Finite-budget optimization of the five-site path

Determine the remaining two-variable finite-budget optimization for the five-site path under a fixed contrast cap, including the active-constraint structure and the minimization of the cooperation threshold over the admissible positive edge weights.

Background

The paper derives the sharp weak-selection threshold infimum for the five-site path and characterizes promoter rays through an outer-ray analysis. Under a contrast cap, this reduces the search to a constrained family involving the outer weight A and the largest weight Q, with the inner ratio q subject to promoter conditions.

The asymptotic contrast cost is established as the threshold approaches seven, but the exact finite-budget optimum at arbitrary contrast is not resolved. Solving it would determine how the optimal finite five-site design interpolates between the asymptotic resource regime and moderate contrast.