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

arXiv:2604.07632v2 Announce Type: replace-cross 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_{ao b}(arepsilon)) is the minimum complexity within a nested Lipschitz-controlled family required to reach error (arepsilon). For a declared local projection family, sheaf-Laplacian obstruction (C_{ao b}(arepsilon)) 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.

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Source: arXiv cs.AI | 2026-07-28