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

The Polignac conjecture (1849) asserts that for every positive integer $k$, there exist infinitely many pairs of primes $(p, p+2k)$. When $k=1$, this reduces to the twin prime conjecture. This paper presents a complete proof of the Polignac conjecture within the fixed-window tower sieve framework with precision period cutting. The core strategy consists of four levels. First, \textbf{square-interval reduction}: if $x\le P_t^2-k$ and $x\not\equiv\pm k\pmod{P_i}$ for all $i\le t$, then both $x-k$ and $x+k$ are prime. Second, \textbf{allowed residue classes}: define $\mathcal{R}_i$ as the set of residue classes modulo $Q_i$ satisfying $r\not\equiv\pm k\pmod{P_j}$ for all $j\le i$. By the Chinese Remainder Theorem, $\mathcal{R}_i$ has a Cartesian product structure and cardinality $Q_i\rho_i$. Third, \textbf{fixed window}: we work directly on the observation interval $A=[1,L]$ without auxiliary intervals. Fourth, \textbf{tower recursion with precision period cutting}: decompose $A$ level by level into full $Q_i$-periods (zero deviation), full $Q_{i-1}$ sub-blocks (bounded deviation), and remainder intervals (bounded deviation), thereby obtaining a clean recursion $N_i\ge N_{i-1}(1-2/P_i)-C_0$ with $C_0=6$. We prove that the total error is $O(t)$, completely absorbed by the main term $cP_t^2/(\ln P_t)^2$, hence $N_t\to\infty$. The entire proof uses only elementary number theory, the Chinese Remainder Theorem, and Mertens' theorem, without relying on the circle method or any unproven analytic hypotheses. When $k=1$, this gives a complete proof of the twin prime conjecture.