A Proof of Polignac's Conjecture via Translational Tower Sieve

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 of this conjecture within the framework of the translational tower sieve.

We first introduce the concept of the set of allowed residue classes $\mathcal{R}_i$, which is the subset of residue classes modulo $Q_i$ defined by the congruence conditions $\not\equiv \pm k \pmod{P_j}$ ($j\le i$), and it has an exact Cartesian product structure. Then we construct a base interval $B=[1,Q_t]$ (a complete residue system) and an observation interval $A=[1,L]$, where $L$ satisfies $P_t^2/2 \le L \le P_t^2 - k$, and define the total interval $U = B \cup (Q_t + A)$.

Using the complete residue system property of $B$, we prove that the number of survivors on $B$ is exactly $Q_t A_t$. Using the periodic decomposition of $C = Q_t + A$, we prove that at each sieving layer, the survivors in complete periods correspond exactly to the allowed residue class set $\mathcal{R}_{i-1}$, so the deviation in complete periods is zero; for incomplete periods, by decomposing them into complete sub-blocks and a remainder, we prove that the deviation is bounded by an absolute constant $C_0 = 6$. This analysis does not rely on arithmetic progression assumptions, only on periodicity and interval decomposition.

From this we establish the recurrence relation $N_i = N_{i-1}(1-2/P_i) - \Delta_i$, where $|\Delta_i| \le C_0$. 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.