Existence of $x$ in $[0,N-2]$ such that $N\pm x$ are both prime --- A Tower Sieve Proof of the Goldbach Conjecture

The Goldbach conjecture asserts that every even number greater than 2 can be expressed as the sum of two primes. In this paper we develop a new sieve method, the Tower Model Sieve, which removes at most two ``bad'' residue classes modulo each prime step by step, precisely controlling the structure of the remaining set at each stage. For any sufficiently large even number $2N$, let $A=[0,N-2]$ and define $B=[0,Q_t-1]$ with $Q_t=\prod_{j=1}^t P_j$, where $t$ satisfies $P_t^2\le 2N-2<P_{t+1}^2$ (i.e., $P_t$ is the largest prime not exceeding $\sqrt{2N-2}$). The complement $C=B\setminus A$ is translated to $C^*=[0,M-1]$ with $M=Q_t-(N-1)$. On the translated interval we apply the tower sieve; the number of complete periods at each step is at least $1$, so the Halberstam–Richert uniform distribution lemma applies strictly and the errors accumulate linearly. The iteration yields an upper bound $N_{C^*}\le M A_t+2t$ for the number of good points in $C^*$, where $A_t=\frac12\prod_{i=2}^t\frac{P_i-2}{P_i}$. Using the fact that the total number of good points in $B$ is $r_t=N_A+N_{C^*}=Q_tA_t$, we obtain a lower bound $N_A\ge (N-1)A_t-2t$ for the number of good points in $A$. Applying explicit lower bounds from Mertens' theorem, we prove rigorously that $N_A>0$ for all $N\ge 5\times10^5$, and that this lower bound tends to infinity as $N$ increases. Hence there exists $x\in[0,N-2]$ such that $N\pm x$ are both prime, i.e., $2N$ is a sum of two primes. The even numbers below that threshold are verified directly, so the Goldbach conjecture holds for all even numbers $>2$. The proof is completely elementary, uses no unproven conjectures, and circumvents the parity obstacle of classical sieve methods.