	0.293	0.261	0.326 $\scriptscriptstyle\pm 0.001$

Appendix

Table 17 : Jaccard@5 (

\uparrow

): full per-pair results, seed mean

\pm

std. Best bold , second underlined ; grouped by source (donor) LLM.

Target	Terminal	Relative depth	CKA	MATCHA	MATCHA-K	MATCHA-FL	MATCHA-avg	MATCHA-no FF	CommandV	HOT
Source: Llama-3.2-1B
Llama-3.2-3B	0.077 $\scriptscriptstyle\pm 0.000$	0.073 $\scriptscriptstyle\pm 0.000$	0.120 $\scriptscriptstyle\pm 0.000$	0.071 $\scriptscriptstyle\pm 0.000$	0.079 $\scriptscriptstyle\pm 0.001$	0.072 $\scriptscriptstyle\pm 0.001$	0.070 $\scriptscriptstyle\pm 0.000$	0.077 $\scriptscriptstyle\pm 0.000$	0.516	0.145
Llama-3.1-8B	0.090 $\scriptscriptstyle\pm 0.000$	0.113 $\scriptscriptstyle\pm 0.000$	0.195 $\scriptscriptstyle\pm 0.002$	0.085 $\scriptscriptstyle\pm 0.000$	0.090 $\scriptscriptstyle\pm 0.000$	0.088 $\scriptscriptstyle\pm 0.001$	0.083 $\scriptscriptstyle\pm 0.000$	0.098 $\scriptscriptstyle\pm 0.000$	0.627	0.180
Qwen2.5-1.5B	0.576 $\scriptscriptstyle\pm 0.000$	0.564 $\scriptscriptstyle\pm 0.001$	1.576 $\scriptscriptstyle\pm 0.020$	0.451 $\scriptscriptstyle\pm 0.002$	0.501 $\scriptscriptstyle\pm 0.002$	0.463 $\scriptscriptstyle\pm 0.003$	0.446 $\scriptscriptstyle\pm 0.002$	0.611 $\scriptscriptstyle\pm 0.021$	16.772	5.424
Qwen2.5-3B	0.932 $\scriptscriptstyle\pm 0.001$	1.481 $\scriptscriptstyle\pm 0.000$	1.655 $\scriptscriptstyle\pm 0.016$	0.724 $\scriptscriptstyle\pm 0.002$	0.797 $\scriptscriptstyle\pm 0.005$	0.738 $\scriptscriptstyle\pm 0.001$	0.718 $\scriptscriptstyle\pm 0.002$	0.953 $\scriptscriptstyle\pm 0.027$	49.502	6.687
Qwen2.5-7B	1.214 $\scriptscriptstyle\pm 0.000$	1.278 $\scriptscriptstyle\pm 0.001$	3.237 $\scriptscriptstyle\pm 0.055$	1.113 $\scriptscriptstyle\pm 0.002$	1.269 $\scriptscriptstyle\pm 0.011$	1.149 $\scriptscriptstyle\pm 0.004$	1.104 $\scriptscriptstyle\pm 0.002$	1.370 $\scriptscriptstyle\pm 0.008$	35.587	5.432

Appendix

Table 18 : MSE loss (

\downarrow

): full per-pair results, seed mean

\pm

std. Lowest bold , second underlined ; grouped by source (donor) LLM.

