Restriction-Consistent Finite Additivity for Sparse-Supervision Visual Counting

Global count supervision can fit an image total while leaving the spatial allocation of count mass unidentified. Fixed-map additivity cannot diagnose this failure because integrating one predicted density over disjoint regions is additive by construction. We introduce restriction-consistent finite additivity: the same predictor independently re-encodes a parent region and its disjoint children, and their predicted counts are trained to compose. ADDCOUNT realizes this principle as a backbone-agnostic partition regularizer, with image totals and sparse point anchors identifying the value of the latent count measure. On ShanghaiTech A, UCF-QNRF, and NWPU-Crowd, ADDCOUNT obtains MAE 50.9, 74.6, and 68.1, compared with 53.7, 79.8, and 72.2 for Point-to-Region; local MAE falls from 10.9 to 9.2 and regional additivity violation from 4.1 to 1.7. The ablations isolate the mechanism: partition consistency controls regional composition, whereas point anchors prevent compositionally coherent but numerically incorrect counts.