Transient Reserves, Sink Dampers, and the Failure of Eigenvalue Reasoning in the Attention Propagator
The attention matrix of a causal transformer is row-stochastic, iterated over depth, and non-normal by construction. For non-normal operators, eigenvalues control only asymptotic behavior; finite-depth behavior is controlled by resolvent quantities such as pseudospectra and Kreiss constants. We test, under pre-registered criteria, whether this resolvent view predicts anything about trained transformers that eigenvalues miss. Two structural facts organize the analysis: the mask pins the Kreiss constant of every causal stochastic matrix at $\sqrt{n}$, and deflating the mask-forced Perron projector factorizes the depth deviation dynamics exactly into a product of deflated operators. Across GPT-2, Pythia-410m, and Llama-3-8B, learned non-normality proves to be signed. A routing minority carries excess transient reserve that tracks previous-token function and doubles when induction heads engage, while the sink majority is suppressed below matched shuffle nulls, so that attention sinks act as transient dampers. On depth products, eigenvalue predictions of surviving deviations err by seven to eleven orders of magnitude, an error absent in matched nulls. Checkpoint censuses date this organization to a consolidation phase after circuit formation, and a clamping intervention on Llama-3-8B establishes a causal chain from three massive activation dimensions through sink attention to transient damping; LayerNorm models implement the same functions elsewhere. A cross-validated contest concludes that resolvent features are required for depth-transient persistence and routing-head identity, and that no single-operator summary of any kind predicts per-head causal criticality.