Affine axis and gauge

For an affine change of reference coordinates $A=(C,a)$,

Moreover,

Thus the affine holonomy fixes the developed particle worldline pointwise in every presentation.

The literal equality $\mathrm{Hol}({\gamma }_s)=J$ uses the declared reference framing. Under an unrestricted common affine change of framing, the invariant statement is the corresponding conjugacy class together with its marked fixed axis.

Because $J^4=I$, a matrix product alone does not establish primitivity. A fourfold meridian is noncontractible but has identity holonomy. Primitivity in Proposition prop:holonomy was proved by the angular lift, independently of the matrix calculation.