Target	Term	Rel-dep	CKA	MATCHA	MATCHA-K	MATCHA-FL	MATCHA-avg	MATCHA-no FF	CmdV	HOT	MLP
Source: Llama-3.2-1B
Llama-3.2-3B	–	0.731 $\scriptscriptstyle\pm 0.004$	0.696 $\scriptscriptstyle\pm 0.005$	0.723 $\scriptscriptstyle\pm 0.003$	0.701 $\scriptscriptstyle\pm 0.009$	0.668 $\scriptscriptstyle\pm 0.014$	0.726 $\scriptscriptstyle\pm 0.003$	0.733 $\scriptscriptstyle\pm 0.003$	0.653	0.734	0.696 $\scriptscriptstyle\pm 0.010$
Llama-3.1-8B	–	0.730 $\scriptscriptstyle\pm 0.002$	0.705 $\scriptscriptstyle\pm 0.004$	0.722 $\scriptscriptstyle\pm 0.005$	0.704 $\scriptscriptstyle\pm 0.006$	0.703 $\scriptscriptstyle\pm 0.012$	0.724 $\scriptscriptstyle\pm 0.002$	0.731 $\scriptscriptstyle\pm 0.003$	0.663	0.743	0.693 $\scriptscriptstyle\pm 0.007$
Qwen2.5-1.5B	–	0.662 $\scriptscriptstyle\pm 0.010$	0.648 $\scriptscriptstyle\pm 0.021$	0.671 $\scriptscriptstyle\pm 0.009$	0.669 $\scriptscriptstyle\pm 0.019$	0.656 $\scriptscriptstyle\pm 0.010$	0.668 $\scriptscriptstyle\pm 0.005$	0.644 $\scriptscriptstyle\pm 0.011$	0.641	0.630	0.659 $\scriptscriptstyle\pm 0.009$
Qwen2.5-3B	–	0.662 $\scriptscriptstyle\pm 0.012$	0.607 $\scriptscriptstyle\pm 0.018$	0.691 $\scriptscriptstyle\pm 0.011$	0.635 $\scriptscriptstyle\pm 0.006$	0.620 $\scriptscriptstyle\pm 0.019$	0.683 $\scriptscriptstyle\pm 0.006$	0.659 $\scriptscriptstyle\pm 0.019$	0.645	0.638	0.662 $\scriptscriptstyle\pm 0.017$
Qwen2.5-7B	–	0.679 $\scriptscriptstyle\pm 0.007$	0.638 $\scriptscriptstyle\pm 0.006$	0.680 $\scriptscriptstyle\pm 0.008$	0.619 $\scriptscriptstyle\pm 0.013$	0.653 $\scriptscriptstyle\pm 0.010$	0.663 $\scriptscriptstyle\pm 0.011$	0.642 $\scriptscriptstyle\pm 0.009$	0.602	0.631	0.660 $\scriptscriptstyle\pm 0.014$

Appendix

Table 19 : Probing — best probe by validation (test AUC

\uparrow

): the receiver probe re-selected by validation on the transferred features. Best bold , second underlined ; grouped by source (donor) LLM.

Target	Term	Rel-dep	CKA	MATCHA	MATCHA-K	MATCHA-FL	MATCHA-avg	MATCHA-no FF	CmdV	HOT	MLP
Source: Llama-3.2-1B
Llama-3.2-3B	–	0.700 $\scriptscriptstyle\pm 0.004$	0.696 $\scriptscriptstyle\pm 0.005$	0.723 $\scriptscriptstyle\pm 0.004$	0.683 $\scriptscriptstyle\pm 0.010$	0.646 $\scriptscriptstyle\pm 0.030$	0.716 $\scriptscriptstyle\pm 0.008$	0.728 $\scriptscriptstyle\pm 0.002$	0.592	0.695	0.678 $\scriptscriptstyle\pm 0.011$
Llama-3.1-8B	–	0.705 $\scriptscriptstyle\pm 0.003$	0.663 $\scriptscriptstyle\pm 0.002$	0.715 $\scriptscriptstyle\pm 0.001$	0.692 $\scriptscriptstyle\pm 0.005$	0.677 $\scriptscriptstyle\pm 0.008$	0.714 $\scriptscriptstyle\pm 0.005$	0.716 $\scriptscriptstyle\pm 0.001$	0.607	0.722	0.672 $\scriptscriptstyle\pm 0.005$
Qwen2.5-1.5B	–	0.555 $\scriptscriptstyle\pm 0.005$	0.592 $\scriptscriptstyle\pm 0.007$	0.619 $\scriptscriptstyle\pm 0.013$	0.589 $\scriptscriptstyle\pm 0.029$	0.579 $\scriptscriptstyle\pm 0.052$	0.623 $\scriptscriptstyle\pm 0.003$	0.596 $\scriptscriptstyle\pm 0.028$	0.606	0.545	0.654 $\scriptscriptstyle\pm 0.017$
Qwen2.5-3B	–	0.633 $\scriptscriptstyle\pm 0.004$	0.599 $\scriptscriptstyle\pm 0.016$	0.681 $\scriptscriptstyle\pm 0.012$	0.633 $\scriptscriptstyle\pm 0.020$	0.605 $\scriptscriptstyle\pm 0.031$	0.665 $\scriptscriptstyle\pm 0.012$	0.657 $\scriptscriptstyle\pm 0.012$	0.571	0.598	0.649 $\scriptscriptstyle\pm 0.019$
Qwen2.5-7B	–	0.640 $\scriptscriptstyle\pm 0.018$	0.622 $\scriptscriptstyle\pm 0.003$	0.662 $\scriptscriptstyle\pm 0.009$	0.580 $\scriptscriptstyle\pm 0.018$	0.616 $\scriptscriptstyle\pm 0.013$	0.638 $\scriptscriptstyle\pm 0.012$	0.647 $\scriptscriptstyle\pm 0.008$	0.562	0.547	0.628 $\scriptscriptstyle\pm 0.003$

