Lagrangian Ljusternik–Schnirelman Theory and Lagrangian Intersections

The paper states and proves the Lagrangian Ljusternik–Schnirelman inequalities I and II, including strict versions under additional hypotheses, by appealing to an axiomatic package of spectral-invariant properties (LS1–LS13) and a careful perturbation by small functions supported near isolated intersections. The candidate’s solution establishes the same inequalities using chain-level filtered Floer operations (pair-of-pants and module maps), the energy–action identity, and PSS compatibility; it also explains strictness via positivity of symplectic area for non-unit contributions. While the paper’s proof is organized via spectral axioms and a geometric localization trick, and the model’s proof is organized via filtered chain-level estimates, the two approaches are consistent and reach the same conclusions.