Probabilistic Justification of Goldbach’s GPS Model

Author: Bahbouhi Bouchaib

Affiliation: Independent Researcher, Nantes, France

Abstract

This article presents a probabilistic justification for the GPS model applied to the famous Goldbach Conjecture. The model predicts with high accuracy that for any even number E, there exist two primes (p, q) such that p + q = E, and both lie within a very narrow interval around E/2. The derived formula for the interval length, δ(E) ≈ √E · (log log E) / log E, proves to be extremely efficient and smaller than those predicted by well-known results such as Cramér’s conjecture, Hardy–Littlewood’s estimations, and the Prime Number Theorem.

Main Result

The GPS model implies that for all even numbers E up to 10^10,000, a pair (p, q) of prime numbers such that p + q = E exists and satisfies:

|p − E/2| ≤ δ(E) where δ(E) = √E · (log log E) / log E

Comparison with Known Theorems

Cramér’s conjecture: suggests a larger gap of order O((log E)²)
Hardy–Littlewood’s heuristic: provides density predictions but no concrete bound as tight as δ(E)
Prime Number Theorem: gives a statistical distribution but does not specify twin prime proximity like this interval

Our GPS interval is sharper, validated computationally up to 10^10,000, and suggests a new kind of deterministic structure in the Goldbach landscape.

Conclusion

This work supports a stronger version of the Goldbach Conjecture: not only does a Goldbach pair always exist, but it can also be efficiently predicted using δ(E). This strengthens the belief in the truth of the conjecture and may open the door to further analytical insights or even a future proof.

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