Definition. Forgetting a weight while retaining its endpoint [fdip-004O]

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For each \(w>0\), let \(\mathcal B_w\) be the finite Gaussian background with vertices \(V=\{0,1\}\), one edge \(0\to 1\) of weight \(w\), counting pairing, mass \(m=1\), retained set \(R=\{0\}\) and eliminated set \(V\setminus R=\{1\}\). The field coordinates remain labelled. The weighted Laplacian and massive precision are \[\begin {aligned} L_w&=w\begin {pmatrix}1&-1\\-1&1\end {pmatrix},\\ Q_w&=I+L_w=\begin {pmatrix}1+w&-w\\-w&1+w\end {pmatrix}. \end {aligned}\] Indeed the incidence row is \((-1,1)\). The vectors \((1,1)\) and \((1,-1)\) have precision eigenvalues \(1\) and \(1+2w\), so \(Q_w>0\). The normalized Gaussian law is therefore well-defined on counting-coordinate volume \(df_0\,df_1\), with \[\begin {aligned} d\mu _w(f)&=Z_w^{-1}\exp (-f^TQ_wf/2)\,df_0\,df_1,\\ Z_w&=\frac {2\pi }{\sqrt {1+2w}}. \end {aligned}\] The normalizer is the two-dimensional specialization of positive Gaussian normalization.

Let \(X=\{\mathcal B_w:w>0\}\) and let \(Y=\{U\}\) be the singleton whose element is the unweighted labelled directed edge \(0\to 1\) with distinguished endpoint \(0\). Define \[\begin {aligned} c:X&\longrightarrow Y,&c(\mathcal B_w)&=U,\\ r:X&\longrightarrow \mathbb R,&r(\mathcal B_w)&=\int f_0^2\,d\mu _w(f). \end {aligned}\] Thus \(c\) forgets the varying weight while retaining the labels, orientation and selected endpoint; the mass and reference pairing are fixed throughout the family. The observable is integrable by Gaussian polynomial integrability. The coordinate has zero mean, so \(r\) is the variance at the retained endpoint. Composing \(c\) with the interval realization gives the same labelled interval with that endpoint for every \(w\).