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

The Polignac Conjecture (1849) asserts that for every positive integer $k$, there are infinitely many pairs of primes $(p, p+2k)$. 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 $x\not\equiv \pm k \pmod{P_j}$ for $j\le i$, where for $j=1$ we only exclude $x\equiv k\pmod2$. We construct the fixed window $A=[1,L]$ with $L=P_t^2-k$. Using the square-interval property, we transform the problem into finding integers $x$ in $A$ satisfying $x\not\equiv k\pmod2$ and $x\not\equiv\pm k\pmod{P_i}$ for $i\ge2$.

Employing the fixed-window tower sieve, we mark bad points layer by layer without shrinking the window. Through multi-level periodic cutting of $A$, we recursively decompose the incomplete interval of each layer into complete sub-blocks, and prove that on any sub-block at arbitrary depth, the deviation of the sum of counts of the two bad-point classes from the expected value is absolutely less than $4$. 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 $N_t\to\infty$, proving the Polignac Conjecture. The entire argument is elementary number theory and successfully bypasses the parity barrier of classical sieve methods.