On the unit four-cycle with \(m=1\), the opposite, adjacent and positive half-turn selectors are pairwise inequivalent under the orthogonal comparison. Consequently no directed or signed graph arrow identifies any pair. Their retained precisions have the following invariants:

The intrinsic squared-norm observable has expectation \[\mathbb E_{\mu _Q}\|P\phi \|^2 =\mathbb E_{\nu _b}\|x\|^2 =\operatorname {tr}_X C_P =\operatorname {tr}_H(PG).\] Indeed, in an orthonormal basis of \(X\) the squared norm is the sum of squares of coordinates, each with variance equal to the corresponding diagonal covariance entry. Since \(C_P=(PGP)|_X\) and \(P^2=P\), cyclicity of the finite-dimensional trace gives the last equality. In the same order as the table, \[\mathbb E\|P\phi \|^2=\frac {14}{15},\quad \frac {14}{15},\quad \frac 65.\] The reciprocal eigenvalues give these values. Thus neither \(\det S_P\) nor this covariance trace alone separates all three selectors. In particular the equal covariance traces of the first two do not mean equal precision traces.

For the two coordinate selectors, named coordinate observations also distinguish the laws. The full covariance in Example [fdip-005F] gives \[\begin {aligned} \frac {\langle \phi _0\phi _2\rangle }{\langle \phi _0^2\rangle } &=\frac 27&&\text {(opposite)},\\ \frac {\langle \phi _0\phi _1\rangle }{\langle \phi _0^2\rangle } &=\frac 37&&\text {(adjacent)}. \end {aligned}\] The marginal expectation identity makes these the corresponding retained expectations. They refer to the stated vertex coordinates; unlike the spectrum and squared norm, such coordinate ratios are not invariants under arbitrary orthogonal changes of retained basis.