Diffusion Geometry for Galaxy Bulk-Flow Reconstruction I: Foundations and Full-Vector Validation

We formulate a diffusion-spectral field-reconstruction framework based on diffusion geometry for reconstructing cosmological vector fields from finite and nonuniformly distributed galaxy point clouds. Low-frequency eigenfunctions of a Markov operator defined on the point cloud are used as a global basis shared by the Cartesian velocity components. Within the same operator framework, we construct generator-spectrum regularization, out-of-graph evaluation through Nyström extension, and a metric-calibrated gradient based on the carré du champ and the coordinate Gram matrix. We validate the method using three controlled mock catalogs constructed from three-dimensional velocity vectors in a cosmological simulation, successively incorporating complete sampling, distance-dependent selection, and direction-dependent selection. Metric calibration recovers coordinate and affine gradients to machine precision. We further show that convergence with increasing retained mode number into the high-mode regime is essential for both method comparison and the interpretation of selection weighting. After convergence is established, the diffusion-spectral estimator outperforms the adaptive local-kernel baseline in every split for Mock-1 and Mock-2, while the apparent benefits of selection weighting seen at low mode numbers in Mock-2 and Mock-3 do not persist. The most stable primary estimator uses an operator-side unweighted geometry together with an unweighted loss. Direction-dependent weighting does not improve the mean reconstruction risk, but acts as a secondary sensitivity model that slightly reduces the dispersion of reconstruction errors across different directions. We deliberately leave the three-dimensional identifiability problem from radial-velocity observations outside the scope of this study. Instead, we establish the preceding methodological foundation for field representation, regularization, local differentiation, and out-of-sample extension on irregular galaxy point clouds. This preprint has been submitted to Physical Review D as part of a joint submission with the companion paper, “Diffusion Geometry for Galaxy Bulk-Flow Reconstruction II: Three-Dimensional Potential-Flow Reconstruction from Radial Velocities.” This record contains the submitted manuscript, not an accepted manuscript or a publisher-issued version. The research code, configuration records, and selected numerical outputs supporting Papers I and II are publicly archived as software version 0.1.0 at https://doi.org/10.5281/zenodo.22898864. The scope of the public archive and access to additional data are described in the manuscript's Data Availability statement.