Appendix

Table 20 : Probing — target’s own probe (test AUC

\uparrow

): the receiver’s globally-best probe applied to the transferred features (transfer-free ceiling). Best bold , second underlined ; grouped by source (donor) LLM.

Target	Term	Rel-dep	CKA	MATCHA	MATCHA-K	MATCHA-FL	MATCHA-avg	MATCHA-no FF	CmdV	HOT	MLP
Source: Llama-3.2-1B
Llama-3.2-3B	–	20	13	13	21	–	–	–	21	19	21
Llama-3.1-8B	–	21	16	18	17	–	–	–	29	18	21
Qwen2.5-1.5B	–	16	18	20	18	–	–	–	21	8	21
Qwen2.5-3B	–	18	27	25	18	–	–	–	27	10	24
Qwen2.5-7B	–	18	5	18	5	–	–	–	21	2	8

Appendix

Table 21 : Winning probe layer (receiver index) per method and pair, i.e. the layer whose test AUC is in Table 19 . Grouped by source (donor) LLM.

Panel of Figure 13 — Figure 13 : Refusal-suppression steering. A direction and native coefficient are selected once for each target model (top). At each decoding step, an unmodified source-model pass provides the complete last-token activation stack. The stack is mapped into target space, edited at the selected target layer, and mapped back both with and without the edit; subtracting the two outputs yields one delta for each source layer (middle). The same token is then processed a second time by the source model, with these deltas added at the corresponding source layers (bottom). The alignment maps are trained beforehand and remain frozen throughout steering.

Figure 14 : Best-by-val transfer, aggregated: mean test ASR per method and dataset (42 pairs), against the no-steering floor and native-adapter ceiling — the same quantity reported in the main results.

Target	Ours	Rel-dep	Term	CKA	CmdV	HOT	MLP
Source: Llama-3.2-1B
Llama-3.2-3B	89.5 $\pm$ 1.1	0.3 $\pm$ 0.0	–	31.5 $\pm$ 4.0	84.0	62.9	80.4 $\pm$ 7.9
Llama-3.1-8B	88.3 $\pm$ 0.4	0.3 $\pm$ 0.0	–	55.0 $\pm$ 21.2	85.1	42.5	82.7 $\pm$ 6.9
Qwen2.5-1.5B	33.8 $\pm$ 5.1	57.6 $\pm$ 3.0	–	67.4 $\pm$ 13.3	72.7	0.5	38.6 $\pm$ 22.8
Qwen2.5-3B	49.1 $\pm$ 7.2	0.3 $\pm$ 0.0	–	34.0 $\pm$ 5.5	33.5	0.5	30.6 $\pm$ 13.3
Qwen2.5-7B	90.5 $\pm$ 0.8	44.2 $\pm$ 12.5	–	35.7 $\pm$ 12.8	77.3	51.3	80.9 $\pm$ 10.1

Appendix

Table 22 : Refusal-steering transfer on AdvBench — best-by-val : per-pair test ASR (%,

\uparrow

), steering coefficient chosen by the highest average held-out validation ASR ( refusal_val ) of the transfer. Each cell is the mean

\pm

std over the pair’s seeds/runs (std blank for the deterministic CommandV/HOT). Methods: Ours

=

our method, Rel-dep

=

Relative depth, Term

=

Terminal, CmdV

=

CommandV. Best method per row in bold , second underlined ; the last two rows count, per column, how many pairs each method won (1st) or came 2nd on. Rows grouped by source (donor) LLM; –

=

run not yet available (and, for Terminal, every smaller

\to

bigger pair, where the terminal map is degenerate).

