Determine the exact second-order critical threshold

Determine the exact second-order minimax learning allowance threshold at the critical center z_c for achieving assessment risk of order E_k, including whether the critical center is attainable and whether the second-order correction has a universal coefficient or depends on nuisance and reference classes.

Background

The paper establishes that achieving assessment risk O(E_k) requires and, under a specific diagnostic-abstention construction, permits a learning allowance within an O(sqrt(ell_k log ell_k)) window around the corrected center z_c = U ell_k/(2 - ell_k/k). However, the results do not identify the exact minimax threshold within this window.

The unresolved issue concerns both attainability at the critical center itself and the precise second-order correction. Resolving it would sharpen the leading-log frontier into a more exact characterization of the learning–assessment tradeoff.

References

Neither its constants nor simulations of a particular procedure locate the minimax threshold inside that window. In particular, a gate's failure at z_c does not rule out another feasible learner--assessor pair there. The bounds permit a smaller-order correction as well as an order-w_k gap; possible dependence of its coefficient on the nuisance or reference class remains unresolved. The finite-budget checks are reported as procedure diagnostics rather than evidence resolving this open threshold.

— Target-Dependent Limits of Causal Repair: A Leading-Log Frontier in a Gaussian Model  (2610.00424 - Cheng et al., 30 Sep 2026) in Appendix, Section 'Which assumptions support which conclusions?', paragraph 'What the critical-window evidence can establish'; related discussion in Section 1, Introduction