Fixed-Window Tower Sieve with Precision Period Cutting: A Proof of Polignac's Conjecture

Polignac's conjecture (1849) asserts that for every positive integer $k$, there exist infinitely many pairs of primes $(p, p+2k)$. In this paper, we introduce the fixed-window tower sieve with precision period cutting to prove this conjecture. Using the square interval property, the problem is reduced to finding integers $x$ in the fixed interval $A=[1,P_t^2-k]$ satisfying $x\not\equiv k\pmod2$ and $x\not\equiv\pm k\pmod{P_i}$ for all $i\ge2$. We apply the tower sieve directly on $A$ and analyze its structure through precision period cutting. We prove that at each sieve level, the deviation on complete periods is zero, while the deviation on incomplete periods is $O((\ln t)^2/\ln\ln t)$. This yields the recurrence $N_i\ge N_{i-1}(1-2/P_i)-C_1(\ln t)^2/\ln\ln t$, which iterates to the lower bound $N_t\gg t^2$, so $N_t\to\infty$. Applying Mertens' theorem gives Polignac's conjecture. The method uses only elementary number theory and successfully bypasses the parity obstacle of classical sieves.