= Solution
{c}
The second coordinate $\zeta^2$ is one-dimensional <Brownian motion>, whose zero set is unbounded almost surely by recurrence. The clock $\tau$ and its inverse are continuous, increasing, and unbounded, so the unwrapped angle $\Theta_t=\zeta^2_{\tau(t)}$ has zeros at arbitrarily large original times.
To ensure the times are <stopping times> for the original Brownian filtration, define directly
$$
T_0=0,\qquad T_n=\inf\{t\geq T_{n-1}+1:\Theta_t=0\}.
$$
The logarithmic lift makes $\Theta$ a continuous adapted process. Hitting a closed set after a stopping time is a stopping time under the usual filtration conditions, and the unbounded zero set proves inductively that each $T_n$ is finite almost surely. Since $T_n\geq T_{n-1}+1$, the sequence tends to infinity, and continuity gives
$$
\boxed{\zeta^2_{\tau(T_n)}=\Theta_{T_n}=0\quad\text{for every }n.}
$$
Geometrically, the Brownian path returns to the positive real ray at arbitrarily late times with \b[zero net unwrapped winding angle] about the origin. The assertion is stronger than principal argument zero, which also allows any integer number of completed turns.
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