Over a commutative unital ring \(k\), give \(M=X\oplus Y\) and \(M'=X'\oplus Y'\) chosen splittings. Suppose isomorphisms \(P:M\to M'\), \(P_X:X\to X'\), \(P_Y:Y\to Y'\) preserve the inclusions and projections: \[\begin {aligned} P\iota _X&=\iota '_XP_X,&P\iota _Y&=\iota '_YP_Y,\\ \pi '_XP&=P_X\pi _X,&\pi '_YP&=P_Y\pi _Y. \end {aligned}\] Let \(Q'=PQP^{-1}\). Write its 4 split components and those of \(Q\) as \(A:X\to X\), \(B:Y\to X\), \(E:X\to Y\), \(C:Y\to Y\), and their primed versions. Then \[\begin {aligned} A'&=P_XAP_X^{-1},&B'&=P_XBP_Y^{-1},\\ E'&=P_YEP_X^{-1},&C'&=P_YCP_Y^{-1}. \end {aligned}\] If \(C\) is invertible, the split Schur maps obey \[S'=P_XSP_X^{-1},\qquad S=A-BC^{-1}E.\] For \(K=C^{-1}E:X\to Y\), \(K'=(C')^{-1}E'\), \[K'P_X=P_YK,\qquad -K'(P_Xx)=P_Y(-Kx).\] Thus the eliminated-equation field \(y_*(x)=-Kx\), the unique solution of the lower equation with zero source, is transported with its sign intact.

When \(Q\) is also invertible, the inverse-compression theorem says \[S^{-1}=\pi _XQ^{-1}\iota _X.\] Conjugating and using the split maps gives \[(S')^{-1}=P_XS^{-1}P_X^{-1} =\pi '_X(Q')^{-1}\iota '_X.\] This uses an actual inverse. No pairing has been assumed; only for an orthogonal split and self-adjoint \(Q\) does one further have \(E=B^\dagger \), by Proposition [fdip-001X]. For an orthogonal split of finite-dimensional real inner-product spaces and a positive self-adjoint \(Q\), this field is also stationary for the quadratic action in the eliminated variable, by the real square-completion argument.

For a vertex bijection \(p:V\to V'\), let \(R'=pR\), \(X=k^R\), \(X'=k^{R'}\), and let \(P_R\) be inverse precomposition by \(p|_R\). At \(p(v)\), \(v\in R\), both sides of \[\pi _{R'}P=P_R\pi _R\] take the value \(f(v)\). The target space is \(k^{pR}\), even if the labels of \(R\) and \(pR\) differ. This equation supplies the moving projection square in Gaussian marginal transport.

If a group action preserves \(R\), its restrictions compose on \(k^R\) and the square becomes \(\pi _RP_g=P_{g,R}\pi _R\). Faithfulness of the full action need not survive restriction: a permutation moving only eliminated vertices acts trivially on retained fields. The opposite-pair half-turn has a separate nontrivial retained action.

Stable subsets and equivariance follow the convention of [milne2025group, p. 58]. The real Schur convention is [boyd2004convex, Appendix A.5.5, pp. 650-651], with the source's first pivot exchanged to the eliminated \(C\) block. The typed transport proof above uses only the chosen splitting, and does not by itself supply any integral or Gaussian normalizer.