Definition. A selected Gaussian background [fdip-005A]
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  • November 1, 2026

A selected Gaussian background is a tuple \[b=(\Pi ,w,m,P),\qquad \Pi =(V,E,s,h),\] where \(\Pi \) is a finite incidence presentation, \(w:E\to \mathbb R_{>0}\), \(m>0\), and \(P\) is an orthogonal projector on the counting inner-product space \(H=\mathbb R^V\), with \[P^2=P=P^*,\qquad 0<\operatorname {rank}P<|V|.\] The selected space is \(X=\operatorname {im}P\), its orthogonal complement is \(Y=\ker P\), and the observed random vector is \(P\phi \in X\). Incidence, the Green operator and the full law are \[\begin {aligned} (d\phi )(e)&=\phi (h(e))-\phi (s(e)),\\ Q&=d^*Wd+m^2I>0,\qquad G=Q^{-1},\\ \phi &\sim \mu _Q. \end {aligned}\] Here \(W=\operatorname {diag}(w)\) and \(\mu _Q\) uses the induced Euclidean volume. These are derived from the tuple, not additional choices.

Three constructions supply such projectors, provided their ranks are nonzero and proper.

  1. A retained subset \(R\subsetneq V\) gives \(P_R\phi =\sum _{v\in R}\phi (v)\delta _v\). Identifying \(\operatorname {im}P_R\) with \(\mathbb R^R\) recovers the coordinate background.
  2. A finite signed graph symmetry \(\rho :\Gamma \to O(H)\) extends to \(\mathbb R[\Gamma ]\). For \(e^2=e=e^*\), where \([g]^*=[g^{-1}]\), \(P=\rho (e)\) is orthogonal because \(\rho (e)^*=\rho (e^*)\). The precision commutes with every \(\rho (g)\), hence with this \(P\).
  3. For a \(\Gamma \)-stable subspace \(X\subset H\), its orthogonal projector commutes with \(\rho (\Gamma )\): orthogonality makes \(X^\perp \) stable too. It need not commute with \(Q\). Within a repeated real isotypic block, the chosen multiplicity subspace may fail to be invariant under the positive multiplicity matrix. These are different constructions: a subrepresentation projector commutes with \(\rho (\Gamma )\) but need not commute with \(Q\); a group-algebra projector commutes with \(Q\) but need not commute with \(\rho (\Gamma )\). Centrality of \(e\) is sufficient for the latter commutation, and hence for a stable image.

Every subspace has a unique orthogonal projector for the specified metric. This makes the selector equivalent to choosing \(X\), without choosing an additional complementary subspace. An oblique idempotent \(T^2=T\) also gives a Gaussian observation \(T\phi \), and orthogonal conjugation does transport it naturally: \(T\mapsto OTO^{-1}\). But it also records a possibly nonorthogonal kernel, so the image alone no longer determines the observation or its covariance \(TGT^*\). General linear observations are outside this definition. No physical rule selecting \(P\) from an environment is assumed.