The Origin of Edge of Stability

Full-batch gradient descent on neural networks drives the largest Hessian eigenvalue to the threshold $2/\eta$, where $\eta$ is the learning rate. This phenomenon, the Edge of Stability, has resisted a unified explanation: existing accounts establish self-regulation near the edge but do not explain why the trajectory is forced toward $2/\eta$ from arbitrary initialization. We introduce the edge coupling, a functional on consecutive iterate pairs whose coefficient is uniquely fixed by the gradient-descent update. Differencing its criticality condition yields a step recurrence with stability boundary $2/\eta$, and a second-order expansion yields a loss-change formula whose telescoping sum forces curvature toward $2/\eta$. The two formulas involve different Hessian averages, but the mean value theorem localizes each to the true Hessian at an interior point of the step segment, yielding exact forcing of the Hessian eigenvalue with no gap. Setting both gradients of the edge coupling to zero classifies fixed points and period-two orbits; near a fixed point, the problem reduces to a function of the half-amplitude alone, which determines which directions support period-two orbits and on which side of the critical learning rate they appear.