Linear instability of periodic orbits of free period Lagrangian systems

The paper’s Theorem 1.3 states a parity-based instability criterion for periodic orbits of free-period Lagrangian systems and proves it via a CLM–Maslov/spectral-flow framework; the argument is coherent and complete, including the degenerate orbit-cylinder case. The candidate solution gives an independent proof outline: it reduces to the transverse map, uses the odd-parity of Conley–Zehnder indices for semisimple elliptic endpoints, and invokes a mod-2 spectral-flow/Maslov identity to reach exactly the same parity obstruction. It is essentially correct for the nondegenerate orbit-cylinder case, but it tacitly assumes away certain endpoint degeneracies; the paper covers these carefully. Overall, both are correct, with the model’s proof being a different (slightly more Hamiltonian/CZ-index) route. See the paper’s statement and proof sketch of Theorem 1.3 and Sect. 4 for the CLM parity reduction and the key parity identity ιgeo = ιTspec + dim ker(A − I), together with the splitting (4.4) and parity bookkeeping in (4.5) leading to Lemma 4.3’s instability trigger .