A Recursive Tower Sieve Method Based on the Uniform Distribution Lemma: Proof of the Twin Prime Conjecture

We develop a recursive tower sieve method based on the uniform distribution lemma and apply it to prove the twin prime conjecture. At each step, at most two fixed residue classes modulo each prime are removed, while the surviving set maintains a complete periodic structure. We rigorously prove the recursive relation $N_i = N_{i-1}(1 - d_i/P_i) + E_i$, where $|E_i| \le d_i$ (i.e., $\le 1$ for $i=1$ and $\le 2$ for $i \ge 2$). Furthermore, we construct the interval $U = [1, Q_t + P_t^2]$ and use translation invariance and complement decomposition to obtain the lower bound $N(A) \ge P_t^2 A_t - 2t$, where $A_t = \frac{1}{2}\prod_{i=2}^t (1 - 2/P_i)$. By Mertens' theorem, the main term is $\sim c t^2$ for some constant $c>0$, which tends to infinity, thus proving the twin prime conjecture. Numerical verification confirms that $|E_i| \le d_i$ holds in all tested cases, with maximum observed error $\approx 1.24$. The proof is elementary, self-contained, and uses no unproven conjectures.