This work proposes a paradigm shift in number theory: moving from theclassical multiplicative language (based on divisors and sieves) to a constructiveadditive theory. We introduce a positive definition of prime numbers asadditive atoms, based on the Additive Regularity Index (ARI(n)). We showthat the entire set N can be constructively built starting from the minimal“seed” {2, 3}, which establishes fundamental additive rules concerning parityand minimal additive rank r(N). The Goldbach conjectures (weak andstrong) are reformulated as tests of the constructive consistency of this system.Additionally, we introduce the concepts of Additive Entropy (Hadd)and Synergy, which describe quantitative and qualitative aspects of thisconsistency. In the final part, we discuss “gappy universes” generated byalternative seeds, interpreting them as numerical semigroups.