The pCN Gaussian reference pair law is symmetric

By [F2], take a finite or countable complete orthonormal basis $(e_j)$ of $H$. Give $H\times H$ the inner product $⟨(u,v),(x,y){⟩}_{\times }=⟨u,x⟩+⟨v,y⟩$. Its induced norm is $\sqrt{\mathrm{\parallel }u{\mathrm{\parallel }}^2+\mathrm{\parallel }v{\mathrm{\parallel }}^2}$. It is complete because a Cauchy sequence has Cauchy coordinate sequences in $H$, whose limits in $H$ give its product limit. Thus this is a Hilbert space. The family $(e_j,0),(0,e_j)$ is orthonormal, and any vector orthogonal to every listed vector has both components orthogonal to every $e_j$, hence both components are zero by completeness; it is a complete orthonormal basis. Let $D\subseteq H$ be countable and dense. The balls $B(d,s)$ with $d\in D$ and positive rational $s$ form a countable base for $H$: for $x\in U$ open, choose $r>0$ with $B(x,r)\subseteq U$, then $d\in D$ with $\mathrm{\parallel }d-x\mathrm{\parallel }<r\mathrm{/}3$ and rational $s$ with $\mathrm{\parallel }d-x\mathrm{\parallel }<s<r-\mathrm{\parallel }d-x\mathrm{\parallel }$. Thus $H$ is second-countable. Then $D\times D$ is countable by [F2] and dense, since each coordinate of a pair can be approximated within $\epsilon \mathrm{/}2$ to make the product norm error less than $\epsilon$. Balls with centers in $D\times D$ and positive rational radii form a countable base: for $z$ in an open set choose $r>0$ with $B(z,r)$ contained there, then a center $d$ with $\mathrm{\parallel }d-z\mathrm{\parallel }<r\mathrm{/}3$ and rational $s$ with $\mathrm{\parallel }d-z\mathrm{\parallel }<s<r-\mathrm{\parallel }d-z\mathrm{\parallel }$ give $z\in B(d,s)\subseteq B(z,r)$. Thus $H\times H$ is second-countable, and [F10] identifies its Borel sigma-algebra with the product Borel sigma-algebra. Since $U$ and $\xi$ are Borel random elements, the pair $(U,\xi )$ is measurable into the product sigma-algebra: the preimage of each measurable rectangle is the intersection of the two coordinate preimages, and rectangles generate the product sigma-algebra. Continuous addition and scalar multiplication then make $(U,V)$ and $(V,U)$ Borel random elements of $H\times H$. Their Borel laws are therefore defined.

Fix a finite list of continuous linear functionals on $H\times H$. Each has the form $(u,v)\mapsto \mathrm{\ell }(u)+m(v)$, because its restrictions to the two coordinate subspaces sum to the functional. Any real linear combination of the listed projections is another such functional. Under $(U,V)$ that combination is $(\mathrm{\ell }+\rho m)(U)+\beta m(\xi )$; under $(V,U)$ it is $(\rho \mathrm{\ell }+m)(U)+\beta \mathrm{\ell }(\xi )$. By [F3], the two summands in each display are centered normal, and [F5] makes them independent; [F6] makes their sum normal, including a zero coefficient. Hence every finite projection vector under either pair is multivariate normal by the projection definition in [F7], with zero mean.

Define the symmetric covariance form $\mathrm{\Gamma }(\mathrm{\ell },m)=\mathbb{E}[\mathrm{\ell }(U)m(U)]$. These products are integrable: [F4] gives finite second moments of $\mathrm{\parallel }U\mathrm{\parallel }$, boundedness of $\mathrm{\ell },m$ gives finite second moments of their evaluations, and $\mathrm{\mid }ab\mathrm{\mid }\le (a^2+b^2)\mathrm{/}2$. The same argument applies to evaluations of $\xi$. For functionals $(\mathrm{\ell },m)$ and $({\mathrm{\ell }}^{\mathrm{\prime }},m^{\mathrm{\prime }})$, the projection under $(U,V)$ is $(\mathrm{\ell }+\rho m)(U)+\beta m(\xi )$. Independence and centering make its mixed covariance terms zero by [F9], so its covariance with the corresponding projection for $({\mathrm{\ell }}^{\mathrm{\prime }},m^{\mathrm{\prime }})$ is $\mathrm{\Gamma }(\mathrm{\ell },{\mathrm{\ell }}^{\mathrm{\prime }})+\rho \mathrm{\Gamma }(\mathrm{\ell },m^{\mathrm{\prime }})+\rho \mathrm{\Gamma }(m,{\mathrm{\ell }}^{\mathrm{\prime }})+\mathrm{\Gamma }(m,m^{\mathrm{\prime }}),$ using ${\rho }^2+{\beta }^2=1$. Under $(V,U)$ the projections are $(\rho \mathrm{\ell }+m)(U)+\beta \mathrm{\ell }(\xi )$ and $(\rho {\mathrm{\ell }}^{\mathrm{\prime }}+m^{\mathrm{\prime }})(U)+\beta {\mathrm{\ell }}^{\mathrm{\prime }}(\xi )$; expansion gives the same expression because $\mathrm{\Gamma }$ is symmetric. Therefore the finite projection vectors from step 2.1 have equal covariance matrices and zero means.

By [F7], the finite projection laws in step 3.1 are equal, in particular the finite coordinate laws for the product basis from step 1.1. That step also shows that $H\times H$ is a separable Hilbert space. By [A1], Countable Choice follows from AC; the coordinate-law result [F8] therefore identifies the two Borel probability laws. Hence $\mathrm{Law}(U,V)=\mathrm{Law}(V,U)$.

Step 4.1 is the claimed invariance of the joint law under coordinate swap, for every $\beta \in [0,1]$, including the identity proposal at $\beta =0$ and the independent-reference proposal at $\beta =1$.