A hidden common cause is observationally indistinguishable from an edge

Let $W_1,\dots ,W_m$ be a finite independent family of scalar normal variables with means ${\mu }_r$ and variances ${\sigma }_r^2$. For real coefficients $a_1,\dots ,a_m,b$, [F3] gives the characteristic function of $b+{\sum }_r{a}_r{W}_r$ as $\exp \text{\hspace{0.17em}⁣}(it(b+{\sum }_r{a}_r{\mu }_r)-\frac{t^2}{2}{\sum }_r{a}_r^2{\sigma }_r^2)$; scalar uniqueness identifies this as the normal law with that mean and variance. Every scalar projection of a finite affine image of $W=(W_1,\dots ,W_m)$ is such a sum, so [F4] proves that every such image is multivariate normal, including when a projection has variance zero.

In the latent model, $\mathbb{E}(X,Y)=(0,0)$, $\mathrm{Var}(X)=\mathrm{Var}(Y)=2$, and $\mathrm{Cov}(X,Y)=1$, by independence of $L,U_X,U_Y$. In the direct-edge model, $\mathbb{E}(X,Y)=(0,0)$, $\mathrm{Var}(X)=2$, $\mathrm{Cov}(X,Y)=\frac{1}{2}\mathrm{Var}(X)=1$, and $\mathrm{Var}(Y)=\frac{1}{4}(2)+\frac{3}{2}=2$. Both vectors are Gaussian by step 1.1 and have the same positive-definite covariance matrix $\mathrm{\Sigma }=\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right)$, whose leading principal minors are $2$ and $3$. By [F5], both vector characteristic functions equal $\exp (-t^T\mathrm{\Sigma }t\mathrm{/}2)$ for every $t\in {\mathbb{R}}^2$; the uniqueness clause of [F5] gives equality of the full observed laws.

The latent model is a finite recursive Markovian SCM with observed set $\{X,Y\}$ and omitted latent vertex $L$, so it does not satisfy [F2]. The direct-edge model has observed set $\{X,Y\}$ and independent exogenous variables; its full-joint law factors as $N(0,2)(dx)N(x\mathrm{/}2,3\mathrm{/}2)(dy)$, the root probability kernel for $X$ followed by the probability kernel for $Y$ given its parent $X$. In its two-vertex graph $X\to Y$, the only pair of nonempty disjoint endpoint sets is $\{X\},\{Y\}$, which is not d-separated because of the direct path; step 2.1 gives $\mathrm{Cov}(X,Y)=1$, so they are not independent. Thus this law is faithful to the direct-edge DAG, and the direct-edge model satisfies the observed-only Markovian factorization and faithfulness conditions in the discovery regime. Since the observed laws agree by step 2.1, no procedure using only that law can distinguish the two causal explanations.

The common observed law has $\mathrm{Cov}(X,Y)=1\mathrm{\ne }0$, so $X,Y$ are dependent and the perfect oracle does not delete their only edge. With only two vertices there is no unshielded triple and none of R1–R3 applies; by [F9] the regime-compliant run terminates, and its computed output is $X-Y$. Reversing the single edge in the direct-edge DAG keeps the same skeleton and no unshielded collider, so [F6] proves the reverse DAG is equivalent; [F7] therefore gives the same undirected CPDAG. The PC output reports adjacency but cannot recover the latent-versus-direct causal distinction.