Exact identities and value distribution of the Craig-Ono functional on symmetric integer partitions

We study the arithmetic properties of the functional E(n) introduced by Craig,van Ittersum, and Ono (2024), which vanishes at prime indices. By evaluating thisfunctional over symmetric partition sets of the form 2k = r+s, we establish an exactidentity for configurations where both r and s are prime, showing that these configurations attain a constant value under a specific normalization. We characterizethe algebraic fluctuations for composite summands and provide an asymptotic concentration scale for the functional values. This note provides a structural analysisof divisor sum combinations under symmetric additive constraints.