One-cluster convergence conjecture for causal attention with a multidimensional dominant eigenspace

Let $Q$ and $K$ be arbitrary matrices, and let $V$ be a matrix whose largest eigenvalue $\lambda_{\max}>0$ is real, with eigenspace $L$ satisfying $\mathop{\rm \dim} L\geq2$. Assume that $Vz\in L^{\perp}$ and $\langle Vz,z\rangle<\lambda_{\max}|z|^2$ for every $z\in L^{\perp}$. For an initialization $(x_1(0),\ldots,x_n(0))$, let $\mathbf{P}_{L^{\perp}}$ denote orthogonal projection onto $L^{\perp}$, set $y_1:=\mathbf{P}_{L^{\perp}}(x_1(0))$, and define $\xi:=y_1/|y_1|$. One-cluster convergence conjecture. For almost every initialization, the causal attention dynamics converge to one cluster, specifically

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

This is the proposed description of the positive-dominant-eigenvalue case when the dominant eigenspace has dimension at least two. The paper presents it as a conjectural generalization because critical manifolds make the corresponding stable-manifold analysis technically difficult; no resolution is given.