Proposition. Jacobian compensation in normalized Gaussian transport [fdip-003M]
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  • October 29, 2026

Let \(M,M'\) be finite-dimensional real positive inner-product spaces with induced Borel Euclidean volumes \(\lambda ,\lambda '\). Let \(Q,Q'\) be positive self-adjoint precisions, and suppose a linear isomorphism \(T:M\to M'\) preserves their actions: \[\mathcal A_{Q'}(Tx)=\mathcal A_Q(x).\] It need not preserve the reference pairings. For its absolute Jacobian \(J(T)>0\), one has \[\begin {aligned} T_*\lambda &=J(T)^{-1}\lambda ',\\ Z_{Q'}&=J(T)Z_Q,\qquad T_*\mu _Q=\mu _{Q'}. \end {aligned}\] For every real or complex Borel observable \(F'\), \[\int |F'|\,d\mu _{Q'}=\int |F'\circ T|\,d\mu _Q\] as extended nonnegative integrals. Hence integrability is equivalent on the two sides, and, under that condition, \(\omega _{Q'}(F')=\omega _Q(F'\circ T)\).

Proof. 

  • October 29, 2026

The reference-volume change formula gives the pushforward of volume, equivalently \(\int h'\,d\lambda '=J(T)\int h'\circ T\,d\lambda \) for every nonnegative Borel \(h'\). Apply it to the Gaussian weight; action preservation gives \(Z_{Q'}=J(T)Z_Q\). Both normalizers are finite and positive by Theorem [fdip-002T]. For a nonnegative Borel \(h'\), divide the same substitution by this equality: \[\begin {aligned} \int h'\,d\mu _{Q'} &=\frac {J(T)}{Z_{Q'}}\int h'(Tx)e^{-\mathcal A_Q(x)}\,d\lambda (x)\\ &=\int h'\circ T\,d\mu _Q. \end {aligned}\] Indicators of Borel sets prove the event-pushforward statement. Taking \(h'=|F'|\) proves the extended absolute-integral equality. Only when finite do positive/negative parts, and for complex functions real/imaginary parts, give the stated expectation identity.

In orthonormal coordinates, action preservation means \(T^TQ'T=Q\), so inversion gives \[(Q')^{-1}=TQ^{-1}T^T.\] This is covariance congruence, not generally operator similarity. The scalar Jacobian is computed in the specified reference volumes; independently ordered orthonormal frames may change the signed determinant but not its absolute value. The quadratic-isometry example shows that this normalized transport can lose both counting-pairing and retained-projection compatibility. Such maps are not enriched background arrows.

The volume convention is [tao2011measure, Exercises 1.2.20-22, p. 41]; the covariance-parameter Gaussian substitution is [friedli2017statistical, Proposition B.55, p. 505]. The inverse-precision covariance theorem identifies its covariance with \(Q^{-1}\) for these reference pairings. TauCeti's TauCeti.Probability.map_affine_multivariateGaussian supports the covariance congruence for a supplied finite real linear map, even rectangular. That broader covariance result does not assert invertibility, endpoint preservation or the retained square used here.