Sample-only access is not coherent sampler access

Let $D$ be a classical procedure that, on each call, returns a bit $x\in \{0,1\}$ with $\mathrm{Pr}(x=1)=p$. Its interface is a classical input/output map from a fresh random seed to a bit; it has no quantum-register input and specifies no unitary $V$.

The law of $D$ does not provide a circuit for $V\mathrm{\mid }0⟩=\sqrt{1-p}\mathrm{\mid }0⟩\mathrm{\mid }0⟩\mathrm{\mid }{w}_0⟩+\sqrt{p}\mathrm{\mid }1⟩\mathrm{\mid }1⟩\mathrm{\mid }{w}_1⟩$, nor circuits for $V^{†}$, controlled $V$, value rotations, or reflections. Thus the sample-only object fails the explicit interface required by [F1] and cannot be substituted directly into QA29's coherent-query bound.

This refutes the statement about the interface alone. It does not rule out constructing a separate coherent preparation circuit from additional information about $p$; such a circuit and all its costs would be new supplied inputs.