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 fixed-window tower sieve and multi-level periodic 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 take $A$ as a fixed window and apply the tower sieve directly on it. By applying multi-level periodic cutting to $A$, we decompose the incomplete part of each layer recursively into full sub‑blocks of lengths $Q_{i-1},Q_{i-2},\dots,Q_2$ and prove that on any sub‑block of any depth the deviation of survivors modulo $P_i$ is absolutely bounded by $2$. 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 $N_t \ge c P_t^2/(\ln P_t)^2 - C_1 t (\ln t)^2/\ln\ln t$, which tends to infinity as $t\to\infty$, thereby proving the cousin prime conjecture. The entire argument uses only elementary number theory and successfully circumvents the parity obstacle of classical sieves.