Two-cluster convergence conjecture for causal transformer dynamics with simple dominant eigendirection

Let $Q$ and $K$ be arbitrary matrices, and let $V$ be diagonalizable with $d$ different positive real eigenvalues. Denote its largest eigenvalue by $\lambda_{\max}$, and let $\xi$ be a unit eigenvector satisfying

$$V\xi=\lambda_{\max}\xi.$$

For the causal transformer dynamics, two-cluster convergence conjecture. For almost every starting point $(x_1(0),\ldots,x_n(0))$ with respect to the volume measure on $(\mathbb{S}^{d-1})^n$,

$$\forall k\in[n],\qquad \lim_{t\to\infty}x_k(t)\in\\{\xi,-\xi\\}.$$

This conjectures that, when the dominant eigenvalue is positive and has a one-dimensional eigendirection under the stated diagonalizability assumptions, causal attention generically drives all tokens into two antipodal clusters. The claim is presented as part of the paper's conjectural extension beyond the proved identity-matrix case; its resolution is not supplied here.