This manuscript presents a formal classification of quadratic arithmetic functionals derived from divisor sum moments . Building upon the recent integration of partition theory and prime-detecting quasimodular forms (Craig, van Ittersum, and Ono, 2024), we establish a Rigidity and Uniqueness Theorem for the structural normalizer . Key Contributions: The Rigidity Theorem: We prove that is the unique minimal-degree polynomial compatible with a rational product–sum symmetry over the divisor lattice. We demonstrate that this normalizer is extremal in preserving the additive coupling of prime factors within their multiplicative interaction. Exact Factorization of Residues: We derive closed-form polynomial identities for the algebraic residue at composite indices. Specifically, for semiprimes , we establish the interaction identity . Deterministic Framework: Unlike probabilistic models of prime distribution, this work treats the residues of composite integers as structured algebraic objects rígidamente defined by their multiplicative constituents. Symmetry and Stability: We establish a "Parallelogram Law" for the normalizer and provide sublinear bounds for the residual deviation, confirming the stability of the characterization across the integer spectrum. This approach is intended to be complementary to established probabilistic models, such as the Hardy-Littlewood circle method, by providing an exact structural perspective on individual divisor residues. The results offer a rigorous basis for the study of Multiple Divisor Sum (MDS) algebras and their connection to additive arithmetic structures. Keywords: Number Theory, Divisor Sums, Prime Numbers, Algebraic Rigidity, Multiple Divisor Sums, Quasimodular Forms, Polynomial Factorization.
Daniel Taraborelli (Wed,) studied this question.