arXiv:2611.13584v1 Announce Type: cross Abstract: Momentum methods with the same slow dynamics can have different worst-case rates on smooth strongly convex functions. We identify a half-step phase barrier in the gradient query that governs this difference. For momentum with extrapolation $\tau_\kappa\to\tau$, $\beta=1-\theta/\sqrt\kappa+o(\kappa^{-1/2})$ and $hL\to y$, where $\kappa=L/\mu$ and $\theta,y>0$, the exact uniform distance factor is $1-c/\sqrt\kappa+o(\kappa^{-1/2})$, with $c=\min\{\theta/3,c_Q(\theta,y)\}$, when $\tau>1/2$ and $y(1+2\tau)<4$; $c_Q$ is the slowest quadratic decay exponent. Quadratically accelerated families with $\limsup\tau_\kappa<1/2$ have divergent trajectories on the full function class. Upper bounds use finite-lag certificates: nonnegative combinations of inequalities for values and gradients at finitely separated query points, together with comparisons to the minimizer. For any prescribed accuracy, spiral mixtures or quadratic trajectories of one finite-dimensional objective give matching lower bounds at all iteration horizons. We resolve the parameter layer near $\tau=1/2$ and classify all tunings of this family attaining the optimal leading constant 1. The flow also bounds the leading constant of every second-order stationary linear family with the same slow parameters $(\theta,y)$ and fixed finite memory.