VANISHING ASYMPTOTIC MASLOV INDEX FOR CONFORMALLY SYMPLECTIC FLOWS

The candidate solution targets exactly Theorem 1.1 of the paper: for a conformally symplectic isotopy (φ_t) with φ_0 = Id on a cotangent bundle and a Lagrangian graph L, there exist a closed 1-form η and a Lipschitz u, C^1 on a full-measure open set U, so that for all q ∈ U, p := φ_1^{-1}(η(q)+du(q)) ∈ L and the dynamical Maslov index along (0,1] vanishes. This is the paper’s main preliminary statement, explicitly stated in the introduction (Theorem 1.1) and proved in Section 4 using graph selectors and a careful reduction by closed 1-forms; see Theorem 1.1 and its proof outline in Section 4.1 and the closing step after Lemma 4.2, where the zero-section case is handled first and the general Lagrangian graph case is obtained by composing with translations T_t by the closed 1-form defining the graph. The statement proven is exactly the one claimed by the candidate solution . However, the model’s proof contains critical gaps. The key misstep is reducing to a Hamiltonian isotopy θ_t and then invoking the existence of generating functions S_t for L_t := θ_t(L). In general, generating functions quadratic at infinity (GFQI) exist for exact Lagrangians Hamiltonianly isotopic to the zero section, but not for arbitrary (non-exact) Lagrangian graphs; Hamiltonian isotopies preserve the Liouville class, so if L is the graph of a non-exact closed 1-form, L_t will typically remain non-exact and need not admit a GFQI. The paper circumvents this by composing with time-dependent translations by closed 1-forms f_t so that ψ_t := f_t ∘ φ_t yields images ψ_t(L_0) that are H-isotopic to the zero section and thus admit GFQI and a graph selector u_t (see the construction in Section 4.1 and eq. (9) linking du_t(q) to ψ_t(L_0)). The candidate’s argument does not implement this exactness correction; it asserts GF existence directly for θ_t(L), which is unjustified in the non-exact case. Additionally, the claimed homotopy-invariance of the Maslov index under fiberwise dilations ρ_{a(t)^{-1}} is not established (endpoints may move in the Lagrangian Grassmannian), and the spectral-flow/Morse-index end-point argument assumes minima at both ends without proving that a compatible family of critical points exists along t. By contrast, the paper uses a robust selector-based argument to show the Maslov index is zero when the Lagrangian path lies in a Lagrangian submanifold whose endpoints project to the selector, avoiding these pitfalls (see Section 3–4 and the final step in Section 4) .