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

The Polignac conjecture (1849) asserts that for any positive integer $k$, there exist infinitely many prime pairs $(p, p+2k)$. In this paper we give a rigorous proof of this conjecture within 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 fixed window $A=[1,L]$ with $L=P_t^2-k$. By the square interval property, the problem reduces to counting integers $x$ in $A$ satisfying $x\not\equiv k\pmod2$ and $x\not\equiv\pm k\pmod{P_i}$ ($i\ge2$). We adopt a fixed-window tower sieve, marking bad numbers layer by layer without shrinking the window, thereby preserving the periodic structure. By performing multi-level periodic cutting on $A$, the incomplete part at each layer is recursively decomposed into full sub-blocks, and we prove that on any sub-block of any depth the deviation of the survivors modulo $P_i$ is absolutely bounded by $C$. 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 $N_t \gg t^2$, hence $N_t\to\infty$, which proves the Polignac conjecture. All arguments are elementary and successfully bypass the parity obstacle of classical sieve methods.