Tower Sieve with Layered Pre-sieving: 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 combine the tower sieve with a layered pre-sieving strategy to present a rigorous proof. Using the square-interval property, we transform the problem into finding odd integers $x$ in the interval $A=[1,L]$ ($L=P_t^2-3$) such that $x\not\equiv\pm2\pmod{P_i}$ for all $i\ge2$. Adopting a layered strategy, the first $K$ primes are handled by exact period cutting, while the remaining primes are handled by the large sieve. We prove that for the first $K$ layers, the deviation on complete periods is zero, and the cumulative deviation on incomplete periods is $O(Q_K)$; in the final stage, the large sieve bounds the error by $L^{1/2+o(1)}$. The main term $L/(\log L)^2$ dominates the error, yielding $N_t\to\infty$. The method uses the Chinese Remainder Theorem and the large sieve, and constructs a framework outside the classical linear sieve paradigm, thereby bypassing the scope of the parity problem.