Recovering a general invariant presymplectic two-form in P^1

Construct an invariant presymplectic two-form for general polygonal dynamics on P^1 that unifies the recurrent structures appearing across systems such as the leapfrog map and cross-ratio dynamics, potentially via a lambda-length interpretation.

Background

In the P1 setting, multiple systems exhibit recurring invariant objects, and the authors identify quantities I, J, K and related forms. They conjecture the existence of a unifying invariant presymplectic two-form but report that they could not retrieve it.

A deeper interpretation via lambda-lengths is proposed as a possible avenue to derive such a form, aligning with cluster-algebra frameworks in discrete geometry.

References

Ideally, we could even retrieve an invariant pre-symplectic two form which keeps reappearing in each polygonal dynamics in $\mathbb{P}1$, but we didn't manage.

— Collapsing in polygonal dynamics (2507.16432 - Jean-Baptiste, 22 Jul 2025) in Section 4, after Theorem (thm:monod_closed)