Ω-N Prime Element Native System: Foundational White Paper v1.0

This is the revised formal preprint v1.0 of the Ω-N Prime Element Native System, a constructive number-theoretic framework built upon Ω-prime primitive units as the irreducible generative units of the natural number domain, grounded in exactly 4 core axioms. This version corrects the proof gap of the Inductive Transfer Lemma (T14), establishes the bi-interpretability and relative consistency between the Ω-system and Peano Arithmetic (T15), and proves the Goldbach Theorem (T16) within the system via constructive strong induction based on the minimal generation axiom. The system provides a synthetic generative perspective for classical number-theoretic propositions, with equivalent proof-theoretic strength to PA.