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

Polignac's conjecture (1849) asserts that for any positive integer $k$, there exist infinitely many pairs of primes $(p, p+2k)$. In this paper, we present a rigorous proof within the framework of the translational tower sieve and precise cutting. We first introduce the concept of the set of allowed residue classes $\mathcal{R}_i$, defined by the congruence conditions $\not\equiv \pm k \pmod{P_j}$ ($j\le i$). Then 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 interval $U=B\cup C$. The observation interval $A=[1,L]$ is translation-equivalent to $C$, where $L=KQ_j$ satisfies $P_t^2/4 \le L \le P_t^2-k$. Using the complete residue system property of $B$, we prove that the number of survivors on $B$ is exactly $Q_tA_t$. Using precise cutting: decompose $C$ modulo $Q_i$ into complete periods and an incomplete interval; the deviation in complete periods is zero. Then further cut the incomplete interval modulo $Q_{i-1}$ into complete sub-blocks and a remainder; the deviation in the remainder is bounded by an absolute constant $C_0=6$. This analysis depends only on periodicity and interval decomposition, not on arithmetic progression assumptions. From this we establish the recurrence $N_i = N_{i-1}(1-2/P_i) - \Delta_i$ with $|\Delta_i|\le C_0$. Iteration yields the lower bound $N_t \ge c P_t^2/(\ln P_t)^2 - C_0 t$, which tends to infinity as $t\to\infty$, thus proving the conjecture.