= Solution
Let $B_\delta=\sqrt{\log(2m)}+\sqrt{\log(1/\delta)}$. The columns of $D^\dagger/L$ have <Euclidean norm> at most one. Under the usual <sub-Gaussian random vector> convention, 3(d) justifies the tuning choice $\lambda=2\sqrt2 L B_\delta/n$, which is independent of $\theta^*$. Also $\|\Pi_1z\|_2=|\langle z,\mathbf1/\sqrt n\rangle|$, so the <Chernoff bound> gives $\|\Pi_1z\|_2^2\leq c_\delta=2\log(2/\delta)$ with <probability> at least $1-\delta$.
On the intersection of these events, of <probability> at least $1-2\delta$, set $a=\|h\|_2/\sqrt n$ and $b=\|\Pi_1z\|_2/\sqrt n$. The weighted <Young inequality> gives $2ab\leq t^2a^2+b^2/t^2$ for $0<t<1$. Applying it to 3(c) and rearranging yields the rigorous bound
$$
\boxed{\frac{\|h\|_2^2}{n}\leq\frac{4\sqrt2\,s(\theta^*)L}{n(1-t^2)}B_\delta+\frac{c_\delta}{nt^2(1-t^2)}.}
$$
If the coefficient-one concentration statement in 3(d) is taken as an extra hypothesis, instead choose $\lambda=2LB_\delta/n$; the identical argument gives precisely the printed coefficient $4$ in the first term. Thus the deterministic argument and its risk rate are sound; its concentration constants depend on correcting or strengthening 3(d).
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