Head 3 OV Circuit

The tested hypothesis

A natural first guess was that Head 3 might be a copy circuit: attend to the maximum digit, then boost the matching output digit. The OV matrix lets us test that directly.

Measured verdict

Head 3 does not look like a clean diagonal digit-copy circuit. In token-only OV, only 2 of 10 source digits have positive diagonal margins.

Token-only Head 3 OV heatmap — Token-only OV effects: rows are source digits, columns are output digit logits.

The matrix multiplication

With d_model = 64, n_heads = 4, and d_head = 16, Head 3 uses the final slice of the concatenated attention output.

Value matrix

W_V3.weight: (16, 64)
W_V3.T:      (64, 16)

Output slice

Head 3 slice: 48:64
W_O3:         (16, 64)

Unembedding

unembed.weight:   (14, 64)
unembed.weight.T: (64, 14)

Combined map

OV_logits_3 = W_V3.T @ W_O3 @ unembed.T
(64,16) @ (16,64) @ (64,14) = (64,14)

Three views of the OV effect

Token-only

tok_embed[d] @ OV_logits_3
(10,64) @ (64,14) = (10,14)

Tests whether digit identity alone is copied into output logits.

Position-only

pos_embed[pos] @ OV_logits_3
(5,64) @ (64,14) = (5,14)

Tests whether Head 3 writes position-dependent output biases.

Token + position

(tok_embed[d] + pos_embed[pos]) @ OV_logits_3
(5,10,64) @ (64,14) = (5,10,14)

Matches the real residual vector at number source positions.

Token-only diagonal margins — A positive bar means the matching output digit is stronger than every off-diagonal digit logit for that source digit.

Why this is not a simple copy circuit

In a clean copy circuit, the row argmax would be [0,1,2,3,4,5,6,7,8,9]. Instead, token-only Head 3 gives:

[0, 0, 0, 0, 6, 6, 6, 6, 6, 6]

That means source digits 0-3 most strongly push output digit 0, and source digits 4-9 most strongly push output digit 6, before considering the rest of the model's residual and other heads.

Position does not rescue the copy hypothesis

Adding position embeddings produces almost the same row-argmax pattern at every number position: low digits point to 0, and most higher digits point to 6. This suggests Head 3's role is more subtle than direct answer copying.

Token plus position mean OV heatmap — Token + position effects averaged over five number positions.

Position-only OV heatmap — Position-only effects are similar across the five number positions.

Interpretation

The current evidence says: Head 3 is probably a max selector on the QK side, but not the complete answer-copy circuit on the OV side.

The next useful step is per-component logit attribution at [ANS]. That will show whether Head 3 provides a coarse high-vs-low signal, while another component or the residual stream separates the exact digit.

Diagonal summary CSV Token-only matrix CSV Metadata JSON