Metric packets and coordinate covariance
A metric packet distinguishes a bare quotient, a quotient equipped with geometric metadata, and a numerical compression of that metadata. This guide describes finite redundant frames and metric-decorated holonomy.
1. Redundant analysis--synthesis presentations
Let
\[
A_i:X\to K_i,\qquad S_i:K_i\to X,\qquad S_iA_i=I_X.
\]
The coefficient gauge is \(\ker S_i\), and the quotient \(K_i/\ker S_i\) represents the physical source space. The positive semidefinite coefficient metric
\[
M_i=S_i^*S_i
\]
is exactly the physical metric pulled back to coefficient space:
\[
\langle c,M_i d\rangle=\langle S_i c,S_i d\rangle.
\]
For two presentations of the same physical space, the canonical transition is
\[
T_{ji}=A_jS_i.
\]
It obeys
\[
S_jT_{ji}=S_i,
\qquad
T_{ji}^*M_jT_{ji}=M_i.
\]
Hence the induced quotient map is an exact isometry. If \(P_i=A_iS_i\), then
\[
T_{ij}T_{ji}=P_i,
\qquad
T_{ji}T_{ij}=P_j,
\]
so the transitions become genuine inverses after quotienting the coefficient gauge directions. They also compose transitively:
\[
T_{kj}T_{ji}=T_{ki}.
\]
These identities require only the left-inverse laws \(S_iA_i=I\). They do not require Parseval, orthogonal, or canonical-dual presentations.
2. Why Euclidean coefficient conditioning is representation dependent
If the quotient metric is discarded and raw Euclidean coefficient norm is used instead, the same physical vector can acquire arbitrarily different coefficient magnitudes under a nonunitary coordinate change.
For an invertible coefficient reparameterization \(G\),
\[
A'=GA,\qquad S'=SG^{-1}
\]
still gives \(S'A'=I\), while
\[
M'=G^{-*}MG^{-1}.
\]
Thus the metric-aware seminorm is unchanged,
\[
\|Gc\|_{M'}=\|c\|_M,
\]
while \(\|Gc\|_2/\|c\|_2\) can be arbitrarily large or small. Scalar condition-number envelopes are therefore a compression of richer covariant metric information, not always the most fundamental packet.
3. Full-linear holonomy with transported metrics
Bare full-linear conjugacy data does not generally admit a continuous scalar that distinguishes every nontrivial holonomy from the identity. Metric metadata supplies a separate coordinate-covariant regime. If a positive-definite Hermitian root metric \(Q\) is part of the represented object and a full-GL coordinate change acts by
\[
H'=gHg^{-1},
\qquad
Q'=g^{-*}Qg^{-1},
\]
then the metric Hilbert--Schmidt residual
\[
\|H-I\|_{HS,Q}^2
=
\operatorname{tr}\!\left(Q^{-1}(H-I)^*Q(H-I)\right)
\]
is invariant in exact arithmetic:
\[
\|H'-I\|_{HS,Q'}=\|H-I\|_{HS,Q}.
\]
The invariant is a function of the pair \((H,Q)\), not of the bare conjugacy class \([H]\). The semantics must therefore distinguish:
- an active gauge symmetry with a fixed metric;
- a coordinate reparameterization that transports the metric.
The bare full_gl numerical certificate fails closed.
metric_gluing_certificate handles the transported-metric regime, and
transport_root_metric supplies the required metric transformation.
4. Conditioning of the metric packet itself
Metric covariance does not eliminate floating-point conditioning. From
\[
Q'=g^{-*}Qg^{-1}
\]
one obtains the standard conservative bound
\[
\kappa_2(Q')\le \kappa_2(Q)\,\kappa_2(g)^2.
\]
Therefore a coordinate change can make a previously well-resolved metric numerically singular. The implementation rejects positive-definite metrics whose smallest eigenvalue is not resolved relative to the declared graph tolerance.
This gives a hierarchy of represented information:
\[
\text{object}
\to
\text{quotient}
\to
\text{task quotient}
\to
\text{covariant geometric metadata}
\to
\text{conditioning envelope}.
\]
The fourth layer can remove coordinate distortion exactly. The fifth layer records when the fourth layer itself can no longer be represented reliably at finite precision.
5. Scope boundary
These contracts use finite-dimensional linear algebra and frame geometry: analysis and synthesis operators, dual frames, frame bounds, quotient spaces, congruence transport of metrics, and matrix condition numbers. Exact covariance requires the stated left-inverse identities and metric transport. Numerical certificates additionally require the metric to be resolved at the declared tolerance.
See redundant-frame algebra, distortion cocycles, and convex packets for related representations.