Target	Ours	Rel-dep	Term	CKA	CmdV	HOT	MLP
Source: Llama-3.2-1B
Llama-3.2-3B	89.1 $\pm$ 1.1	2.3 $\pm$ 0.4	–	49.6 $\pm$ 4.8	87.5	63.0	79.4 $\pm$ 9.0
Llama-3.1-8B	89.2 $\pm$ 1.7	2.4 $\pm$ 0.2	–	65.4 $\pm$ 15.3	82.0	51.0	83.8 $\pm$ 6.4
Qwen2.5-1.5B	37.3 $\pm$ 4.8	68.8 $\pm$ 6.2	–	57.8 $\pm$ 7.4	61.0	3.0	54.0 $\pm$ 12.3
Qwen2.5-3B	51.1 $\pm$ 7.5	2.5 $\pm$ 0.2	–	44.0 $\pm$ 7.2	40.0	3.5	49.1 $\pm$ 11.7
Qwen2.5-7B	89.9 $\pm$ 1.1	61.3 $\pm$ 4.4	–	48.1 $\pm$ 7.4	73.5	65.0	80.4 $\pm$ 11.5

Appendix

Table 23 : Refusal-steering transfer on HarmBench — best-by-val : per-pair test ASR (%,

\uparrow

), steering coefficient chosen by the highest average held-out validation ASR ( refusal_val ) of the transfer. Each cell is the mean

\pm

std over the pair’s seeds/runs (std blank for the deterministic CommandV/HOT). Methods: Ours

=

our method, Rel-dep

=

Relative depth, Term

=

Terminal, CmdV

=

CommandV. Best method per row in bold , second underlined ; the last two rows count, per column, how many pairs each method won (1st) or came 2nd on. Rows grouped by source (donor) LLM; –

=

run not yet available (and, for Terminal, every smaller

\to

bigger pair, where the terminal map is degenerate).

Target	Ours	Rel-dep	Term	CKA	CmdV	HOT	MLP
Source: Llama-3.2-1B
Llama-3.2-3B	83.6 $\pm$ 1.6	1.7 $\pm$ 0.5	–	39.0 $\pm$ 3.8	86.0	55.0	75.2 $\pm$ 9.2
Llama-3.1-8B	84.7 $\pm$ 1.2	1.3 $\pm$ 0.5	–	57.7 $\pm$ 15.6	82.0	45.0	77.9 $\pm$ 5.9
Qwen2.5-1.5B	36.3 $\pm$ 5.5	63.0 $\pm$ 4.3	–	57.3 $\pm$ 11.0	61.0	1.0	44.4 $\pm$ 14.4
Qwen2.5-3B	47.1 $\pm$ 8.8	1.4 $\pm$ 0.5	–	40.1 $\pm$ 6.0	33.0	1.0	40.2 $\pm$ 12.4
Qwen2.5-7B	84.0 $\pm$ 1.3	49.6 $\pm$ 6.6	–	43.3 $\pm$ 7.6	64.0	61.0	74.3 $\pm$ 10.4

Appendix

Table 24 : Refusal-steering transfer on JailbreakBench — best-by-val : per-pair test ASR (%,

\uparrow

), steering coefficient chosen by the highest average held-out validation ASR ( refusal_val ) of the transfer. Each cell is the mean

\pm

std over the pair’s seeds/runs (std blank for the deterministic CommandV/HOT). Methods: Ours

=

our method, Rel-dep

=

Relative depth, Term

=

Terminal, CmdV

=

CommandV. Best method per row in bold , second underlined ; the last two rows count, per column, how many pairs each method won (1st) or came 2nd on. Rows grouped by source (donor) LLM; –

=

run not yet available (and, for Terminal, every smaller

\to

bigger pair, where the terminal map is degenerate).

Target	Ours	Rel-dep	Term	CKA	CmdV	HOT	MLP
Source: Llama-3.2-1B
Llama-3.2-3B	89.5 $\pm$ 1.1	0.3 $\pm$ 0.0	–	42.3 $\pm$ 16.3	82.2	62.9	80.4 $\pm$ 7.9
Llama-3.1-8B	87.0 $\pm$ 1.3	0.3 $\pm$ 0.0	–	40.7 $\pm$ 23.3	76.0	4.1	79.6 $\pm$ 7.9
Qwen2.5-1.5B	15.3 $\pm$ 13.6	38.5 $\pm$ 5.6	–	41.4 $\pm$ 7.7	18.8	0.0	31.1 $\pm$ 18.7
Qwen2.5-3B	17.4 $\pm$ 10.8	0.3 $\pm$ 0.0	–	32.9 $\pm$ 8.5	12.1	0.3	21.9 $\pm$ 13.3
Qwen2.5-7B	19.0 $\pm$ 5.9	44.2 $\pm$ 12.5	–	3.7 $\pm$ 1.5	77.3	0.8	66.8 $\pm$ 13.3

