Translational Tower Sieve and Precise Cutting: A Proof of the Cousin Prime Conjecture

The cousin prime conjecture asserts that there exist infinitely many pairs of primes differing by $4$. In this paper, we present a rigorous proof within the framework of the translational tower sieve and precise cutting. Using the square interval property, we transform the problem into finding odd integers $x$ in the interval $A=[1,P_t^2-3]$ such that $x\not\equiv\pm2\pmod{P_i}$ for all $i\ge2$. 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 $U=B\cup C$. Using the complete residue system property of $B$, we prove that the number of survivors on $B$ is exactly $Q_tA_t$. Using the periodic decomposition and precise cutting of $C$, we prove that at each sieving layer, the deviation in complete periods is zero, while the deviation in incomplete periods is bounded by an absolute constant $C_0=6$. From this we establish the recurrence $N_i\ge N_{i-1}(1-2/P_i)-C_0$, which iterates to give the lower bound $N_t\ge cP_t^2/(\ln P_t)^2-C_0t$. As $t\to\infty$ this tends to infinity, thus proving the cousin prime conjecture.