Take the unit-weight cycle on \(\mathbb Z/(2N)\), with \(N\ge 3\) and precision \(Q=L_{C_{2N}}+aI\), where \(a=m^2>0\). The group \(G=C_N\) acts by two-site rotations. It preserves the even retained vertices and the odd eliminated vertices. Write \(x_j=\phi _{2j}\), \(y_j=\phi _{2j+1}\) and \(q=2+a\); indices \(j\) lie in \(\mathbb Z/N\). For the shift \((Tz)_j=z_{j-1}\), the blocks are \[A=C=qI,\qquad B=-(I+T),\qquad B^*=-(I+T^{-1}).\] Thus the equivariant Schur formula gives \[\begin {aligned} S&=qI-q^{-1}(2I+T+T^{-1})\\ &=w'L_{C_N}+a'I,\\ w'&=\frac 1{2+a},\qquad a'=(2+a)-\frac 4{2+a} =\frac {a(a+4)}{a+2}>0. \end {aligned}\] Here \(L_{C_N}=2I-T-T^{-1}\). The restriction \(Q_{RR}=qI\) has no off-diagonal entries, while \(S\) has entries \(-1/q\) between adjacent retained vertices. The retained set is stable under \(G\), but the marginal precision is induced by elimination.

The six-cycle illustrates \(N=3\). Shaded even vertices remain; dashed odd vertices are integrated out. Each triangle edge has weight \(1/q\), and the mass squared changes from \(a\) to \(q-4/q\), with \(q=2+a\).

The finite Gaussian normalization is retained exactly: \(Z_Q=Z_CZ_S\) with \(Z_C=(2\pi/q)^{N/2}\), and the retained law is \(\mu _S\). The effective weight and mass are both part of this law. Dividing \(S\) by \(w'\) is useful for comparing its parameters, but changes the field covariance by a scalar unless the corresponding field rescaling is also made.

Put \(q=2\cosh \omega \) with \(\omega >0\). Then \[2+\frac {a'}{w'}=q^2-2=2\cosh (2\omega ).\] The doubled parameter has an exact finite-ring interpretation. For \(M\ge 3\), the covariance of precision \(w(2\cosh \eta \,I-T-T^{-1})\) on \(\mathbb Z/M\), with \(w,\eta >0\), is the periodic kernel \[\begin {aligned} K_{M,w,\eta }(r) &=\frac {e^{-\eta r}+e^{-\eta (M-r)}} {2w\sinh \eta \,(1-e^{-M\eta })},\\ &\hspace {24mm}0\le r\le M-1. \end {aligned}\] For the original and retained fields this gives \[(Q^{-1})_{0,2r} =K_{2N,1,\omega }(2r) =K_{N,w',2\omega }(r) =(S^{-1})_{0,r}.\] The equality is between covariances with their actual weights.

In the convention that correlation length is the reciprocal decay rate of these periodized exponential images, it changes from \(\xi =1/\omega \) fine lattice units to \(\xi '=1/(2\omega )=\xi / 2\) coarse lattice units. The finite ring has two winding directions, so its covariance is not a single exponential in separation. If the old spacing is \(\ell \), the new spacing is \(2\ell \); the physical decay length \(\ell / \omega =(2\ell )/(2\omega )\) is unchanged.

The bound \(N\ge 3\) keeps the reduced graph a simple cycle. At \(N=2\), \(T=T^{-1}\) and two eliminated vertices connect the same retained pair, producing two parallel edges of weight \(1/q\), or one edge of weight \(2/q\). At \(a=0\), the algebraic formula still gives \(w'=1/2\) and \(a'=0\), the two-resistors-in-series limit. The full precision then has a constant zero mode and no normalized full-space Gaussian law. Pinning or a mean-zero constraint must be specified as in constrained massless laws; the positive-mass covariance formula is not a finite limit on the full field space.

This is a direct cycle calculation within Kron reduction, the Schur reduction of graph Laplacians with possible positive diagonal terms. Closure follows from [dorfler2011kron, Lemma 2.1, pp. 5-6]. Successive and simultaneous eliminations agree by [dorfler2011kron, Lemma 3.3, p. 14]. In that source the diagonal mass corresponds to a shunt conductance or graph self-loop. In the incidence convention used here an edge with identical endpoints has zero difference; the mass is therefore kept as the separate term \(aI\). The source supplies the general reduction context, while the periodic covariance and scale statements are proved above.