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

Polignac's conjecture (1849) asserts that for every positive integer $k$, there exist infinitely many prime pairs $(p, p+2k)$. In this paper, we give a rigorous proof of this conjecture under the framework of the fixed-window tower sieve and multi-level periodic cutting. We first introduce the set $\mathcal{R}_i$ of admissible residue classes defined by the congruence conditions $\not\equiv \pm k \pmod{P_j}$ ($j\le i$). We construct a base interval $B=[1,Q_t]$ (a complete residue system) and a translated interval $C=Q_t+A$, and define the total window $U=B\cup C$. The observation interval $A=[1,L]$ is translation-equivalent to $C$ with $L=P_t^2-k$. Using the complete residue property of $B$, we prove that the number of survivors on $B$ is exactly $Q_tA_t$. By applying multi-level periodic cutting, we decompose $C$ modulo $Q_i$ into full periods and an incomplete interval $R_i$; the deviation on full periods is zero. Then we recursively decompose $R_i$ into full sub‑blocks of lengths $Q_{i-1},Q_{i-2},\dots,Q_2$ and prove that on any sub‑block of any depth the deviation of survivors modulo $P_i$ is absolutely bounded by $2$. This yields the recurrence $N_i \ge N_{i-1}(1-2/P_i) - C_1 (\ln t)^2/\ln\ln t$. Iteration gives the lower bound $N_t \ge c P_t^2/(\ln P_t)^2 - C_1 t (\ln t)^2/\ln\ln t$, which tends to infinity as $t\to\infty$, thereby proving Polignac's conjecture. The entire argument uses only elementary number theory and successfully circumvents the parity obstacle of classical sieves.