Complement Cutting with Translation: 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 using \textbf{complement cutting with translation}. 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 $i\ge2$. We embed $A$ into the complete period $B=[1,Q_t]$, take the complement $C=[L+1,Q_t]$, and use $N_A=N_B-N_C$ to reduce the problem to sieving the complement $C$. Applying a right-to-left two-level periodic decomposition to $C$, we prove through the interleaving argument that survivors in the remainder interval are uniformly distributed among residue classes modulo $P_i$ with deviation less than $2$. This yields the recurrence $N_i \ge N_{i-1}(1-2/P_i)-C_1(\ln t)^2/\ln\ln t$. Iteration gives $N_A \gg Q_t/(\ln t)^2 \to \infty$, thereby proving the cousin prime conjecture. The entire argument uses only elementary number theory and successfully circumvents the parity obstacle of classical sieves.