= Solution
For the normalized vector defining the <orthogonal projection> $P=|\phi\rangle\langle\phi|$, $P^k=P$ for every $k\geq1$. Thus
$$
e^{-i\pi P}=I+\sum_{k=1}^{\infty}\frac{(-i\pi)^k}{k!}P
=I+(e^{-i\pi}-1)P,
$$
which proves the <projector phase rotation> identity in this case:
$$
\boxed{U_P=I-2P.}
$$
It changes the sign of the component along $\phi$ and fixes the orthogonal complement, so it is a <reflection operator>.
For a marked <computational basis> vector $w$, $U_{P_w}=I-2|w\rangle\langle w|$ is the <marked-state phase oracle>. With $|\psi\rangle=H^{\otimes N}|0^N\rangle$, the other reflection is
$$
U_{P_\psi}=H^{\otimes N}(I-2|0^N\rangle\langle0^N|)H^{\otimes N}.
$$
The <Hadamard gates> thus implement its change of basis from the known all-zero state. The usual <Grover diffusion operator> is $D=2|\psi\rangle\langle\psi|-I=-U_{P_\psi}$. Consequently a <Grover search algorithm> iteration is
$$
\boxed{G=D\,U_{P_w}=-U_{P_\psi}U_{P_w}.}
$$
The minus sign is an irrelevant <global phase>. These two reflections rotate the search state in the span of the marked vector and the unmarked uniform vector, with the <Grover rotation angle> controlled by the initial marked amplitude. Iterating them approximately $\pi\sqrt{2^N}/4$ times gives high-probability search. The same pair of projectors appears additively in the continuous-time Hamiltonian, but the product of full reflections and evolution under their sum are distinct dynamics.
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