The discrete-time Kalman conjecture for Lurye systems

Let $G$ be a discrete-time, linear time-invariant plant in negative feedback with a nonlinearity $\phi\in S_{0,k}$. Let $k_{AS}$ be the supremum of the values of $k$ for which the Lurye system is stable for all nonlinearities in $S_{0,k}$, and let $k_N$ be the supremum of the gains $k$ such that the feedback interconnection between $G$ and every linear gain $K\in[0,k]$ is stable. Discrete-time Kalman conjecture. The Lurye system with $G$ and any $\phi\in S_{0,k}$ is stable if and only if $k<k_N$, equivalently $k_N=k_{AS}$. The conjecture asserts that testing stability on linear gains suffices to characterize absolute stability for the full nonlinearity class. The paper constructs periodic counterexamples, so the conjecture is not valid in general; however, the supplied parser does not provide explicit resolution evidence for this candidate.