Appendix

Table 25 : Refusal-steering transfer on AdvBench — best-target : per-pair test ASR (%,

\uparrow

), steering coefficient the target model’s own native steering coefficient (chosen on the target’s native validation, no transfer-side tuning). Each cell is the mean

\pm

std over the pair’s seeds/runs (std blank for the deterministic CommandV/HOT). Methods: Ours

=

our method, Rel-dep

=

Relative depth, Term

=

Terminal, CmdV

=

CommandV. Best method per row in bold , second underlined ; the last two rows count, per column, how many pairs each method won (1st) or came 2nd on. Rows grouped by source (donor) LLM; –

=

run not yet available (and, for Terminal, every smaller

\to

bigger pair, where the terminal map is degenerate).

Target	Ours	Rel-dep	Term	CKA	CmdV	HOT	MLP
Source: Llama-3.2-1B
Llama-3.2-3B	89.1 $\pm$ 1.1	2.3 $\pm$ 0.4	–	50.5 $\pm$ 6.6	83.0	63.0	79.4 $\pm$ 9.0
Llama-3.1-8B	84.0 $\pm$ 1.3	2.4 $\pm$ 0.2	–	57.9 $\pm$ 14.9	71.0	12.0	79.9 $\pm$ 6.8
Qwen2.5-1.5B	20.4 $\pm$ 15.1	51.8 $\pm$ 4.6	–	42.4 $\pm$ 6.2	28.5	2.5	43.8 $\pm$ 12.2
Qwen2.5-3B	28.6 $\pm$ 11.4	2.5 $\pm$ 0.2	–	38.7 $\pm$ 8.4	23.5	0.0	35.4 $\pm$ 12.8
Qwen2.5-7B	31.7 $\pm$ 6.1	61.3 $\pm$ 4.4	–	14.4 $\pm$ 3.4	73.5	4.5	65.2 $\pm$ 10.2

Appendix

Table 26 : Refusal-steering transfer on HarmBench — best-target : per-pair test ASR (%,

\uparrow

), steering coefficient the target model’s own native steering coefficient (chosen on the target’s native validation, no transfer-side tuning). Each cell is the mean

\pm

std over the pair’s seeds/runs (std blank for the deterministic CommandV/HOT). Methods: Ours

=

our method, Rel-dep

=

Relative depth, Term

=

Terminal, CmdV

=

CommandV. Best method per row in bold , second underlined ; the last two rows count, per column, how many pairs each method won (1st) or came 2nd on. Rows grouped by source (donor) LLM; –

=

run not yet available (and, for Terminal, every smaller

\to

bigger pair, where the terminal map is degenerate).

Target	Ours	Rel-dep	Term	CKA	CmdV	HOT	MLP
Source: Llama-3.2-1B
Llama-3.2-3B	83.6 $\pm$ 1.6	1.7 $\pm$ 0.5	–	45.3 $\pm$ 11.2	86.0	55.0	75.2 $\pm$ 9.2
Llama-3.1-8B	80.2 $\pm$ 3.2	1.3 $\pm$ 0.5	–	47.7 $\pm$ 15.3	65.0	9.0	74.7 $\pm$ 8.1
Qwen2.5-1.5B	20.0 $\pm$ 14.1	48.9 $\pm$ 6.8	–	40.4 $\pm$ 6.8	25.0	1.0	35.2 $\pm$ 15.9
Qwen2.5-3B	24.7 $\pm$ 15.2	1.4 $\pm$ 0.5	–	40.3 $\pm$ 5.8	12.0	1.0	26.4 $\pm$ 14.7
Qwen2.5-7B	30.0 $\pm$ 7.0	49.6 $\pm$ 6.6	–	9.2 $\pm$ 2.4	64.0	1.0	59.0 $\pm$ 9.2

Appendix

Table 27 : Refusal-steering transfer on JailbreakBench — best-target : per-pair test ASR (%,

\uparrow

), steering coefficient the target model’s own native steering coefficient (chosen on the target’s native validation, no transfer-side tuning). Each cell is the mean

\pm

std over the pair’s seeds/runs (std blank for the deterministic CommandV/HOT). Methods: Ours

=

our method, Rel-dep

=

Relative depth, Term

=

Terminal, CmdV

=

CommandV. Best method per row in bold , second underlined ; the last two rows count, per column, how many pairs each method won (1st) or came 2nd on. Rows grouped by source (donor) LLM; –

=

run not yet available (and, for Terminal, every smaller

\to

bigger pair, where the terminal map is degenerate).

