 Known-state success dilution for exact amplitude amplification

For known good probability $p={\sin }^2\theta$, choose $j=\lceil \pi /(4\theta )-1/2\rceil$. A known reversible preparation of one state with good probability $p_{\ast }$ permits $j$ ordinary amplitude amplification iterations to reach the good subspace exactly. With ancillary marking access, prepare a flag with one-probability $q=p_{\ast }/p$ and define joint success as good data and flag one. Reflection about the prepared product state is known, while the joint phase test requires a controlled marking oracle or a computed Boolean output. This changes the success projector and enlarges the state space; it does not evade the obstruction to uniform probability lowering by a unitary. Query costs of preparation, its inverse and marking must all be counted.