= Solution
An oracle can recognize a solution without disclosing it in advance. If an efficiently evaluable or black-box predicate has the promised unique marked input, write $f(x)=1$ exactly for that input. A <Boolean quantum oracle> computes this predicate reversibly as
$$
|x\rangle|b\rangle\longmapsto|x\rangle|b\oplus f(x)\rangle.
$$
Its diagonal projector on the search register is $\sum_x f(x)|x\rangle\langle x|=|w\rangle\langle w|$. To implement a selectable phase, compute $f(x)$ into a zero <quantum ancilla>, apply a known phase to its $1$ component, and uncompute. The search register then undergoes
$$
|x\rangle\longmapsto e^{-i\Delta t f(x)}|x\rangle
=e^{-i\Delta tP_w}|x\rangle.
$$
This requires evaluation access to the predicate, not classical knowledge of which input passes it. The $P_\psi$ term is known from the <Hadamard transform>. With Hamiltonian-oracle access, or by controlled short-time simulations of the two terms, their sum can realize <continuous-time quantum search>.
\b[The unknown projector is supplied through an oracle or a physically encoded recognition rule.] It cannot be manufactured for an arbitrary hidden $w$ with no such input-dependent resource. The oracle assumption and the fixed energy scale $\Delta$ are essential when interpreting the square-root search time as a computational advantage.
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