A Solution to Goldbach's Conjecture Based on Differential-Algebraic Finite Representation Theory

This paper reestablishes the framework for solving Goldbach’s Conjecture based on Differential-Algebraic Finite Representation Theory. Unlike traditional analytic number theory which relies on infinite processes, we employ differentialalgebraic finite representation methods to transform the problem into a tractable form of finite operations. The core innovations are: (1) constructing an exact differential-algebraic finite prime detection function 1PDA(n) through complete differential-algebraic formalization of the AKS primality test; (2) establishing the differential-algebraic finite representation of the Goldbach counting function GDA(N) itself and deriving the finite-order linear recurrence relation it satisfies; (3) proposing a deterministic decision algorithm based on finite initial value verification and recurrence computation, theoretically completing a rigorous proof of the existence of Goldbach representations for every even number N ≥ 4. All results are self-contained within the differential- algebraic finite representation framework, realizing a paradigm shift from infinite asymptotic analysis to finite constructive verification.