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

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 the graph Dirichlet energy of a vector-valued parameter field; the sheaf formulation identifies zero energy with successful gluing and extends to edge-dependent transports and heterogeneous parameter spaces. We relate obstruction to the site spectral gap and excess global-map error, and construct ReLU examples showing non-transitive compatibility and a quadratic separation between staged and direct width. Controlled synthetic calibrations recover the predicted hardness separation, cut-induced obstruction scaling, and sensitivity to the fixed site graph.