Sharp Certificates for Worst-Case Controlled Cost Sensitivity

We study exact identification and finite-sample certification of the worst-case controlled contrast $M_i={\sup }_z|{\Delta }_i(z)|$. The access contract is explicit: fresh bounded joint interventions fix every nuisance coordinate, and a valid global Lipschitz constant is supplied externally. For any feasible collection of simultaneous grid-value intervals, McShane–Whitney envelopes give the exact identified interval for $M_i$; at zero trace its unavoidable geometric width is $\min \{L_i{\rho }_i,1\}$. Thresholding its confidence refinement yields family-wise valid sensitive, insensitive, or unresolved reports. For adaptive placement we prove a KL-weighted cap- packing lower bound and a doubling-tree upper bound. On $[-1,1{]}^d$, a converse valid for every $L$-Lipschitz insensitive base, without the lower comparison's strict-slope or interiority assumptions, bounds the packing functional by $2(3/2{)}^{d+2}\max \{2L,1{\}}^d\int (\tau -|\Delta |{)}^{-(d+2)}$. The reverse bound, and hence the two-sided comparison, still require an $(L/16)$-Lipschitz base and interior contrasts; corrected constants improve by factors 8, 16, and 32 for $d=1,2,3$ but remain loose. On the declared synthetic bank, Lipschitz placement remains the strongest equal-cap comparator (1,155 versus 1,037 stagewise decisions at 1,048,576 pairs), and understating $L$ causes false-low reports. A retained $d=2$ execution is reported only as an aggregate theorem audit, not a competitive evaluation. The result certifies controlled sensitivity under this oracle contract; it does not learn $L$, identify causal parents, improve a controller, or establish policy safety. \