Example. The complete sector law of the alternating four-cycle [fdip-003S]
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October 29, 2026

For the alternating four-cycle with \(t,m>0\), the two real half-turn sectors are independent under the Gaussian law. Their precision and covariance blocks vary with \(t,m\), although the action and its projectors do not. Put \(a=m^2\) and \(s=1+t+a\). The counting quadratic form is \[\begin {aligned} \langle \phi ,Q\phi \rangle &=(\phi _1-\phi _0)^2+t(\phi _2-\phi _1)^2\\ &\quad +(\phi _3-\phi _2)^2+t(\phi _0-\phi _3)^2 +a\sum _i\phi _i^2. \end {aligned}\] It is strictly positive for nonzero \(\phi \), since \(a>0\). The weighted incidence construction is self-adjoint, and the half-turn reindexes the 4 edge terms without changing their weights. Thus \(QP=PQ\), as required by the intrinsic product law. In vertex order \((0,1,2,3)\), the massive precision is \[Q=\begin {pmatrix} s&-1&0&-t\\-1&s&-t&0\\0&-t&s&-1\\-t&0&-1&s \end {pmatrix}.\] Use the orthonormal vectors \[\begin {aligned} v_{\varepsilon ,0}&=(\delta _0+\varepsilon \delta _2)/\sqrt 2,\\ v_{\varepsilon ,1}&=(\delta _1+\varepsilon \delta _3)/\sqrt 2, \qquad \varepsilon \in \{1,-1\}. \end {aligned}\] Write \(\phi =\sum _\varepsilon (x_\varepsilon v_{\varepsilon ,0}+y_\varepsilon v_{\varepsilon ,1})\) and \(z_\varepsilon =(x_\varepsilon ,y_\varepsilon )\). These are normalized sector coordinates, rather than the coefficients in the unnormalized free bases of Definition [fdip-000L]. In particular \[\begin {aligned} \phi _0&=(x_++x_-)/\sqrt 2,&\phi _2&=(x_+-x_-)/\sqrt 2,\\ \phi _1&=(y_++y_-)/\sqrt 2,&\phi _3&=(y_+-y_-)/\sqrt 2. \end {aligned}\]

In the order \((v_{+,0},v_{+,1},v_{-,0},v_{-,1})\), the precision is \(\operatorname {diag}(Q_+,Q_-)\) and its inverse is \(\operatorname {diag}(G_+,G_-)\), where \[\begin {aligned} Q_+&=\begin {pmatrix}s&-(1+t)\\-(1+t)&s\end {pmatrix},\\ Q_-&=\begin {pmatrix}s&t-1\\t-1&s\end {pmatrix},\\ G_+&=\tfrac 1{a(a+2+2t)} \begin {pmatrix}s&1+t\\1+t&s\end {pmatrix},\\ G_-&=\tfrac 1{(a+2)(a+2t)} \begin {pmatrix}s&1-t\\1-t&s\end {pmatrix}. \end {aligned}\] The covariance in these coordinates is given by the displayed inverse blocks. If \(T\phi =(z_+,z_-)\), then \[T_*\mu _Q=\mu _{Q_+}\otimes \mu _{Q_-}.\] The product on the right is the product probability measure. In particular the sector random vectors \(z_+,z_-\) are independent.

[friedli2017statistical, Proposition B.55, p. 505] gives the Gaussian covariance and coordinate-change convention. The positive Gaussian normalization and inverse-precision covariance apply to each block. The product assertion follows from the explicit density calculation below, rather than from the algebraic direct sum alone.

Proof.

October 29, 2026

Each unnormalized vector has squared norm two, and the pairings between distinct vectors are zero. Thus the basis matrix \(O\) with the displayed columns is orthogonal, \(T=O^T\), and \(|\det O|=1\). Applying \(Q\) gives \[\begin {aligned} Qv_{+,0}&=sv_{+,0}-(1+t)v_{+,1},\\ Qv_{+,1}&=-(1+t)v_{+,0}+sv_{+,1},\\ Qv_{-,0}&=sv_{-,0}+(t-1)v_{-,1},\\ Qv_{-,1}&=(t-1)v_{-,0}+sv_{-,1}. \end {aligned}\] These are precisely the two asserted blocks. Their sum and difference vectors have eigenvalues \[\begin {aligned} Q_+&:\quad a,\ a+2+2t,\\ Q_-&:\quad a+2t,\ a+2. \end {aligned}\] They are all positive. Multiplication by the displayed \(G_\pm \) gives the identity: the diagonal numerators are respectively \(s^2-(1+t)^2=a(a+2+2t)\) and \(s^2-(t-1)^2=(a+2)(a+2t)\), and the off-diagonal numerators vanish. This proves the inverse formulas.

