Md. Asif Uddin

Proposition 331 of 39 in the corpus

Layers write into a shared stream rather than replacing it.

Every sublayer adds its output to the running representation. This is why gradients reach the first layer, and why a state from partway up the stack can be read with the output head.

Depends on

The residual streamA single horizontal line running the width of the figure, from the embedding to the output head. Five blocks sit beside it; each reads from the line and adds its result back onto it rather than replacing it.the residual streamembedunembedblock 1readwriteblock 2readwriteblock 3readwriteblock 4readwriteblock 5readwriteBecause every block adds, the gradient reaches layer one along a path of additions, not a product.And because every block writes into the same space, a vector taken partway up the stack can be readwith the output head — which is what makes a transformer legible from the inside at all.
Fig. 3 — The residual stream. Blocks read from it and add back into it rather than replacing it, which is both why gradients reach layer one and why intermediate states can be read.

Demonstration

The addition in x ← x + f(Norm(x)) looks like a minor architectural convenience. It is the load-bearing element of the whole design, for two separate reasons.

The gradient has an additive path. Differentiate: ∂xout/∂x = I + ∂f/∂x. The identity term means the gradient arriving at a layer is passed back undiminished, plus a correction. Compare this with Chapter IV, Proposition 2, where the path was a product of Jacobians and decayed exponentially with depth. Here the path is a sum, and ninety-six layers is not qualitatively harder than twelve. Residual connections are why deep stacks train at all.

The stream is a single shared space. Because every block reads from and adds to the same vector, all layers communicate in one basis. Nothing overwrites; the representation at layer k is the embedding plus the sum of every sublayer output so far. This makes a transformer unusually legible from the inside: the logit lens takes the stream at an intermediate layer, applies the final normalisation and the unembedding, and reads out what the model would predict if it stopped there. That this produces sensible, progressively refining distributions is a direct consequence of the additive structure, and it would be meaningless in an architecture where each layer replaced its input.

The circuits framing follows from the same fact. Heads in early layers write information into the stream that heads in later layers read out, forming compositions that can be traced. The stream is the communication channel, and its width — d — is the total bandwidth every component competes for.

Corollary

The width d is a shared resource, not a per-layer one. Every head and every MLP in every layer reads from and writes to the same d dimensions, which is one explanation for superposition: with far more features to represent than dimensions to represent them in, the model packs them into overlapping directions rather than dedicating one each.

Sources

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