בדיקה שלילית של פונקציית ליאפונוב לפקטוריזציה של מטריצה דרגה 1

תקציר מקורי באנגליתarXiv:2604.26993v3 Announce Type: replace-cross Abstract: We develop a state-dependent Lyapunov framework for gradient descent on rank-1 matrix factorization. A parameterized quadratic certificate $I(\delta;\,\cdot)$ generates strictly nested sublevel sets. Their ordering assigns each point a state $\delta$, while a boundary-inward property makes this state monotone along gradient-descent trajectories. Together with internal chain transitivity, this geometry identifies the limiting dynamics even when all relevant stationary points are unstable. For scalar and rank-1 factorization, it yields convergence to a global minimizer below the stability threshold and to a balanced period-$2$ orbit in a post-critical interval, for almost every initialization in an explicit region. We formalize the me