A Central Window Algorithm for Explicit Goldbach Representations Certified Examples Up To 10^700

Abstract

This article presents a constructive and algorithmic framework for producing explicit Goldbach representations of very large even integers. The method is based on a central window strategy centered at E divided by two, combined with a sieve applied to a deviation parameter, probabilistic primality screening and deterministic certification using elliptic curve primality proving. The results reported here are punctual certified examples and not exhaustive verifications. Fully certified prime-prime Goldbach representations are exhibited for selected even integers up to 10^700. Beyond this range, the same method produces high-confidence probable prime examples. The contribution of this work is methodological. It demonstrates a scalable and reproducible approach for constructing Goldbach representations at extreme numerical scales, while clearly separating certified proofs from probabilistic evidence.