Constructive Additive Theory of Natural Numbers

This work proposes a paradigm shift in number theory: moving from the
classical multiplicative language (based on divisors and sieves) to a constructive
additive theory. We introduce a positive definition of prime numbers as
additive atoms, based on the Additive Regularity Index (ARI(n)). We show
that the entire set N can be constructively built starting from the minimal
“seed” {2, 3}, which establishes fundamental additive rules concerning parity
and minimal additive rank r(N). The Goldbach conjectures (weak and
strong) 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 this
consistency. In the final part, we discuss “gappy universes” generated by
alternative seeds, interpreting them as numerical semigroups.