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

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.