Self-Attention as Transport: Limits of Symmetric Spectral Diagnostics

Abstract

Every attention head defines a degree-normalized transport operator, and a growing family of diagnostics reads model behavior (hallucination among them) from its spectrum. We ask what such diagnostics can and cannot infer. The operator splits orthogonally into a symmetric part governing transport \emph{capacity} and an antisymmetric part encoding \emph{orientation}. We prove an identifiability limit: every transpose-invariant spectral diagnostic is \emph{orientation-blind} (unable to distinguish an operator from its transpose, hence blind to the orientation of information flow), with a transpose-stability bound limiting any Lipschitz diagnostic's transpose sensitivity by the asymmetry coefficient $G$. This bounds what spectral diagnostics of the attention operator can resolve (e.g.\ LapEigvals and the attention-spectral branch of LLM-Check). On the surviving axis, a closed-form bipartite-Cheeger landscape shows uniform causal attention obeys an $n$-independent \emph{temporal-cut} floor $\phi \ge 1/5$ while window attention pierces it as $O(w/n)$; the floor is an idealized benchmark, not an empirical attractor, and the fraction of real heads falling below it is itself an empirically stable architectural descriptor. The two-axis diagnostic ($\phi$ for capacity, $G$ for asymmetry magnitude) yields a falsifiable polarity prediction, borne out \emph{in sign} under length-controlled, forced-scoring evaluation across decoder-only, encoder-only, and encoder--decoder models (capacity-axis signal 0.62--0.84 LC-AUROC): polarity reverses between HaluEval and MedHallu, directionally as predicted though asymmetric in strength, with decision polarity calibrated per regime.