Invariance of G_k for general P^1 polygonal dynamics

Prove that the projectively invariant quantities G_k(P), defined as sums of multiratios for 1 ≤ k ≤ ⌊n/2⌋, are invariant under the dynamics for any polygonal dynamic on P^1, beyond the specific cases (flat cross-ratio and staircase cross-ratio) where this invariance has been established.

Background

The quantities G_k(P) arise in the flat cross-ratio dynamics and are linked to lambda-lengths and alternated perimeters of ideal polygons. The authors verify their invariance for the staircase cross-ratio dynamic via extensive computation but cannot establish it in full generality.

Demonstrating invariance for all P1 polygonal dynamics would strengthen the integrable structure and unify invariants across systems.

References

A long and painful computation (not reproduced here) shows that these quantities are also invariant for the staircase cross-ratio dynamic. But for the general case of polygonal dynamic over $\mathbb{P}1$, we didn't manage to prove the invariance.

— Collapsing in polygonal dynamics (2507.16432 - Jean-Baptiste, 22 Jul 2025) in Open question, Section 4