Example. Quadratic isometry forgets the reference pairing and projection [fdip-000Z]

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The enriched classification changes if one keeps only a positive quadratic form and permits every linear isometry. Any two positive symmetric matrices \(Q_1,Q_2\) of the same size define isometric quadratic spaces, even when their counting-relative spectra and retained observables differ.

[boyd2004convex, Appendix A.5.2, pp. 646-647] gives the symmetric spectral decomposition and positive square root used below to construct the isometry.

For \(Q=O\operatorname {diag}(\lambda _i)O^T\), with \(O\) orthogonal and every \(\lambda _i>0\), define \[\begin {aligned} Q^{1/2}&=O\operatorname {diag}(\sqrt {\lambda _i})O^T,\\ A=Q^{-1/2}&=O\operatorname {diag}(\lambda _i^{-1/2})O^T. \end {aligned}\] Direct multiplication gives \(A^TQA=I\). More generally, \[\begin {aligned} T&=Q_2^{-1/2}Q_1^{1/2},\\ T^TQ_2T &=Q_1^{1/2}Q_2^{-1/2}Q_2Q_2^{-1/2}Q_1^{1/2}\\ &=Q_1. \end {aligned}\] The map \(T\) is invertible. This calculation does not require \(Q_1Q_2=Q_2Q_1\). Thus positive quadratic forms have one linear-isometry class in each fixed dimension.

The reference pairing changes under the same substitution \(\phi =Az\): its matrix becomes \[H'=A^TIA=Q^{-1},\] while the action form has matrix \(I\). If that transported pairing is retained, its relative operator is \((H')^{-1}I=Q\), preserving the original spectrum. In general simultaneous transport of a positive reference pairing \(H\) and an action form \(Q\) gives \[\begin {aligned} H'&=A^THA,\qquad Q'=A^TQA,\\ (H')^{-1}Q'&=A^{-1}(H^{-1}Q)A. \end {aligned}\] Similarity preserves the spectrum. Replacing \(H'\) by \(I\) after normalizing \(Q\) discards this data; it is not simultaneous transport of the pair.

For the four-cycle with \(a=m=1\), compare \(b=1\) and \(b=2\). The spectral calculation gives respectively \[\begin {aligned} \operatorname {spec}(Q_1)&=\{1,3,3,5\},\\ \operatorname {spec}(Q_2)&=\{1,3,5,7\}. \end {aligned}\] These forms are linearly isometric by \(T\), but cannot be conjugate by a counting-orthogonal map, since their spectra differ. In particular they are not compatible relabelings. Their trace ratios also differ by \(1/72\).

The same isometry gives a concrete obstruction to the retained projection, even though it preserves the named half-turn. In the common orthonormal spectral basis \(u_0,u_1,u_2,u_3\), the matrix of \(T\) is \[\operatorname {diag}\bigl (1,1,\sqrt {3/5},\sqrt {5/7}\bigr ).\] The half-turn is diagonal in this basis as well, so \(T\rho (h)=\rho (h)T\). Returning to vertex coordinates gives \[T_{01}=\frac {\sqrt {3/5}-\sqrt {5/7}}4\ne 0,\] because \(3/5\ne 5/7\). For \(R=R_0\), one has \(\pi _R\delta _1=0\), whereas the zeroth component of \(\pi _RT\delta _1\) is nonzero. Hence no linear map \(T_R\) can satisfy \(\pi _RT=T_R\pi _R\). Also \(T^TT\) has eigenvalues \(1,1,3/5,5/7\), so \(T\) fails to preserve the counting pairing. These are separate failures of the enriched-arrow conditions.

A Jacobian magnitude different from one does not obstruct normalized Gaussian transport. If \(T^TQ_2T=Q_1\), then \[\det Q_1=(\det T)^2\det Q_2.\] For \(y=T\phi \), the exponent in the normalized density becomes \(-y^TQ_2y/2\); its coefficient after the reference change of variables is \[\frac {\sqrt {\det Q_1}}{(2\pi )^{n/2}|\det T|} =\frac {\sqrt {\det Q_2}}{(2\pi )^{n/2}}.\] Therefore \(T_*\mu _{Q_1}=\mu _{Q_2}\). Equivalently, \(Q_2^{-1}=TQ_1^{-1}T^T\) is the covariance transport law. For \(A=Q^{-1/2}\), the Jacobian magnitude \(|\det A|=(\det Q)^{-1/2}\) is exactly absorbed by normalization, even when it differs from one. The loss is the discarded pairing, carrier and retained projection, not a failure of normalized probability transport.

TauCeti's Matrix.PosDef.exists_generalLinearGroup_mul_transpose_eq provides an invertible factor \(C\) with \(Q=CC^T\) for a real positive-definite matrix on a finite index set. For a bare witness of that conclusion the normalizer is \(C^{-T}\), since \[(C^{-T})^TQ C^{-T}=C^{-1}CC^TC^{-T}=I.\] The theorem's proof chooses the symmetric positive square root, but its existential conclusion does not assert symmetry of \(C\). One therefore cannot infer that \(C^{-1}\) is the normalizer from the factorization alone. Its separate TauCeti.Probability.map_affine_multivariateGaussian supports Gaussian transport in covariance parameters, after the precision-density identification. Neither declaration supplies the enriched classification or projection obstruction.

By Clifford functoriality, an isometry of these configuration forms also induces an algebra isomorphism. Since the positive forms already lie in one isometry class, keeping only the abstract quadratic space, or its isometry-functorial Clifford algebra, cannot recover the discarded reference pairing, carrier and projection. This argument does not infer a quadratic isometry from an arbitrary Clifford-algebra isomorphism. Retaining the enriched objects and their specified relabeling arrows prevents the demonstrated collapse; it supplies no continuum or physical selection map.