Exponential inefficiency of restarts

Establish whether maintaining a fixed fraction of the finite-depth ceiling $\delta^*$ requires a restart budget satisfying $\ln B=\Omega(n)$ for the peaked-circuit optimization landscape.

Background

The authors measure the expected best peakedness obtained from restart budgets up to $B=800$ and observe persistent growth with the logarithm of the budget, while the absolute reach decreases with system size.

They formulate C-reach as a conjecture that restarts are exponentially inefficient substitutes for optimization signal. The fraction is left qualitative because the true finite-depth ceiling $\delta^*$ has only been measured directly at $n=8$, so the asymptotic budget requirement remains unresolved.