The orthogonal change of variables preserves coordinate Lebesgue measure and writes the Euclidean action as \[\mathcal A(\phi )=\tfrac 12 z_+^TQ_+z_+ +\tfrac 12 z_-^TQ_-z_-.\] Each factor has a finite positive normalizer \[Z_\pm =\frac {2\pi }{\sqrt {\det Q_\pm }}.\] Tonelli's theorem applied to the nonnegative density gives \(Z_Q=Z_+Z_-\). Division by this product yields the product of the two normalized densities. Integrating over every Borel rectangle \(A_+\times A_-\) gives \[\begin {aligned} &\mu _Q\{z_+\in A_+,\ z_-\in A_-\}\\ &\qquad =\mu _{Q_+}(A_+)\mu _{Q_-}(A_-). \end {aligned}\] Rectangles generate the product Borel sigma-algebra, so uniqueness of probability measures identifies the product law and proves independence. In particular it does not assume that zero covariance alone implies independence. Polynomial moments are integrable by Proposition [fdip-002W].

Writing \[\begin {aligned} d_+&=a(a+2+2t),&d_-&=(a+2)(a+2t),\\ \Delta &=d_+d_-=a(a+2)(a+2t)(a+2+2t)>0, \end {aligned}\] the full normalizer is \(Z_Q=4\pi ^2/\sqrt \Delta \).

Retain \(R=(0,2)\). Its orthonormal sector vectors are \[w_\varepsilon =(\delta _0^R+\varepsilon \delta _2^R)/\sqrt 2.\] The projection satisfies \[\begin {aligned} \pi _Rv_{\varepsilon ,0}&=w_\varepsilon ,\\ \pi _Rv_{\varepsilon ,1}&=0. \end {aligned}\] Thus the retained sector coordinates are exactly \(x_+,x_-\), with no additional factor of \(\sqrt 2\). In these coordinates, the retained precision has positive scalar blocks \[\begin {aligned} \sigma _+&=s-\frac {(1+t)^2}{s} =\frac {a(a+2+2t)}s,\\ \sigma _-&=s-\frac {(t-1)^2}{s} =\frac {(a+2)(a+2t)}s. \end {aligned}\] The retained coordinates are independent centered Gaussians with variances \(\sigma _+^{-1},\sigma _-^{-1}\). Decomposing into sectors and taking this marginal therefore give the same law in either order, as in the map and measure square.

Proof.

October 29, 2026

A sector has precision \(\begin {pmatrix}s&c\\c&s\end {pmatrix}\), with \(c=-(1+t)\) or \(c=t-1\). Complete the square: \[\begin {aligned} sx^2+2cxy+sy^2 &=s(y+cx/s)^2\\ &{}+(s-c^2/s)x^2. \end {aligned}\] The scalar Gaussian integral gives \[\begin {aligned} &\int _{\mathbb R} e^{-(sx^2+2cxy+sy^2)/2}\,dy\\ &\qquad =\sqrt {\frac {2\pi }{s}} e^{-(s-c^2/s)x^2/2}. \end {aligned}\] Each eliminated sector coordinate contributes \(\sqrt {2\pi/s}\); the two factors multiply to \(2\pi/s\). This is exactly the factor for the original eliminated block \(C=sI_2\) in Example [fdip-000F]. The retained orthogonal change of coordinates also has absolute determinant one. Normalizing the two scalar densities proves the claimed retained product law.

Equivalently, in retained/eliminated vertex order \(X=(0,2)\), \(Y=(1,3)\), the typed arrays are \[A=C=sI_2,\qquad B=E^T=-\begin {pmatrix}1&t\\t&1\end {pmatrix}.\] Thus \(BC^{-1}E=s^{-1}\begin {pmatrix}1+t^2&2t\\2t&1+t^2\end {pmatrix}\), and the effective precision in vertex coordinates is \[\begin {aligned} S&=\begin {pmatrix}\alpha &-\beta \\-\beta &\alpha \end {pmatrix},\\ \alpha &=s-\frac {1+t^2}{s},\qquad \beta =\frac {2t}{s}. \end {aligned}\] It acts on \(w_+\) by \(\alpha -\beta =\sigma _+\), and on \(w_-\) by \(\alpha +\beta =\sigma _-\). Thus each sector's Schur complement agrees with the corresponding block of \(S\).

