The controlled input-consistency circuit realizes the exact reflection

Let H=span{|start⟩}⊕(C^N⊗K⊗C²). The algorithm input requires |start⟩ to be unit and orthogonal to every query mode; hence |start⟩⟨start| is the orthogonal projection onto its span. It also labels the standard address basis by |i⟩=e_{i−1}, i∈[N]. Under the standard coordinate pairing in [F25], ⟨e_j,e_k⟩=Σ_rδ_{rj}δ_{rk}=δ_{jk}, so these address vectors and the response basis vectors are orthonormal. Therefore the query summands |i⟩⟨i|⊗I_K⊗|x_i⟩⟨x_i| are mutually orthogonal projections and are orthogonal to the start projection. Thus Π_x is an orthogonal projection.

Append a clean scratch bit u=0. In query mode, call O_x at address i to write u=x_i. In start mode route the oracle address to a fixed i_0∈[N] and apply H to u so the target is |+⟩; since X^{x_{i_0}}|+⟩=|+⟩, the unconditional oracle call leaves start unchanged. Apply a fixed phase −1 exactly when c=1, the mode is a query state, and z≠u; apply +1 on the start mode and when c=0. On query basis states this phase is +1 exactly when z=x_i, and −1 otherwise.

Call O_x again at the same address to reset u to 0 in query mode, using O_x²=I from [F22]. In start mode the second call again fixes |+⟩, and the final controlled H returns u to |0⟩. For c=1 the induced operator on H is 2Π_x-I; for c=0 it is I. Thus this fixed circuit realizes controlled R_{Π_x} with the plus-on-range sign specified in [F24], without a controlled unknown-oracle call. By linearity the same control/data action holds for arbitrary superpositions with the scratch returned clean.