Calibrated selected sample size and an uncalibrated weight change

Put $\mathrm{\Omega }=\{0,1{\}}^4$ with the uniform law and let $I,B_0,B_2,B_3$ be the independent fair-bit coordinates. Define $p_j=1\mathrm{/}2$ when $B_j=0$ and $p_j=1$ when $B_j=1$, for $j\in \{0,2,3\}$. Each $p_j$ is measurable with values in $[0,1]$, and its null CDF is $0$ for $u<1\mathrm{/}2$, $1\mathrm{/}2$ for $1\mathrm{/}2\le u<1$, and $1$ at $u=1$, so each is a valid p-value by [F3]. Let ${\mathcal{F}}_1=\sigma (I)$, select $N=2$ on $I=0$ and $N=3$ on $I=1$, and take the original rejection event $R_0=\{p_0\le 1\mathrm{/}2\}$. Independence and [A1, F4] give its conditional error $A=E_0[{\mathbf{1}}_{R_0}\mid {\mathcal{F}}_1]=1\mathrm{/}2$ almost surely; [F5] makes the resulting event probabilities version-independent. For each $m\in \{2,3\}$ and $H\in {\mathcal{F}}_1$, independence of $B_m$ from $I$ gives $P_0(H\cap \{N=m\}\cap \{p_N\le u\})=c(u)P_0(H\cap \{N=m\})$, where $c(u)=0$ for $u<1\mathrm{/}2$, $1\mathrm{/}2$ for $1\mathrm{/}2\le u<1$, and $1$ for $u=1$; in every case $c(u)\le u$. Thus [F1] makes $p_N$ conditionally superuniform. Rejection at $A=1\mathrm{/}2$ is exactly the event that the selected bit is zero. Since $P_0(R_0)=P_0(B_0=0)=1\mathrm{/}2$, set $\alpha =1\mathrm{/}2$ in the overall conclusion of [F2] to obtain $P_0(p_N\le A)\le 1\mathrm{/}2$; the selected bit is fair, so equality holds and the original budget is attained.

The positive density makes $\mathrm{\Phi }$ continuous and strictly increasing. Define the one-sided stage p-values $p_i=1-\mathrm{\Phi }(Z_i)$, so $Z_i={\mathrm{\Phi }}^{-1}(1-p_i)$; for $0<u<1$, $P_0(p_i\le u)=P_0(Z_i\ge {\mathrm{\Phi }}^{-1}(1-u))=u$, and the endpoint inequalities are immediate, so they are valid by [F3]. Fix $c>0$ and set $\alpha ={\int }_c^{\mathrm{\infty }}\varphi (t)dt$, with $0<\alpha <1\mathrm{/}2$ by positivity and evenness. Under the fixed equal-weight inverse-normal rule, reject when $W=(Z_1+Z_2)\mathrm{/}\sqrt{2}\ge c$. The orthogonal change of variables $w=(z_1+z_2)\mathrm{/}\sqrt{2}$, $v=(z_1-z_2)\mathrm{/}\sqrt{2}$ has absolute Jacobian $1$ and satisfies $z_1^2+z_2^2=w^2+v^2$, so the joint density transforms to $\varphi (w)\varphi (v)$ and integration over $w\ge c$ gives the fixed-rule null rejection probability $\alpha$.

Choose $N_2=1$ when $Z_1\ge 0$ and $N_2=2$ when $Z_1<0$. Without recalibrating $c$, use weights $w(N_2)=(1,0)$ in the first case and $(0,1)$ in the second; each has squared norm one. Since $c>0$, the adapted rejection event is the disjoint union $\{Z_1\ge c\}\cup \{Z_1<0,Z_2\ge c\}$, whose probability by independence is $\alpha +(1\mathrm{/}2)\alpha =3\alpha \mathrm{/}2>\alpha$. Thus unit-norm weights selected with the stage-2 size do not by themselves preserve the fixed-rule error probability.