Example. A trace ratio separates two weighted cycles [fdip-000I]
Example. A trace ratio separates two weighted cycles [fdip-000I]
The alternating cycles at \(t=1\) and \(t=2\) remain different even under the positive-scaling Laplacian comparison \(L'=cPLP^{-1}\), with \(c>0\) and \(P\) a vertex permutation. Use the trace ratio \[R(L)=\frac {\operatorname {tr}(L^2)}{(\operatorname {tr}L)^2},\] where the trace is the intrinsic endomorphism trace. The positive graph energy gives a nonzero positive-semidefinite \(L_t\) for \(t>0\), so its trace is positive and the ratio is defined. The cited proposition proves invariance by trace conjugacy and scalar cancellation. It compares counting-relative operators, not bilinear-form arrays under arbitrary congruence.
[boyd2004convex, Appendix A.5.2, pp. 646-647] gives the symmetric spectral and trace formulas. The spectral cycle calculation supplies the concrete basis; the following traces and comparisons are direct computations.
For the alternating four-cycle, \(t>0\), the orthonormal spectral basis gives eigenvalues \(0,2,2t,2+2t\), including at \(t=1\). Consequently \[\begin {aligned} \operatorname {tr}L_t&=4(1+t),\\ \operatorname {tr}(L_t^2) &=4+4t^2+4(1+t)^2\\ &=8(t^2+t+1),\\ R(t):=R(L_t)&=\frac {t^2+t+1}{2(t+1)^2}. \end {aligned}\] In particular \[\begin {aligned} R(1)&=\frac 38,\qquad R(2)=\frac 7{18},\\ R(2)-R(1)&=\frac 1{72}>0. \end {aligned}\] There can be no \(c>0\) and vertex permutation \(P\) with \(L_2=cPL_1P^T\). Thus the associated finite Gaussian backgrounds cannot be exactly relabeled into each other either. This calculation separates the stated pair; it does not claim a classification of all backgrounds.
The reciprocal identity \(R(t)=R(1/t)\) is consistent with this comparison. A one-step rotation \(p(i)=i+1\pmod 4\) exchanges the alternating edge weights \(1,t\) and leaves the common diagonal \(1+t\) unchanged. Its coordinate permutation therefore satisfies \[PL_tP^T=tL_{1/t}.\] Since \(1/t>0\), the reciprocal Laplacians are equivalent under \(\approx \). Their equal ratios do not demonstrate a failure of completeness. On the strict enriched slice with weights \((1,t,1,t)\), fixed mass and fixed opposite retained pair, the weight product is \(t^2\), whereas at parameter \(1/t\) it is \(t^{-2}\). A weight-preserving edge bijection would preserve this product; for positive \(t\) equality forces \(t=1\). Thus the broader reciprocal comparison is not a strict background isomorphism for \(t\ne 1\).
A fixed-mass calculation shows what changes in the induced state. Set \(m=1\) and retain the same labeled vertices \(R=\{0,2\}\). The alternate-vertex elimination has \(s=t+2\), giving \[\begin {aligned} \alpha (t)&=s-\frac {1+t^2}{s}=\frac {4t+3}{t+2},\\ \beta (t)&=\frac {2t}{t+2},\\ S_t&=\begin {pmatrix}\alpha (t)&-\beta (t)\\ -\beta (t)&\alpha (t)\end {pmatrix}. \end {aligned}\] Hence the effective coupling changes from \(\beta (1)=2/3\) to \(\beta (2)=1\), and the two precisions are \[\begin {aligned} S_1&=\begin {pmatrix}7/3&-2/3\\-2/3&7/3\end {pmatrix},\\ S_2&=\begin {pmatrix}11/4&-1\\-1&11/4\end {pmatrix}. \end {aligned}\] Their determinants are \(5\) and \(105/16\). The off-diagonal entry of \(S_t^{-1}\) is \(\beta (t)/\det S_t\). By the marginal expectation identity, the same labeled observable therefore has different normalized expectations: \[\begin {aligned} \langle \phi _0\phi _2\rangle _{t=1,m=1}&=\frac {2}{15},\\ \langle \phi _0\phi _2\rangle _{t=2,m=1}&=\frac {16}{105}. \end {aligned}\] This comparison fixes the mass, coordinates, reference pairing and partition. The labeled observable is not itself asserted invariant under every relabeling; the trace ratio already established the broader inequivalence.
Scale invariance of \(R\) does not mean equality of fixed-mass states. On an eigenvector of \(L\) with eigenvalue \(\lambda >0\), replacing \(L\) by \(cL\) changes the covariance eigenvalue from \(1/(\lambda +m^2)\) to \(1/(c\lambda +m^2)\) when \(c\ne 1\). Scaling all of \(Q\) instead gives covariance \(c^{-1}G\) and Schur map \(cS\), by inversion and the Schur formula. Changing mass can also leave \(R(L)\) unchanged while changing the law. The exact finite differences above do not construct a physical map from an environment to the parameter \(t\).
The trace identities and their formalization correspondence are at Definition [fdip-003G] and Proposition [fdip-003H]. Their unit-domain hypotheses specialize here to positive real trace; the values and normalized expectation comparisons above are additional cycle calculations.