Additive Squares in Random Integer Compositions: Exact Enumeration and a Brownian Occupation-Density Limit

An additive square in an integer composition consists of two consecutive blocks containing the same number of parts and having the same total sum. We study the number of such occurrences in random compositions of a positive integer. For uniform compositions, we obtain an exact generating function and closed formulas for the total number of occurrences. For a fixed Bernoulli cut parameter p∈(0,1)p∈(0,1), we derive an exact generating function for the mean, which grows on the n3/2n3/2 scale. The main result shows that the fluctuations live on the same scale: after normalization by n3/2n3/2, the number of additive-square occurrences converges in distribution to a nondegenerate Brownian occupation-density functional. The first two moments converge as well, so the variance is of order n3n^3. The same Brownian functional also governs positional abelian-square occurrences in random binary words. The connection is explained through the cut representation of compositions, local limit estimates, and a Brownian second-increment field. Preprint version v2.3, 22 September 2026.