Fixed-one Grover search with candidate verification

Input: An integer $N\ge 1$, standard Boolean XOR-query access $O_x\mathrm{\mid }q,b,w⟩=\mathrm{\mid }q,b\oplus x_{q+1},w⟩$ on the ordered $N$-level address register of Boolean oracle query model for search lower bounds, with $x\in P_N$ from All-zero versus exactly-one-marked search promise, and no information about whether $x=0^N$ or $x=e_i$. Any auxiliary registers used by the procedure start in their declared clean states; the procedure returns all subroutine workspace clean before reuse or termination.

Gate model: Inherit the nonoracle model of One Grover iterate in the standard bit-query model: exact $H$ and $X$ are callable, with each occurrence charged as one elementary gate, including the $m$ Hadamards preparing the uniform address state and all response preparations. The exact reflection $R_s$ on the $M$ padded addresses is supplied either as an explicitly costed primitive or by a declared costed exact circuit over the selected alphabet. All preparation and reflection costs are counted separately from raw oracle queries.

Output: The symbol no and no candidate if verification fails; otherwise yes and a verified marked coordinate.

Instructions:

1. Set $m=\lceil {\log }_{2}N\rceil$, $M=2^m$, so $N\le M<2N$, and hard-code the schedule parameter $t_0=1$. This is a fixed schedule; it is not an assertion that the actual input has one mark.
2. Set $K=\lceil \sqrt{M}\rceil$ and $q=1-2\mathrm{/}M$. Compute exactly $T_0(q)=1$, $T_1(q)=q$, and $T_{r+1}(q)=2q{T}_r(q)-T_{r-1}(q)$ for $1\le r<2K+1$. For $k=0,\dots ,K$, compute $p_k=(1-T_{2k+1}(q))\mathrm{/}2$ and choose the least $k$ maximizing this finite list.
3. Prepare the uniform state on $M$ addresses and apply the Grover iterate of One Grover iterate in the standard bit-query model exactly $k$ times, implementing each padded bit query by Reversible zero-completion of an N-address bit query. Allocate the clean raw-address, comparison, and routing workspace required by each call; return it clean before the next call. The iterate's phase-kickback response is supplied in $\mathrm{\mid }-⟩$ as specified by that iterate.
4. Measure the padded address $j$. Prepare a fresh response bit in $\mathrm{\mid }0⟩$ and use Reversible zero-completion of an N-address bit query for the candidate-verification query, with its required clean workspace. Measure the response bit. If $j<N$ and this bit is $1$, output yes and original coordinate $j+1$. Otherwise output no and no candidate. Return all clean workspace to zero.

The schedule is defined on every input in $P_N$ and never receives an actual marked count. The recurrence and gate costs are classical/nonoracle costs; every simulated Grover or verification query uses one standard raw bit query.