Structural Number Theory: Research on Prime Number Rules and Unified Proofs of Several Number-Theoretic Conjectures — An Original Theory and Complete Demonstration Based on Structural Covering (Part 1: Proofs of Goldbach's Conjecture, Polignac's Conjecture, Twin Prime Conjecture and Related Conjectures)

This paper is one output in the research series Structural Number Theory: Research on Prime Distribution Laws and Unified Proofs of Several Number-Theoretic Conjectures — An Original Theory and Complete Argument Based on Structural Covering. Built upon the original structural covering theory, the entire series adopts a unified proof paradigm to deduce various number-theoretic propositions and verify the universal applicability of the proposed theory. This paper for the first time proposes the concepts of covering and structural covering, together with the definition of number-set correspondence. Taking structural covering as the core and number-set correspondence as the supporting skeleton, this paper constructs the theoretical framework of structural covering. Within a unified structural framework, this paper aims to find common solutions to multiple propositions relying on structural covering. It delivers integrated proofs for several classical number-theoretic conjectures and achieves a unified resolution for diverse number-theoretic problems. The structural covering theory opens up an innovative research path. It clearly uncovers the intrinsic connections between prime-related number-theoretic propositions, establishes a unified operational logic framework for primes, investigates the fundamental structural rules governing prime distribution, and is of great significance to advancing the development of foundational number theory.