Example. The retained variance detects the forgotten weight [fdip-004P]

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For the two-vertex family, eliminate the coordinate at vertex \(1\). The positive Schur formula and retained covariance give \[\begin {aligned} S_w&=(1+w)-\frac {w^2}{1+w} =\frac {1+2w}{1+w}>0,\\ r(\mathcal B_w)&=S_w^{-1}=\frac {1+w}{1+2w}. \end {aligned}\] In particular, \[r(\mathcal B_1)=\frac 23\ne \frac 35=r(\mathcal B_2), \qquad c(\mathcal B_1)=c(\mathcal B_2)=U.\] The fiber-constancy criterion excludes a function \(F:c(X)\to \mathbb R\) satisfying \(r=F\circ c\), even on a restricted family containing these two backgrounds. Consequently the unweighted interval with its retained endpoint also cannot recover this variance on that family. This is an obstruction for the specified forgetting and observable, not for every geometric realization.

This two-coordinate calculation is a teaching variant of the weighted-cycle comparison, which already gives unequal retained correlations \(2/15\) and \(16/105\) at fixed mass and fixed retained pair. The same principle is visible with one retained coordinate: incidence and the selected endpoint do not retain the weighted precision. By contrast, weight-preserving relabelings, with the retained projection and observable transported together, preserve expectations by Gaussian transport. Enlarging their comparison to forget weights need not preserve them.