Theorem: Conditional sensitization and unavoidable failure

For (eq:affine),

For $q_{\ast }\in (0,1)$,

If $d_t-u_t$ is nondecreasing, ${\beta }_t^{\ast }$ is nonincreasing. It decreases strictly when the untruncated numerator decreases and both compared critical values are positive.

Every permitted compensation fails at zero bias if

Relative to the initial $d_0=u_0=0$ reference at the same challenge, accuracy is strictly lower exactly when $d_t>u_t$.

Proof. Evaluate the Gaussian probability of a positive margin. Solve $Q_t(\beta )<q_{\ast }$ for $\beta \ge 0$ and take the failure-set infimum. The monotonicity statements follow from the numerator. The largest admissible zero-bias margin uses $u_t=U$, giving (eq:unavoidable). Strict monotonicity of ${\Phi }_{\mathrm{G}}$ proves the final comparison.