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arXiv · 2609.31098

The Residual Stream's Effective Depth

Barak Gahtan·Ido Galil·Alex M. Bronstein
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Abstract

We introduce \emph{effective depth} ($\Deff$), a scalar diagnostic that treats the layer-wise residual stream of a transformer as a discrete-time process, measures how representation similarity decays with layer distance, and aggregates that profile into one number. Across sixteen decoder-only language models, $\Deff$ separates a structural consequence of residual accumulation from an empirical one: even maximally diverse orthogonal updates have the closed-form reference $F_L = 2L/(L+1)<2$, yet fifteen of sixteen default measurements lie below $F_L$ (Qwen3.5: 32--44\%, OLMo-2: 40--41\%, Pythia: 23--28\%). Matched references show that the gap is not caused by the persistent initial state or update-size imbalance, but is largely a calibrated signature of correlated residual updates rather than evidence that depth is unused. Symmetric position-0, token-normalisation, and top-PC controls show the regime is not reducible to BOS or top-PC artefacts: the lone above-reference default outlier joins the same regime, and all sixteen models are sub-reference after token-normalisation or top-1-PC removal. Intermediate checkpoints show that the regime is established early in OLMo-2 and stable through 5T tokens, while Pythia-1.4B follows a distinct decreasing trajectory. A controlled residual-carry intervention supports the mechanism, and $\Deff$ is best read as a \emph{global} accumulated-state diagnostic, not as a capability score or pruning method.

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Barak Gahtan, Ido Galil, Alex M. Bronstein. 2026-09-25. The Residual Stream's Effective Depth. https://arxiv.org/abs/2609.31098

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