Tower Model Sieve and Translated Complement: An Elementary Proof of the Twin Prime Conjecture

The twin prime conjecture asserts that there are infinitely many primes $p$ such that $p+2$ is also prime. 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. To avoid the loss of error control caused by the period length being larger than the interval length when sieving a short interval directly, we introduce a translation of the complement: the original complement $C=(P_t^2-2,\,Q_t]$ is translated to $C^*=[1,M]$, where $Q_t=\prod_{j=1}^t P_j$ and $M=Q_t-(P_t^2-2)$. On the translated interval we apply the tower sieve in increasing order; for $t\ge5$ we have $Q_{t-1}>P_t^2$, so the number of complete periods at each step satisfies $c_i\ge1$. Using the exact counting enabled by the periodic structure, we show that the deviation of the number of points in each residue class from the average is at most $1$, thus strictly controlling the error. 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 total number of good points in the full period $B=[1,Q_t]$, namely $r_t=N_A+N_{C^*}$, we obtain a lower bound $N_A\ge (P_t^2-2)A_t-2t$ for the number of good points in $A=[1,P_t^2-2]$. Applying explicit lower bounds from Mertens' theorem, we prove that for $t\ge10^4$ we have $N_A>0$ and $N_A\to\infty$, hence there are infinitely many twin primes. For $t\le4$ the existence of twin primes can be verified directly (e.g., $(3,5), (5,7), (11,13)$), so the conclusion is unaffected. Our method completely resolves the problem of error blow‑up when $c=0$ in the classical tower sieve, and the whole proof uses only elementary number theory and classical sieve lemmas.