Oblivious ancilla reflection

Let $A$ be the full coefficient and PREPARE-work ancilla register in LCU decomposition and access model, let $S$ be the system, and let $J:S\to A\otimes S$ be the clean injection of Clean-ancilla zero-branch postselection. Set ${\mathrm{\Pi }}_0:=J{J}^{†}$; An isometric quantum encoding gives an orthogonal code projector makes this the orthogonal projection onto the image of $J$. Its basis formula is ${\mathrm{\Pi }}_0=\mathrm{\mid }{0}_A⟩⟨0_A\mathrm{\mid }\otimes I_S$, where $\mathrm{\mid }{0}_A⟩$ is unit by The computational basis is orthonormal and the tensor action is that of Tensor products of finitely many linear maps. Define $R_0:=I-2{\mathrm{\Pi }}_0\mathrm{.}$ Thus $R_0$ acts as $-I$ on the clean-ancilla subspace and as $+I$ on its orthogonal complement; it acts identically on the unknown system state. In the convention of Orthogonal-Projector Reflection and Access Contract, $R_0=S_{{\mathrm{\Pi }}_0}$, while that item writes $R_{{\mathrm{\Pi }}_0}=2{\mathrm{\Pi }}_0-I=-R_0$. These signs are distinct inside controlled blocks: if only the controlled $R_{{\mathrm{\Pi }}_0}$ is supplied, implement controlled $R_0$ with the exact control-$Z$ correction of Projector Reflections Are Hermitian Unitary Involutions and charge that gate. A projector-reflection call and its controlled form are explicit access operations, not free consequences of a state vector.