In this talk, I will discuss gradient descent for rank-1 matrix factorization at large step sizes. The main idea is to construct a parameterized quadratic certificate $I(\delta;\cdot)$ whose level sets shrink along the discrete-time dynamics, thereby producing a monotone state variable $\delta_t$. This state-dependent Lyapunov perspective gives a geometric mechanism for convergence in the certified regime and explains why, in the post-critical regime, trajectories are driven toward a balanced terminal manifold. I will also describe how these certificates can be derived from structural monotonicity axioms: in the scalar case, the certificate is uniquely determined, and the same local Lagrange-multiplier analysis constrains rank-1 extensions through their signal and noise blocks. Finally, I will present numerical evidence suggesting that the same certificate mechanism may extend beyond the proved settings, including two-dimensional rank-1 approximation and quartic perturbations of scalar factorization.