Spectral contraction of the one-directional token transfer map

Prove that, for every slowdown factor <\alpha<1, the damped transfer map \(M_\alpha=A^{-1}B_\alpha\) associated with a single one-directional token has spectral radius \(\rho(M_\alpha)<1\), thereby establishing convergence \(u\to0\) of the centered collision-abscissa dynamics when \(M_\alpha\) is repeatedly applied.

Background

For the undamped heterogeneous bucket-brigade system, the collision-abscissa update is represented in centered coordinates by $u'=Mu$, where $M$ is a $G$-isometry. This preserves a quadratic energy and prevents convergence to the velocity-proportional equilibrium unless the system starts there.

The one-directional token modifies the first collision relation and yields the transfer map $M_\alpha=A^{-1}B_\alpha$. The paper proves only determinant contraction, namely $|\det M_\alpha|<1$ for $0<\alpha<1$. Determinant contraction alone does not imply that every eigenvalue lies strictly inside the unit disk, so a full spectral contraction argument is needed to establish convergence. The authors leave that proof unresolved and formulate spectral contraction as a conjecture supported by simulations.