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Distortion cocycles and metric completion

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The distortion API measures finite maps between spaces with positive metrics and records obstructions to a common metric completion. Its conventions build on metric packets.

1. Intrinsic anisotropy packet

For an invertible finite-dimensional map

\[ T:(V,Q_s)\longrightarrow(W,Q_t) \]

with positive-definite Hermitian metrics, choose factors \(Q_s=R_s^*R_s\) and \(Q_t=R_t^*R_t\). Define

\[ B_Q(T)=R_tTR_s^{-1} \]

and

\[ D_Q(T)=\kappa_2(B_Q(T)) =\frac{\sigma_{\max}(B_Q(T))}{\sigma_{\min}(B_Q(T))}\ge 1. \]

The singular values of B_Q(T) are independent of the chosen metric factors. If coordinates change by G_s,G_t and the metrics are transported covariantly, the new representative differs by unitary factors. Therefore D_Q(T) and the geometric-mean volume scale are coordinate invariants of the metric-decorated map.

Import the implementation explicitly from komorebi_math.nonlinear.distortion.

2. Distortion cocycle law

For composable metric maps,

\[ D(T_2T_1)\le D(T_2)D(T_1), \]

hence

\[ \delta(T)=\log D(T) \]

is subadditive:

\[ \delta(T_2T_1)\le\delta(T_2)+\delta(T_1). \]

For a transport path, intermediate metric factors cancel exactly, so the same law applies edge by edge.

For a closed word with holonomy H, every metric satisfies

\[ D_Q(H)\ge \frac{\max_i |\lambda_i(H)|}{\min_i |\lambda_i(H)|}. \]

Thus, with

\[ \omega(H)=\log\frac{\max_i|\lambda_i(H)|}{\min_i|\lambda_i(H)|}, \]

any edge distortion budget obeys

\[ \sum_{e\in\gamma} \delta_e \ge \omega(H_\gamma), \]

where traversal multiplicity is counted without orientation cancellation. This is a necessary cycle/word obstruction.

3. Fundamental cycles alone are insufficient

Let

\[ A=\begin{pmatrix}0&-1\\1&0\end{pmatrix},\qquad B_t=\begin{pmatrix}0&-t\\1/t&0\end{pmatrix},\quad t>1. \]

Both A and B_t have eigenvalues +i,-i; each is individually unitarizable and has spectral distortion lower bound one. But

\[ AB_t=\operatorname{diag}(-1/t,-t), \]

so

\[ \omega(AB_t)=2\log t. \]

Consequently, for every common metric Q,

\[ D_Q(A)D_Q(B_t)\ge t^2, \]

and therefore

\[ \max(D_Q(A),D_Q(B_t))\ge t. \]

The bound is attained by a metric with coordinate ratio 1/t, so

\[ \inf_Q\max(D_Q(A),D_Q(B_t))=t. \]

For t=9, the exact optimum is 9. A generating-cycle test can therefore miss an obstruction that appears only after word composition.

4. Conformal metric completion

A conformal metric completion of an invertible transport graph consists of positive metrics Q_v and positive edge length scales lambda_e satisfying

\[ T_e^*Q_{t(e)}T_e=\lambda_e^2Q_{s(e)}. \]

Rescale every vertex metric to determinant one and define

\[ \widehat T_e=|\det T_e|^{-1/n}T_e. \]

Then the conformal equations are equivalent to

\[ \widehat T_e^*Q_{t(e)}\widehat T_e=Q_{s(e)}. \]

After spanning-tree propagation this reduces to one root metric Q satisfying

\[ \widehat H_\gamma^*Q\widehat H_\gamma=Q \]

for every closed word. Therefore:

Conformal metric-completion criterion. A finite invertible transport graph admits a conformal positive-definite metric completion if and only if its determinant-normalized holonomy representation is unitarizable. Equivalently, its normalized holonomy image is conjugate into a unitary group; in finite dimensions this is equivalent to relative compactness of that image.

The equivalence with relative compactness uses the classical Haar-averaging construction for a compact group.

5. Metric-completion trichotomy

Normalize candidate root metrics by det Q = 1, and for a finite set of normalized holonomy generators define

\[ F(Q)=\max_j\log D_Q(H_j),\qquad \Delta_*=\inf_{Q>0,\det Q=1}F(Q). \]

There are three operational cases.

  1. Completion: Delta_* = 0 and the infimum is attained. Then the normalized representation preserves a positive metric.
  2. Gap obstruction: Delta_* > 0. Some anisotropy remains under every positive metric.
  3. Escape obstruction: Delta_* = 0 but the infimum is not attained. Every sequence Q_k with F(Q_k) -> 0 must have cond(Q_k) -> infinity along a subsequence, and in fact no such near-minimizing sequence can have uniformly bounded condition number.

The third claim is a compactness theorem. If det Q_k=1 and cond(Q_k) were uniformly bounded, all eigenvalues would be bounded above and away from zero. A convergent subsequence would then limit to a positive-definite Q_*; by continuity F(Q_*)=0, contradicting nonattainment.

This turns numerical conditioning into an obstruction certificate: a system may be arbitrarily close to metric completion only by sending its metric packet to the boundary of the positive cone.

Unipotent model

For

\[ H=\begin{pmatrix}1&1\\0&1\end{pmatrix}, \]

all eigenvalues have modulus one, so the spectral lower bound is exactly one. There is nevertheless no positive-definite invariant metric. Taking

\[ Q_s=\operatorname{diag}(s^2,1) \]

gives

\[ D_{Q_s}(H)\downarrow1\quad(s\downarrow0), \qquad \kappa(Q_s)=s^{-2}\to\infty. \]

For sampled small positive s, numerical evaluation gives the diagnostic trend

\[ (D_{Q_s}(H)-1)\sqrt{\kappa(Q_s)}\to1. \]

Thus the unipotent obstruction is a zero-gap/nonattainment obstruction rather than a positive spectral gap.

6. Curvature and metric compatibility

Metric compatibility is distinct from flatness. A connection can preserve a positive metric while having nonzero curvature.

On a trivial real rank-three bundle over a planar base, use the constant connection matrices

\[ A_x=\begin{pmatrix}0&-1&0\\1&0&0\\0&0&0\end{pmatrix},\qquad A_y=\begin{pmatrix}0&0&0\\0&0&-1\\0&1&0\end{pmatrix}. \]

Both are skew-symmetric, so the Euclidean metric is parallel. Yet

\[ F_{xy}=[A_x,A_y] =\begin{pmatrix}0&0&1\\0&0&0\\-1&0&0\end{pmatrix}\ne0. \]

A finite rectangular loop at side length 0.2 produced nontrivial holonomy with Frobenius distance 0.05637743157883952 from the identity, while its orthogonality residual was only 4.624480671433126e-17.

So nonzero curvature and nontrivial holonomy are compatible with exact metric transport. For a fixed connection the relevant question is instead whether its holonomy preserves a positive metric. Curvature contributes to the holonomy algebra, but flatness is much stronger than metric compatibility.

7. Verification

The regression suite checks:

Deterministic stress cases additionally compare coordinate covariance and composition bounds across varied finite matrices and positive metrics. These checks concern the represented numerical inputs and declared tolerances.

8. Scope