A Unified Structure Theory of Prime Numbers - Core Logic and Structural Proofs

This paper establishes the core structural logic of the unified prime theory proposed in the first paper. By introducing matrix structures with side length 2 and modular constraints in base-2 and base-10 systems, we reveal that prime numbers exhibit stable periodic shadows under such structures, which correspond to the non-trivial zeros of the Riemann zeta function. We prove that any 2-sided matrix must correspond to an even multiple of a prime, forming a closed structural bridge between prime distribution, the Goldbach conjecture, and the Riemann hypothesis. The framework provides a unified, self-consistent, and finitistic path toward resolving classical problems in number theory.