Sheaf-Laplacian Obstruction and Projection Hardness for Cross-Modal Compatibility on a Modality-Independent Site

תקציר מקורי באנגליתarXiv:2604.07632v2 Announce Type: replace Abstract: Cross-modal representations vary in how easily they can be aligned, and compatibility is generally non-transitive: two modalities may align through an intermediate modality at lower complexity than through a direct map. We introduce a reference formalism that evaluates all modalities on a fixed neighborhood site and defines two directed invariants. Projection hardness \(H_{a\to b}(\varepsilon)\) is the minimum complexity within a nested Lipschitz-controlled family required to reach error \(\varepsilon\). For a declared local projection family, sheaf-Laplacian obstruction \(C_{a\to b}(\varepsilon)\) is the minimum variation of locally fitted projection parameters required to reach the same error. Under identity restrictions, obstruction is