Example. The two-site reflection Gram matrix [fdip-007P]

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Take two real field coordinates \(x,y\), interchanged by reflection, and the centered Gaussian precision \[Q=\begin {pmatrix}a&-w\\-w&a\end {pmatrix}, \qquad a>|w|.\] The positive-half observables depend on \(x\). The covariance is \[G=Q^{-1} =\frac 1{a^2-w^2}\begin {pmatrix}a&w\\w&a\end {pmatrix}.\] For \(a=2,w=1\), this becomes \[G=\begin {pmatrix}2/3&1/3\\1/3&2/3\end {pmatrix}.\] The energy has positive mass squared \(a-w=1\) and crossing weight \(w=1\), so the same cutoff criterion proves reflection positivity.

For \(f_0=1,f_1=x,f_2=x^2\), the reflection Gram matrix is \[\left (B_\mu (f_j,f_k)\right )_{j,k=0}^2 =\begin {pmatrix} 1&0&2/3\\ 0&1/3&0\\ 2/3&0&2/3 \end {pmatrix}.\] Odd Gaussian moments vanish, and \(\mathbb E[x^2y^2]=(2/3)^2+2(1/3)^2=2/3\) by the four-factor Gaussian calculation. The leading principal minors are \(1\), \(1/3\) and \(2/27\), so this finite Gram matrix is positive definite. Positivity of this one matrix is an illustration; the criterion proves the claim for all positive-half observables.

TauCeti's TauCeti.Probability.integral_mul_mul_mul_multivariateGaussian proves the fourth-moment identity for a finite coordinate index set and a positive-semidefinite covariance, including repeated indices. Apply it with the displayed \(G\) and indices \(x,x,y,y\). This correspondence supports the moment calculation; the reflected pairing and its Gram-matrix positivity are computed here.

If instead \(a=2,w=-1\), the precision remains positive definite and its Gaussian remains reflection invariant. But \[B_\mu (x,x)=\mathbb E[xy]=-\frac 13<0.\] Thus reflection invariance and ordinary Gaussian positivity do not imply reflection positivity. The witness \(x\) is square-integrable; bounded value truncations also have a negative pairing once the cutoff is large enough, by continuity of \(B_\mu \).