A multidimensional cube of numbers reveals its maximum capacity through just two geometric slices.

Additive combinatorics describes the structure of number sets where no two elements can add up to a third element in that same set. Mathematicians long suspected that the most efficient way to pack these sets into a lattice cube was by slicing the cube twice, but they lacked a proof for higher dimensions. This resolution confirms that the limiting density of these sets follows a rigid geometric rule regardless of how many dimensions the cube has. These sum-free sets are more than just puzzles, as they form the basis for error-correcting codes used in digital transmissions. Determining these exact bounds allows engineers to understand the ultimate limits of data packing and signal clarity. This proof finally closes a major chapter in the study of discrete structures and how they scale as they get more complex.