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 with precision period 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 precision period cutting to $A$, we decompose the incomplete part of each layer into complete sub-blocks of length $Q_{i-1}$ and a remainder interval, and through the interleaving argument prove that on the remainder interval the survivors are uniformly distributed among the residue classes modulo $P_i$ with deviation less than $2$. This yields the recurrence $N_i \ge N_{i-1}(1-2/P_i) - C_1\ln t$. Iteration gives the lower bound $N_t \to \infty$, thereby proving the cousin prime conjecture. The entire argument uses only elementary number theory and successfully circumvents the parity obstacle of classical sieves.