An abelian variety has a group law: points can be added. Projective space has no group law, but it has additive combinatorics — sumsets, energies, and the machinery of arithmetic structure theory. When you map an abelian variety to projective space, the group structure on the source imposes constraints on the additive combinatorics of the image.
The paper on additive rigidity for images of rational points (arXiv: 2603.24340) proves that for a simple abelian variety A with a finite morphism to projective space, the sumset and energy of any finite subset X of a finite-rank subgroup satisfy quadratic bounds in |X|.
Quadratic energy means the set has maximal additive structure — the number of quadruples (a,b,c,d) with a+b = c+d is ~|X|². Quadratic sumset growth means |X+X| ~ |X|². These are the bounds you'd expect from a group coset, not from a generic subset of projective space. The proof uses the uniform Mordell–Lang conjecture to control how the group law interacts with the projective embedding.
The through-claim: the group structure on the source survives in the additive combinatorics of the image. The morphism to projective space forgets the group law — projective space has no addition. But the image of a finite-rank subgroup retains the additive structure of the group, visible through energy and sumset bounds. The group law is invisible in the geometry but detectable in the combinatorics.
2603.24340. Number theory / abelian varieties / additive combinatorics / Mordell–Lang / sumset bounds.