A Regularized Vorticity-to-Divergence Screening Statistic for Gaia Proper-Motion Fields

We examine a lightweight statistic for exploratory screening of localized structure in two-dimensional Gaia proper-motion fields. The statistic, previously named the Topological Shear Index (TSI), is the regularized ratio of the spatial mean absolute vorticity to the spatial mean absolute divergence in a locally projected proper-motion field. Version 4 corrects the interpretation of the statistic: its numerator measures projected vorticity rather than tensor shear, and the ratio is not a topological invariant. The mathematical core also has prior use in velocity-field analysis, although, to the best of our knowledge, we have not found an earlier application of the regularized mean-absolute ratio as a local Gaia screening statistic for stellar-stream candidates. Results previously reported for the GD-1 region, a comparison field, and the Large Magellanic Cloud are retained only as exploratory case studies. They do not establish sensitivity, specificity, a false-positive rate, or superiority to derivative-based diagnostics and established stream-finding algorithms. No new data reduction or code revision was performed for Version 4. The archived code is therefore retained as a legacy exploratory implementation, not as a fully validated realization of every statistical procedure described in earlier versions. The defensible contribution is a computationally inexpensive candidate-ranking heuristic whose detection performance remains open pending labelled Gaia-like simulations, injection--recovery tests, independent null fields, and preregistered head-to-head comparisons.Keywords: astrometry; Gaia DR3; proper motions; vorticity; divergence; stellar streams; exploratory screening; validation.