A Recursive Sifting Framework for the Goldbach Conjecture via Cross-Addition Density

By partitioning the number line into a computationally verified base (Zone 1) and an analytically bounded asymptotic domain (Zone 2) governed by advanced sieve-theoretic error bounds, this paper examines the cross- addition sumsets generated iteratively between a growing base set of primes and newly emerging sift primes within intervals dictated by Bertrand’s Postulate.

This paper proves that the density of cross-addition prime pairs scales at a super-linear rate, and through explicit sieve remainder estimates, decadal modular closure, and structural sift-prime resolution, local residual voids are shown to be permanently eliminated for all p > N. Furthermore, the foundational mechanics of this framework are formally verified using the Lean 4 interactive theorem prover to eliminate circularity and post-hoc test critiques.</p>