100. Debug Decoder I (NumPy)

The mini-decoder forward pass below was working at one point, but a refactor broke something. On the very first forward pass the function crashes with a shape-mismatch error when adding embeddings.

Find and fix the bug(s) so the function correctly maps token ids to logits.

Signature: def decoder_forward(token_ids, tok_emb, pos_emb_flat, W_out, b_out)

• token_ids: list of length T, integer token ids in [0, vocab_size)
• tok_emb: token-embedding weight, shape (vocab_size, d_model)
• pos_emb_flat: positional-embedding weight as a flat list of length max_seq_len * d_model. Your code is responsible for reshaping it correctly before lookup.
• W_out: output projection, shape (d_model, vocab_size)
• b_out: output bias, shape (vocab_size,)

The model uses vocab_size = 8, max_seq_len = 4, d_model = 8. The pipeline is:

1. Look up token + positional embeddings and add them
2. Apply LayerNorm across the model dimension
3. Project to vocabulary logits

Return the logits as a nested list of shape (T, vocab_size).