The arithmetic of the integers is governed by two fundamental operations, addition and multiplication, whose interaction lies at the core of many deep problems in number theory. While multiplication preserves prime factorization in a rigid and conservative manner, addition typically destroys multiplicative structure and generates new prime content.In this work, we develop a unified structural framework that explains this asymmetry through spectral and operator-theoretic principles. By embedding the integers into a Hilbert space, we show that multiplication acts as a diagonal, layer-preserving operator in the prime spectral basis, whereas addition acts as a non-local, mixing operator driven by carry propagation. This spectral incompatibility leads to an arithmetic uncertainty principle, forbidding simultaneous localization in additive and multiplicative bases.Building on this structure, we introduce additive innovation as a quantitative measure of the new prime information created by a sum. We prove that the only obstruction to innovation arises from smoothness and $S$-unit phenomena in the coprime core. Using classical results on smooth numbers, we show that additive innovation is typically large, yielding unconditional abc-type inequalities in density.Finally, we develop an information-theoretic perspective, showing that addition produces entropy across prime scales while multiplication remains information-preserving. These results provide a structural explanation for the sum-product phenomenon and reframe classical problems as manifestations of the intrinsic incompatibility between additive and multiplicative spectral structures.