A Study on the Goldbach Conjecture via Recursive Interval Covering

This paper reformulates the Goldbach conjecture into a recursive constructive framework: starting from the smallest prime, we cover even intervals step by step, incorporating newly discovered primes into the known set. By introducing an independent coverage boundary variable, the state definition no longer contains unproved assertions. We rigorously establish four properties of this framework — progress existence, capture completeness, prime set completeness, and covering continuity — all independent of the conjecture itself. The core lemma states that every even number N in the verification interval I_k = [2A_k+2, 3A_k+1] can be paired with a known prime p such that N-p is prime; the lemma is proved by contradiction using Selberg's sieve lower bound. The recursion never terminates, hence the Goldbach conjecture follows. Computational experiments show that the initial exhaustive verification is negligible in engineering terms.