= Solution
Set $Z_t=\mathbb E[\phi'(W_T)\mid\mathcal F_t]$. The <Brownian martingale representation theorem> applied to the bounded terminal variable $\phi'(W_T)$ supplies a continuous adapted version of $Z$, so it is <predictable>. Hence
$$
\gamma_t=Z_t\mathbf1_{\{t\le T\}}
$$
is predictable and satisfies $\mathbb E\int\gamma_t^2dt\le T\|\phi'\|_\infty^2$. For every square-integrable <predictable process> $\alpha$, conditioning at each deterministic time and using <Fubini theorem> gives
$$
\mathbb E\int_0^T\phi'(W_T)\alpha_t\,dt=\mathbb E\int_0^T Z_t\alpha_t\,dt.
$$
Thus part (d) says $\mathbb E\int(\beta_t-\gamma_t)\alpha_tdt=0$. Taking $\alpha=\beta-\gamma$, which is an allowed predictable square-integrable process, makes its squared norm zero. We conclude
$$
\boxed{\beta_t=\mathbb E[\phi'(W_T)\mid\mathcal F_t]\mathbf1_{\{t\le T\}}\quad d\mathbb P\,dt\text{-almost everywhere}.}
$$
This is the <Clark-Ocone formula for a smooth Brownian terminal payoff>, with exactly the uniqueness established in part (c).
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