A Proof of Polignac's Conjecture via Translational Tower Sieve and Precise Cutting

Polignac's conjecture (also known as the Star-Moon conjecture) asserts that for any fixed even positive integer $2m$, there are infinitely many prime pairs $(p, p+2m)$. In this paper we propose a new method——the translational tower sieve with precise periodic cutting. Using a square interval property, the problem is reduced to finding integers $x$ in $A=[1,P_t^2-m]$ satisfying $x\not\equiv m\pmod2$ and $x\not\equiv\pm m\pmod{P_i}$ for all $2\le i\le t$. We construct a base interval $B=[1,Q_t]$ (a complete residue system) and a translated interval $C=Q_t+A$. By exact counting on $B$ and a careful deviation analysis on $C$ using periodic cutting, we establish the recurrence $N_i \ge N_{i-1}(1-2/P_i)-C_1\ln t$ for $i\ge3$, where $N_i=|S_i\cap C|$ and $C_1$ is an absolute constant. Iteration together with Mertens' theorem gives $N_t\to\infty$, proving the conjecture. The method uses only elementary number theory and successfully circumvents the parity obstacle.