A BQP promise pair with unrestricted off-promise inputs

The pair is a promise problem because $Y$ and $N$ are disjoint singletons of distinct one-bit words, as required by [F2]. For $n\ge 1$, use input count $n$, total wire count $m=n$, gate count zero, and output tuple $(0)$. For $n=0$, use input count zero, total wire count one, gate count zero, and output tuple $(0)$, whose wire is initialized to zero. These are valid descriptions under [F3]. One fixed one-tape generator writes the two unary counts by repeated sweeps over the $n$ input marks and then appends the constant gate-count and output fields; at most $O((n+1{)}^2)$ transitions suffice. The total predicate is $1$ exactly when $x$ is the one-bit word $0$ and $z=0$, and $0$ otherwise, so it is deterministic and polynomial time.

On the YES input $x=0$, the identity circuit leaves the measured output in $\mathrm{\mid }0⟩$, so acceptance is $1\ge 2\mathrm{/}3$. On the NO input $x=1$, it leaves the measured output in $\mathrm{\mid }1⟩$, so acceptance is $0\le 1\mathrm{/}3$. Every other string is outside $Y\cup N$, where [F1] imposes no condition. Hence this promise problem is in BQP in the stated model.