Goldbach’s Conjecture as a Direct Deduction of Dynamic Symmetry and Invariant Prime Windows

Goldbach’s conjecture asserts that every even integer greater than two can be expressed as the sum of two prime numbers. Despite its elementary formulation and extensive numerical verification, the conjecture has resisted proof for nearly three centuries. Classical analytic approaches have focused primarily on prime density and average distribution, beginning with the Prime Number Theorem and culminating in refined correlation results. While these methods successfully describe global behavior, they have struggled to guarantee simultaneous primality in symmetric additive configurations, a difficulty often associated with parity and covariance obstructions.In this work, we introduce a dynamic and variational framework that reformulates the problem in terms of symmetric motion rather than static inspection. The approach is based on a two-ball model in which symmetric integers around a central point are explored dynamically within an invariant admissible window. This window is shown empirically and structurally to persist under the growth of prime gaps, either as a continuous interval or as a symmetric split domain whose total extent remains invariant.A central role is played by a logarithmic density proxy, denoted λ, which induces a symmetry defect measuring imbalance between symmetric configurations. The evolution of this defect under motion defines a variational landscape whose stable minima correspond to prime–prime configurations, while composite–composite configurations are shown to be dynamically unstable. This mechanism provides a structural explanation for why composite obstructions cannot persist and why symmetric prime pairs are dynamically selected.The results integrate classical analytic insights on prime density and gaps with ideas from dynamical systems and stability theory. Goldbach’s statement emerges not as an assumed condition but as a consequence of symmetry, invariance, and variational stability. While the work does not claim a traditional formal proof in the axiomatic sense, it clarifies the underlying architecture of the problem and offers a coherent pathway toward its analytic resolution.