Diffusion Geometry for Galaxy Bulk-Flow Reconstruction II: Three-Dimensional Potential-Flow Reconstruction from Radial Velocities

Abstract

Reconstructing a three-dimensional velocity field from radial peculiar velocities is non-identifiable because the radial observation operator has a large null space, even when a finite design matrix is formally full rank. Building on Paper I, we construct a potential-flow-constrained estimator from diffusion eigenfunctions and metric-calibrated gradients on galaxy point clouds, with explicit affine bulk modes. Controlled mocks show that an unconstrained Cartesian model can predict radial velocities accurately while catastrophically misreconstructing the hidden tangential field. For fine, fiducial, and coarse targets, the potential model reduces three-dimensional normalized root-mean-square error from 3.840, 1.841, and 1.425 to 0.663, 0.560, and 0.472. High-mode analyses under distance- and direction-dependent selection show that inverse-selection weighting loses its apparent benefit after practical convergence; unweighted geometry and an unweighted selection loss are most stable. A one-standard-error rule combined with a practical plateau criterion suppresses capacity over-selection by the raw radial-validation minimum. Under homoscedastic noise, validation-supported capacity and effective degrees of freedom decrease with noise. Under heteroscedastic noise, correctly specified generalized least squares improves hidden three-dimensional recovery at moderate and high noise but does not always minimize equally weighted latent radial risk. Measurement-only intervals under-cover at low noise because they omit structural completion uncertainty. For the controlled setting of fixed positions, known selection, and independent Gaussian errors, the practical prescription is to calibrate the potential-flow family in survey-specific simulations, use covariance-aware radial fitting, select complexity only within that family, and report fields on common support while separating measurement from structural uncertainty. 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 I: Foundations and Full-Vector Validation.” 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.