Target	Ours	Rel-dep	Term	CKA	CmdV	HOT	MLP
Source: Llama-3.2-1B
Llama-3.2-3B	-10	-0.5	–	-4	-6	-10	-10
Llama-3.1-8B	-8	-0.5	–	-6	-2	-8	-8
Qwen2.5-1.5B	6	8	–	-8	1.5	-0.5	8
Qwen2.5-3B	-8	-0.5	–	6	2	-0.5	-8
Qwen2.5-7B	-4	-1	–	-6	-1	-8	-8

Appendix

Table 28 : Best-by-val chosen steering coefficient per (pair, method): the coefficient with the highest average validation ASR ( refusal_val ), used for the val table (Table 22 ). Negative coefficients suppress the refusal direction. Rows grouped by source (donor) LLM; – = not yet available (and, for Terminal, every smaller

\to

bigger pair, where the terminal map is degenerate).

Figure 17 : Per-pair Token-Jaccard@5 difference, MATCHA $-$ Relative depth. Axes as in Figure 16 .

Panel of Figure 18 — Figure 18 : Assumption 1 as a condition on the keys. Entry $(j,r)$ of $XW_{k}$ is the score $\langle w_{r},x_{j}(X)\rangle$ that detector $r$ assigns to source layer $j$ . Reading down any of the leading $L_{s}$ columns, the entry at $j=r$ (dark) must lead every other by $\gamma$ , uniformly over $X\in\mathcal{X}$ ; off-diagonal magnitudes and the remaining $d-L_{s}$ columns are unconstrained.

Panel of Figure 19 — Figure 19 : Proposition 2 on a concrete instance: $L_{t}=3$ , $L_{s}=4$ , and $\pi(1)=3$ , $\pi(2)=2$ , $\pi(3)=4$ . Each query is one-hot at $\pi(\ell)$ and vanishes on the last $d-L_{s}$ coordinates, so the product annihilates the unconstrained rows of $(XW_{k})^{\!\top}$ and copies its row $\pi(\ell)$ into row $\ell$ of $S$ (orange). By Assumption 1 each such row is largest at its own layer (dark), so after the softmax every row of $\alpha$ concentrates on $\pi(\ell)$ . With $\gamma=1.0$ and $c=3$ , Lemma 1 gives $\|\alpha-P_{\pi}\|_{1,\infty}\leq 2(L_{s}-1)e^{-3}\approx 0.30$ ; larger $c$ drives $\alpha\to P_{\pi}$ exponentially with $W_{k}$ unchanged. Note that detector $1$ is selected by no target layer, and that $\pi$ need not be injective.

Panel of Figure 20 — Figure 20 : Assumption 2 as a condition on the source stacks. Reading each layer’s activation with each of the $L_{s}$ decoding directions must return exactly one on the matching layer and exactly zero elsewhere — that is, $XU=I_{L_{s}}$ , with the same identity matrix for every input. This is strictly stronger than Assumption 1 ( Figure 18 ), which constrains only the ordering within each column and leaves all magnitudes free.

Panel of Figure 21 — Figure 21 : Proposition 4 on a concrete instance, with $L_{t}=3$ , $L_{s}=4$ and $d=L_{s}$ , for which the construction takes the transparent form $W_{k}=U$ and $Q=\sqrt{d}\,\Lambda$ . Assumption 2 makes the key matrix $XW_{k}=XU$ equal to the identity for every source stack, so the input drops out of the product entirely and the logits are $\Lambda=\log\alpha^{\star}$ regardless of $X$ ; the softmax then returns $\alpha^{\star}$ itself. Unlike the one-to-one case, the target here is soft — no row concentrates on a single source layer — and is reached exactly rather than in the limit of a large logit scale. For general $d\geq\min(L_{t},L_{s})$ the same argument runs with $Q$ and $W_{k}$ obtained from a rank factorization of $\sqrt{d}\,M$ , as in Step 2 of the proof.

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