= Solution
The cross-model target must include a model-order prior $\rho_k$. Write its joint, unnormalized density as
$$
t_k(a,v)=\rho_k\,p(y\mid a,v,k)\,p(a\mid k)\,p(v\mid k).
$$
Here the prior densities are normalized within each model. In particular their $k$-dependent determinants and powers of $2\pi$ cannot be dropped merely because the fixed-model posterior in part (i) was given up to proportionality.
For explicit <birth and death moves for Bayesian variable selection>, choose a birth with probability $b_k$, generate $u$ from a density $g_k(u\mid a,v)$, append $u$ as the new covariate coefficient, and leave $v$ unchanged. The map from $(a,v,u)$ to $(a',v')=((a,u),v)$ is invertible under the reverse deletion and has absolute Jacobian one. It matches dimensions: the smaller-model coefficients and <variance> have dimension $k+2$, and the one auxiliary draw supplies the larger-model dimension $k+3$. If the reverse death is selected with probability $d_{k+1}$, accept with
$$
\boxed{\alpha_b=\min\left\{1,
\frac{t_{k+1}((a,u),v)d_{k+1}}
{t_k(a,v)b_k g_k(u\mid a,v)}\right\}.}
$$
For a death, delete the last coefficient $u$, retain the others and the <variance>, and use
$$
\boxed{\alpha_d=\min\left\{1,
\frac{t_k(a,v)b_k g_k(u\mid a,v)}
{t_{k+1}((a,u),v)d_{k+1}}\right\}.}
$$
To verify <detailed balance>, multiply the birth acceptance by $t_kb_kg_k$ and the reverse death acceptance by $t_{k+1}d_{k+1}$. Both equal the minimum of these two quantities in the matched coordinates. This proves reversibility without merely naming a dimension-changing algorithm. A more general bijection introduces its absolute Jacobian, and random choices among several covariates introduce the corresponding forward and reverse selection probabilities. Add within-model updates so that coefficients and <variance> can explore each model. These are <reversible-jump Markov chain Monte Carlo> updates; the unspecified model prior and proposal density must be supplied to implement them numerically.
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