The Constant C and Local Stability in Goldbach’s Conjecture

Abstract

Goldbach’s conjecture has traditionally been approached through global additive methods and density arguments on the prime numbers. In this work, we propose a different perspective based on local stability near the midpoint of even integers. We introduce the notion of invariant logarithmic-square windows centred at E/2 and show that the existence of symmetric prime pairs is governed by a single local parameter measuring midpoint instability. A key contribution is the identification of a split-window mechanism, which demonstrates that large prime gaps at the midpoint do not obstruct symmetry: when the midpoint lies inside a prime desert, the invariant window divides into two equal sub windows on either side, preserving additive structure. This eliminates the classical midpoint-gap objection to Goldbach’s conjecture. We prove that Goldbach’s conjecture for sufficiently large even integers is logically equivalent to a localized, translation-invariant analogue of Bertrand’s postulate at logarithmic-square scale. This reduction isolates a single analytic inequality as the only remaining obstacle to an absolute proof. While this inequality is supported by extensive numerical evidence and heuristic models, it is not currently derivable from known theorems. The results reframe Goldbach’s conjecture as a corollary of a deeper local stability principle governing the prime population and precisely locate the frontier separating structural understanding from analytic proof.