= Solution
A <reversible-jump Markov chain Monte Carlo> calculation compares normalized model-specific densities. The fixed-order posterior written only up to proportionality in part (a) omits normalizing factors that depend on $k$ and therefore cannot simply be canceled between dimensions. Moreover, the printed question does not specify the model-order <prior distribution> or the probabilities of selecting birth and death moves. Denote those by $p_k$, $b_k$ and $d_k$, respectively, retaining them in the answer.
Put $s=\sigma^2$, $c=n^{-1}\sum_i x_i^{k+1}$, $v_i=x_i^{k+1}-c$, and $r=y-X_k\beta$. The <centered polynomial birth move in reversible-jump sampling> changes the fitted vector by $zv$, so its new residual is $r'=r-zv$. Therefore
$$
\|r'\|^2-\|r\|^2=-2zv^Tr+z^2v^Tv.
$$
The map from $(\beta_0,\ldots,\beta_k,z)$ to $(\beta'_0,\ldots,\beta'_{k+1})$ has ones down the diagonal and a single additional entry $-c$ in the intercept row. Its <Jacobian determinant> is one.
Write $Q_k=(\beta-\mu_k)^T\Sigma_k^{-1}(\beta-\mu_k)$ and $Q_{k+1}=(\beta'-\mu_{k+1})^T\Sigma_{k+1}^{-1}(\beta'-\mu_{k+1})$. The normalized <multivariate normal distribution> prior ratio is
$$
\frac{\pi_{k+1}(\beta')}{\pi_k(\beta)}
=(2\pi)^{-1/2}\sqrt{\frac{|\Sigma_k|}{|\Sigma_{k+1}|}}
\exp\left[-\tfrac12(Q_{k+1}-Q_k)\right].
$$
Divide also by the forward proposal density $q(z)=(2\pi\sigma_\beta^2)^{-1/2}e^{-z^2/(2\sigma_\beta^2)}$. The common <inverse-gamma distribution> prior for $s$ and the common normal-likelihood normalization cancel, since $s$ stays fixed. The forward acceptance ratio is consequently
$$
R_+=\frac{p_{k+1}d_{k+1}}{p_kb_k}\,
\sigma_\beta\sqrt{\frac{|\Sigma_k|}{|\Sigma_{k+1}|}}
\exp\left[
\frac{zv^Tr-z^2v^Tv/2}{s}
-\frac{Q_{k+1}-Q_k}{2}
+\frac{z^2}{2\sigma_\beta^2}
\right],\qquad
\boxed{A_+=\min(1,R_+).}
$$
Here $\sigma_\beta>0$ is the proposal standard deviation. If equal model priors and symmetric birth/death selection are explicitly adopted, their ratios become one; they are not determined by the PDF. The expression assumes the same hyperparameters for the variance prior in both models, as in part (a). A dimension-dependent variance prior would contribute its own ratio.
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