BranchReplay: Certifying the Minimum Feedback Depth for Adaptive Measurement Plans

Scientific measurement plans may fix all budgeted measurements in advance for parallel execution or use early responses to choose later measurements. We ask for the fewest response-guided one-measurement stages needed before fixing the remaining budget as one final batch while keeping the best achievable population risk within a prespecified tolerance $\tau$ of the fully sequential optimum. The cited active-acquisition methods learn, optimize, or evaluate adaptive policies but do not certify this depth. BranchReplay uses complete panels containing every candidate response and a separate scoring target. An exact Bellman recursion computes the minimum empirical risk for each depth class without enumerating policy trees. Full-Class certifies the smallest qualifying declared class depth. Frozen-Library selects from a library fixed before independent replay and certifies a policy within $\tau$ of the library minimum at the smallest predeclared worst-case depth among qualifying policies. A replay contract states when complete-panel replay evaluates deployment risk. At a positive-probability history, a necessary-and-sufficient criterion determines when response-guided selection from a finite continuation set lowers conditional expected terminal loss under arbitrary response distributions and within-profile dependence. For 12 binary-response candidates and budget 6, a depth-5 calculation evaluates 259,524 next-measurement or final-batch choices instead of approximately $1.94 \times 10^{56}$ policy-tree representations. In 2,400 prespecified synthetic worlds at $\tau = 0.02$, Frozen-Library returned a policy in 2,103 worlds (87.6%); all returned policies satisfied both certified library properties, and all returned declared depths matched the exact Full-Class population target. At $\tau = 0.02$, six train–selection–test assignments used the 5,376-reaction complete-grid subset of a measured 5,760-reaction Suzuki–Miyaura screen. On the unused test group, the selected policy remained within $\tau$ of the library minimum and retained the smallest qualifying depth in 70 of 84 target-specific evaluations and all six aggregate evaluations. These finite-screen results are not population guarantees. BranchReplay returns a certified minimum class depth or a library policy certified to be within $\tau$ of the library minimum at the smallest qualifying declared depth, and abstains when evidence is insufficient.