Direct multiplication also gives \[\begin {aligned} S^{-1}&=\frac {1}{\alpha ^2-\beta ^2} \begin {pmatrix}\alpha &\beta \\\beta &\alpha \end {pmatrix},\\ \alpha ^2-\beta ^2&=\sigma _+\sigma _-=\Delta/s^2>0,\\ Z_S&=\frac {2\pi }{\sqrt {\sigma _+\sigma _-}} =\frac {2\pi s}{\sqrt \Delta }. \end {aligned}\] The full eliminated factor \(2\pi/s\) times \(Z_S\) is exactly \(Z_Q\); both the pointwise fiber constants and the normalized law agree.

See [boyd2004convex, Appendix A.5.5, pp. 650-651] for the general block formula. Restricting \(Q\) to a full sector still leaves two coordinates. Integrating out its second coordinate takes the Schur complement computed above.

Transforming the retained coordinates back gives \[\begin {aligned} \operatorname {Var}(\phi _0) =\operatorname {Var}(\phi _2) &=\tfrac 12(\sigma _+^{-1}+\sigma _-^{-1}),\\ \langle \phi _0\phi _2\rangle _Q &=\tfrac 12(\sigma _+^{-1}-\sigma _-^{-1}). \end {aligned}\] For \(m=t=1\), the sector blocks and their inverses are \[\begin {aligned} Q_+&=\begin {pmatrix}3&-2\\-2&3\end {pmatrix},&Q_-&=3I_2,\\ G_+&=\tfrac 15\begin {pmatrix}3&2\\2&3\end {pmatrix},&G_-&=I_2/3. \end {aligned}\] Hence \(\sigma _+=5/3\), \(\sigma _-=3\); the retained variance is \(7/15\) and the cross covariance is \(2/15\). Directly, \[\begin {aligned} S&=\tfrac 13\begin {pmatrix}7&-2\\-2&7\end {pmatrix},& S^{-1}&=\tfrac 1{15}\begin {pmatrix}7&2\\2&7\end {pmatrix},\\ Z_Q&=\frac {4\pi ^2}{3\sqrt 5},&Z_S&=\frac {2\pi }{\sqrt 5}. \end {aligned}\] The eliminated factor is \(2\pi/3\). The specified conditional kernel at eliminated field zero instead gives precision \(A=3I_2\) and covariance \(I_2/3\), whose zero cross covariance differs from \(2/15\). For \(m=1,t=2\), \[\begin {aligned} Q_+&=\begin {pmatrix}4&-3\\-3&4\end {pmatrix},\\ Q_-&=\begin {pmatrix}4&1\\1&4\end {pmatrix},\\ G_+&=\tfrac 17\begin {pmatrix}4&3\\3&4\end {pmatrix},\\ G_-&=\tfrac 1{15}\begin {pmatrix}4&-1\\-1&4\end {pmatrix}. \end {aligned}\] Now \(\sigma _+=7/4\), \(\sigma _-=15/4\); the retained variance is \(44/105\) and the cross covariance is \(16/105\). Here \[\begin {aligned} S&=\begin {pmatrix}11/4&-1\\-1&11/4\end {pmatrix},\\ S^{-1}&=\tfrac 1{105}\begin {pmatrix}44&16\\16&44\end {pmatrix},\\ Z_Q&=\frac {4\pi ^2}{\sqrt {105}},&Z_S&=\frac {8\pi }{\sqrt {105}}. \end {aligned}\] The eliminated factor is \(\pi/2\). Both examples have the same \(P,J,E_\pm ,F_\pm \), but different covariance matrices and hence different Gaussian states: their polynomial functionals differ on \(\phi _0\phi _2\).

The half-turn commutant explains why these two-dimensional precision blocks may mix coordinates while respecting the same symmetry.

Mathlib's OrthonormalBasis.measurePreserving_repr supplies the volume-preserving passage to orthonormal coordinates. TauCeti's TauCeti.Probability.indepFun_sumEquivProd_multivariateGaussian_iff characterizes independence of complementary coordinate blocks by vanishing cross covariance. Index each sector by \(\{0,1\}\) and the full coordinates by the disjoint union of these sets. Take mean zero and covariance \(\Sigma =\operatorname {diag}(G_+,G_-)\). The positive eigenvalues above make \(\Sigma \) positive definite, hence positive semidefinite, and its cross block is zero. After identifying the normalized sector density with this Gaussian law, the theorem gives independence. The formalization references discussed with the inverse-precision covariance theorem use \(G_\pm =Q_\pm ^{-1}\) as their parameters. The concrete matrices, density identification and sectorwise Schur calculation are supplied by the proofs above.