Acceptance is invariant under an unconditional global phase

Let ${\psi }_x=U_C(\mathrm{\mid }x⟩\otimes \mathrm{\mid }{0}^{a(n)}⟩)$. Replacing the entire unconditional unitary $U_C$ by $\lambda U_C$ replaces ${\psi }_x$ by $\lambda {\psi }_x$. Since the root pairing is linear in its first argument, every coordinate $⟨P_O{\psi }_x,e_{z,w}{⟩}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}$ is multiplied by $\lambda$. Its squared modulus is unchanged because $\mathrm{\mid }\lambda {\mathrm{\mid }}^2=1$.

On a control-first pair of qubits, define $C(U)=\left(\begin{array}{cc}I& 0\\ 0& U\end{array}\right)$. For target $U=I_2$ and phase $\lambda =-1$, $C(U)=I_4$ while $C(\lambda U)=\mathrm{diag}(I_2,-I_2)=Z\otimes I_2$. By [F2], this second matrix is implemented by applying $T$ four times to the control wire. On input $\mathrm{\mid }00⟩$, apply $H$ to the control, then respectively $C(I_2)$ or $C(-I_2)$, then $H$ to the control again. The first gate list maps $\mathrm{\mid }00⟩$ to $\mathrm{\mid }00⟩$ because $H^2=I_2$; the second maps it to $\mathrm{\mid }10⟩$ because direct matrix multiplication gives $HZH=X$ and $X\mathrm{\mid }0⟩=\mathrm{\mid }1⟩$. Each list uses only the fixed finite gates and the second is an explicit circuit realizing the target phase inside the control block.

The finite sums over $w$ and then over the accepting strings $z$ are therefore unchanged term by term. Thus both $p_C(z\mid x)$ and $p_C(x)$ are unchanged under an unconditional global phase.

Use these as the length-two circuits in two finite families, taking gate-free circuits with empty output tuples at other lengths. Let the promise pair be $Y=\{00\}$ and $N=\mathrm{\varnothing }$, and let the total predicate accept exactly when $n=2$, $x=00$, and the measured control bit is $0$. Designate wire $0$ as the length-two output. The two families differ only by whether the target block is $I_2$ or $-I_2$. By [F1], the first has acceptance probability $1$ and the second has acceptance probability $0$, since their length-two outputs are the distinct basis vectors $\mathrm{\mid }00⟩$ and $\mathrm{\mid }10⟩$. Thus a phase applied only inside a controlled target block can change acceptance, even though the same scalar on an entire unitary leaves acceptance invariant.