The Residual Stream's Effective Depth

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 o...

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