Tower Sieve with Bidirectional Sieving: A Proof of the Twin Prime Conjecture

The twin prime conjecture asserts that there are infinitely many pairs of primes differing by 2. In this paper, we propose a tower sieve with bidirectional sieving method, together with a two-level exact period construction, and present a complete proof of the conjecture.

The core idea is as follows. By the square-interval property, we establish a sufficient sieve condition: if an even integer $x$ satisfies $x \not\equiv \pm 1 \pmod p$ for every prime $p \le P_t$, then both $x-1$ and $x+1$ are prime.

We set the observation interval $A = [0, P_{t-1}P_t - 1]$. This length $L = P_{t-1}P_t$ is simultaneously a multiple of both $P_t$ and $P_{t-1}$. We then sieve from the largest primes first: the first layer (mod $P_t$) and the second layer (mod $P_{t-1}$) are exactly precise with zero deviation.

We prove a key lemma: for each remaining layer $i$, the deviation is bounded by an absolute constant $C$. This gives a total error of $O(t)$, which is dominated by the main term $C_2 t^2$. Combined with Mertens' theorem, we prove that the number of good points $N_A \to \infty$. Each good point corresponds to a pair of twin primes, hence the twin prime conjecture follows.

The entire proof uses only elementary number theory and standard analytic number theory tools (Mertens' theorem), and does not rely on any unproved conjectures.