Polyhedral selective normal pivot

Assume AC. In the setup of Polyhedral selection event and contrast interval, let $Y\sim N_n(\mu ,{\sigma }^2{I}_n)$, where $\sigma >0$ is known (Multivariate normal law, including singular covariance). Put $\theta ={\eta }^T\mu$, $s=\sigma \mathrm{\parallel }\eta \mathrm{\parallel }$, and let $\mathrm{\Phi }$ be the standard normal CDF from Standard normal and normal laws. Extend it by $\mathrm{\Phi }(-\mathrm{\infty })=0$ and $\mathrm{\Phi }(+\mathrm{\infty })=1$. For a candidate contrast value $\vartheta \in \mathbb{R}$, define ${\mathrm{\Delta }}_{\vartheta }(z)=\mathrm{\Phi }\text{\hspace{0.17em}⁣}\left(\frac{U(z)-\vartheta }{s}\right)-\mathrm{\Phi }\text{\hspace{0.17em}⁣}\left(\frac{L(z)-\vartheta }{s}\right)\mathrm{.}$ When $G(z)$ holds, $L(z)<U(z)$, and ${\mathrm{\Delta }}_{\vartheta }(z)>0$, define $q_{\vartheta }(t,z)=\frac{\mathrm{\Phi }((t-\vartheta )\mathrm{/}s)-\mathrm{\Phi }((L(z)-\vartheta )\mathrm{/}s)}{{\mathrm{\Delta }}_{\vartheta }(z)}(L(z)\le t\le U(z))\mathrm{.}$ For a total statistic on ${\mathbb{R}}^n$, set $q_{\vartheta }(y)=0$ when $y\mathrm{\notin }S$ or when the denominator is zero; on the remaining part of $S$, set $q_{\vartheta }(y)=q_{\vartheta }(T(y),Z(y))$. The value assigned on zero-normalizer fibers is only a measurable extension; when $P_{\mu }(S)>0$, the conditional law given selection gives those fibers probability zero.