A Symmetric Two-Ball Dynamical Framework for Goldbach’s Conjecture From Static Additivity to Deterministic Non-Avoidance

Goldbach’s conjecture asserts that every even integer can be expressed as the sum of two prime numbers. Despite its simple formulation, the conjecture has resisted proof for nearly three centuries. The principal difficulty lies not in the scarcity of primes, but in guaranteeing simultaneous primality at two symmetric locations.This work introduces a deterministic two-ball dynamical framework that reformulates Goldbach’s conjecture as a problem of symmetric motion and non-avoidance. Two arithmetic “balls” move symmetrically around the midpoint of a fixed even integer, generating an infinite sequence of candidate decompositions. Rather than testing static offsets, the framework studies whether such a symmetric, recurrent, and non-periodic motion can avoid all prime–prime configurations indefinitely.By encoding symmetry, direction reversals, and arithmetic dispersion into the motion, the problem is reduced to a structural question: whether invariant obstructions can exist under infinite symmetric exploration. Using classical results on prime density and modern insights from dynamics and ergodic theory, the work demonstrates that permanent avoidance is structurally unstable. Goldbach’s conjecture is thus transformed into a conditional theorem governed by deterministic motion rather than probabilistic assumptions, isolating a single, well-defined step remaining before an unconditional proof.