Explicit Upper and Lower Bounds for the Goldbach Partition Function \(r_{2}^{\text{avg}}(x)\) with Error Term \(O(\sqrt{x})\) and the Mapping Translation Mechanism

This paper investigates the local average \(r_{2}^{\text{avg}}(x)\) of the Goldbach partition function \(r_{2}(x)\), and under the assumption that the Riemann hypothesis holds, we construct explicit upper and lower bounds for \(r_{2}^{\text{avg}}(x)\) with asymptotic expansion \(r_{2}^{\text{avg}}(x)=x/\log^{2}x+(2\log 2+1)x/\log^{3}x+O(x/\log^{4}x)\), and the interval width satisfies \(U_{\text{avg}}(x)-L_{\text{avg}}(x)=O(\sqrt{x})\), indicating that the leading term of \(r_{2}^{\text{avg}}(x)\) is \(x/\log^{2}x\) with an error term of order \(O(\sqrt{x})\); the mapping model provides the cumulative sum formula for prime pairs, while the translation model shows that for any interval of length equal to the maximal prime gap \(\max_{k} g_{k}\), the number of prime pairs satisfies \(\sum_{n=m-L}^{m} r_{2}(n) \ge \pi(x)-\pi(x/2)\), with \(\pi(x)-\pi(x/2) \sim x/(2\log x)\), and moreover, the collection of prime gaps forms a continuous coverage mechanism for prime pairs; since the relative deviation caused by prime gap oscillations decreases as \(x\) increases and the prime density decreases, we do not introduce the classical singular series correction, and this work provides a new analytical framework for the structural study of prime pair representations of even numbers.