= Solution
Let $\pi$ be the normalized density $\widetilde\pi$ in the transformed coordinates of (iv). Its fixed Gaussian increment gives a symmetric <proposal distribution>: $q(\theta'\mid\theta)=q(\theta\mid\theta')$. For distinct states, the accepted transition density is $K(\theta,\theta')=q(\theta'\mid\theta)\alpha(\theta,\theta')$, where $\alpha=\min\{1,\pi(\theta')/\pi(\theta)\}$. Then
$$
\pi(\theta)K(\theta,\theta')=q(\theta'\mid\theta)\min\{\pi(\theta),\pi(\theta')\}=\pi(\theta')K(\theta',\theta).
$$
The rejection probability gives a mass on the diagonal, which trivially satisfies the same balance identity. Thus $\boxed{\pi(d\theta)K(\theta,d\theta')=\pi(d\theta')K(\theta',d\theta)}$, the <detailed balance> condition, and integrating it proves invariance of the target.
The transformed density includes its Jacobian, so mapping back preserves the desired posterior law. Warmup adaptation is excluded from this fixed-kernel proof. Independent augmentation with $H_0$ is itself a reversible conditional update; alternatively the original joint augmented space can be sampled directly. This is the <Metropolis–Hastings algorithm> balance argument.
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