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 prime pairs $(p, p+2k)$. In this paper, we present a rigorous proof of this conjecture within the tower sieve framework. We introduce the admissible residue class set $\mathcal{R}_i$, defined by the congruence conditions $\not\equiv \pm k \pmod{P_j}$ ($j\le i$), which possesses an exact Cartesian product structure. The observation interval is taken as $A=[1,L]$, where $L=mQ_j$ satisfies $P_t^2/4 \le L \le P_t^2-k$. The key tool is the \textbf{CRT single-class upper bound lemma}: in any complete period, the count of a single bad residue class $a$ satisfies$$N(a)\le \frac{2|\mathcal{R}_{i-1}|}{P_i}+1.$$This bound is rigorously proven by elementary enumeration ($i\le 10$) and asymptotic estimates ($i\ge11$). Using this lemma, we prove the core upper bound$$D_i \le \frac{6N_{i-1}}{P_i}.$$This yields the tower recurrence $N_i \ge N_{i-1}(1-6/P_i)$, whose iteration gives$$N_t \ge c\frac{P_t^2}{(\ln P_t)^6}\to\infty,$$thereby proving the Polignac Conjecture. The entire proof uses only elementary number theory, the Chinese Remainder Theorem, and Mertens' theorem.