Proposition: Constraint words and affine edge lengths

A polygonal slice induced from a fixed flat conical spacetime and its component-indexed atlas has ordered transition products equal to the corresponding monodromies.

On an interval of fixed incidence, suppose normals are constant, offsets are affine in the parameter, and endpoints solve nonsingular constant-coefficient linear intersection systems. If an edge has a fixed spacelike direction, its length is affine on subintervals where orientation does not reverse.

Proof. The first statement is chart continuation. An endpoint satisfying $Az=b_0+s{b}_1$, with $A$ constant and invertible, is affine. An edge difference with fixed unit spacelike direction is $(c_0+s{c}_1)e$, of length $|c_0+s{c}_1|$.