An Exact Memory Kernel and Pulse-Duration-Dependent Switching in a Three-State Dynamical System

We study an explicitly defined three-dimensional autonomous nonlinear system with a single forced coordinate. Eliminating its two linear hidden coordinates yields an exact scalar Volterra equation with a closed-form memory kernel and a separate contribution from hidden initial conditions. The integral of this kernel gives the critical parameter for a pair of nonzero equilibria. For one frozen, dimensionless parameter set, independently reproduced numerical computations identify the thresholds of rectangular pulses that switch between the two locally asymptotically stable equilibria. Inputs with the same integral but different durations can yield different limiting states. A statically matched one-dimensional bistable control exhibits the same qualitative phenomenon, so unequal pulse-duration outcomes alone do not identify the memory kernel as its unique cause. The note reports the algebra, numerical data, negative control, and executable source without claiming an empirical realization or a new general theory of pulse switching.