INTERSECTION NUMBERS BETWEEN HORIZONTAL FOLIATIONS OF QUADRATIC DIFFERENTIALS

The paper proves finiteness via Minsky’s inequality i(μφ, μψ)^2 ≤ ∫|φ| ∫|ψ| and establishes joint L1-continuity of the intersection number with a detailed geometric argument (Theorem 6.3) that handles all surface types, including groups of the first and second kind . The candidate solution hinges on a nontrivial identity i(μφ, μψ) = ∫X |Im(√(φ/ψ))| |ψ|, justified by an incorrect application of holonomy invariance across the other foliation’s leaves; the subsequent Hölder-1/2 continuity claims depend entirely on this identity. Because this key step is unproven and the invariance claim is unsound in general, the model’s proof is not reliable, whereas the paper’s arguments are consistent and complete under their stated framework (definition via quadrilateral partitions, strip decompositions, and doubling) .