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