Charts and ordered words

For a spatial gradient $g$, $|g|<1$, let

with $\mathcal{B}(0)=I_3$. In the first sector,

Their norms are $2\sqrt{2}/3,3/5,3/5$, and their Lorentz factors are $3,5/4,5/4$.

With $O_i=\mathrm{diag}(1,\mathrm{Rot}(i\alpha ))$, the adapted charts are

The first three rows flatten their plane intercepts to zero. The identity charts on horizontal faces retain the constant face height $2+s$; adaptation does not require the same intercept in every chart.

Canonical overlaps are eq:canonical. For a closed ordered word based in $f_0$, telescoping gives

The integer $N(\gamma )$ is supplied by the winding evidence, not inferred from the matrix alone. For example, at ${\mu }_0=\pi /2$, different windings can have the same rotational matrix.

Within an epoch, linear chart parts remain fixed, while translations and plane intercepts change. Thus orientations remain fixed, not literal spacetime locations.