Controlled input-consistency reflection from a bit query

Inputs. A Boolean query problem with $N\ge 1$, promised input $x\in D$ accessed by its standard XOR oracle, a finite work space $K$, a one-bit control $c$, and a data register with orthogonal decomposition $\mathcal{H}=\mathrm{span}\{\mathrm{\mid }start⟩\}\oplus ({\mathbb{C}}^N\otimes K\otimes {\mathbb{C}}^2),$ where the address basis is $\mathrm{\mid }i⟩=e_{i-1}$ for $i\in [N]$ and $\mathrm{\mid }start⟩$ is a specified unit vector orthogonal to the query summand, and one clean scratch bit $u=0$. Fix a known address $i_0\in [N]$. Define the consistency sum ${\mathrm{\Pi }}_x=\mathrm{\mid }start⟩⟨start\mathrm{\mid }+{\sum }_{i=1}^N\mathrm{\mid }i⟩⟨i\mathrm{\mid }\otimes I_K\otimes \mathrm{\mid }{x}_i⟩⟨x_i\mathrm{\mid }\mathrm{.}$

Output. A gate list for the controlled plus-on-range reflection $2{\mathrm{\Pi }}_x-I$ on the data register, returning the scratch bit to zero.

Instructions. In query mode route address $i$ to the oracle and use $u$ as its target. In start mode route the oracle address to the fixed $i_0$ and apply a Hadamard to $u$, so its query target is $\mathrm{\mid }+⟩$. Make one unconditional standard XOR oracle call. Apply phase $-1$ exactly when $c=1$, the mode is query, and the response bit $z$ differs from $u$; disable this phase on start and when $c=0$. Make a second unconditional XOR call with the same address routing. In start mode apply the Hadamard to $u$ again. Return the data and control registers and the scratch. No controlled unknown-oracle call is invoked.