Restarting Heavy Ball: Sharp Intervals for Momentum Retention

Resetting momentum too frequently can destabilize a convergent heavy-ball method. We characterize restart intervals over the complete scalar class with spectral radius at most $s<1$. Building on the known repeated-root peak effect, we show that full reset is uniformly stable exactly when $s^h[1+h(1+s)]<1$. Retaining a fixed fraction of momentum changes the exact answer: the optimal retention tends to $(3-\sqrt{5})/2$, and we determine the restart interval through its constant-order asymptotic term. Exact rational exclusions and independently checked interval certificates establish exact results at finite margins: at $1-s=0.05,0.02,0.01$, partial retention permits shorter optimal common intervals than full reset. A secondary gradient-only checkpoint construction preserves an accelerated exponent on fixed quadratics. The experiments separate rejected work from delays in checking the stopping target: observed-target stopping removes all gradient-query overhead when the proposal equals the fallback. A delayed-gradient handoff illustrates a distinct use for velocity control: endpoint loss can be small while a subsequent position limit is violated. Controlled optimization experiments find no general speed advantage from joint-energy certification. The main contribution is a sharp restart design rule, with the scope and costs of additional certificates made explicit.